Recent zbMATH articles in MSC 65L https://zbmath.org/atom/cc/65L 2021-07-26T21:45:41.944397Z Werkzeug Three iterative methods for solving Jeffery-Hamel flow problem https://zbmath.org/1463.34051 2021-07-26T21:45:41.944397Z "AL-Jawary, Majeed" https://zbmath.org/authors/?q=ai:al-jawary.majeed-ahmed "Nabi, AL-Zahraa J. Abdul" https://zbmath.org/authors/?q=ai:nabi.al-zahraa-j-abdul Summary: In this article, the nonlinear problem of Jeffery-Hamel flow has been solved analytically and numerically by using reliable iterative and numerical methods. The approximate solutions obtained by using the Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM) and Banach contraction method (BCM). The obtained solutions are discussed numerically, in comparison with other numerical solutions obtained from the fourth order Runge-Kutta (RK4), Euler and previous analytic methods available in literature. In addition, the convergence of the proposed methods is given based on the Banach fixed point theorem. The results reveal that the presented methods are reliable, effective and applicable to solve other nonlinear problems. The computational work to evaluate the terms in the iterative processes was carried out using the computer algebra system MATHEMATICA10. Dynamics and numerical approximations for a fractional-order SIS epidemic model with saturating contact rate https://zbmath.org/1463.34180 2021-07-26T21:45:41.944397Z "Hoang, Manh Tuan" https://zbmath.org/authors/?q=ai:hoang.manh-tuan "Zafar, Zain Ul Abadin" https://zbmath.org/authors/?q=ai:zafar.zain-ul-abadin "Ngo, Thi Kim Quy" https://zbmath.org/authors/?q=ai:ngo.thi-kim-quy Summary: The aim of this paper is to propose and analyze a fractional-order SIS epidemic model with saturating contact rate that is a generalization of a recognized deterministic SIS epidemic model. First, we investigate positivity, boundedness, and asymptotic stability of the proposed fractional-order model. Secondly, we construct positivity-preserving nonstandard finite difference (NSFD) schemes for the model using the Mickens' methodology. We prove theoretically and confirm by numerical simulations that the proposed NSFD schemes are unconditionally positive. Consequently, we obtain NSFD schemes preserving not only the positivity but also essential dynamical properties of the fractional-order model for all finite step sizes. Meanwhile, standard schemes fail to correctly reflect the essential properties of the continuous model for a given finite step size, and therefore, they can generate numerical approximations which are completely different from the solutions of the continuous model. Finally, a set of numerical simulations are performed to support and confirm the validity of theoretical results as well as advantages and superiority of the constructed NSFD schemes. The results indicate that there is a good agreement between the numerical simulations and the theoretical results and the NSFD schemes are appropriate and effective to solve the fractional-order model. Analysis of continuous collocation solutions for nonlinear functional equations with vanishing delays https://zbmath.org/1463.34261 2021-07-26T21:45:41.944397Z "Tian, Jian" https://zbmath.org/authors/?q=ai:tian.jian "Xie, Hehu" https://zbmath.org/authors/?q=ai:xie.hehu "Yang, Kai" https://zbmath.org/authors/?q=ai:yang.kai "Zhang, Ran" https://zbmath.org/authors/?q=ai:zhang.ran.1 Summary: In this paper, we study the existence, uniqueness, and regularity of solutions for the nonlinear functional equations $$y(t) = b(t,y(\theta (t))) +f(t)$$, $$t\in [0,T]$$, with vanishing delay $$\theta (t)$$. We then present a collocation method to solve this equation, and analyze the convergence properties of the piecewise polynomial collocation approximations. Several numerical experiments are used to illustrate the theoretical results. The coupled nonlinear Schrödinger equations describing power and phase for modeling phase-sensitive parametric amplification in silicon waveguides https://zbmath.org/1463.35465 2021-07-26T21:45:41.944397Z "Li, Xuefeng" https://zbmath.org/authors/?q=ai:li.xuefeng "Wang, Zhaolu" https://zbmath.org/authors/?q=ai:wang.zhaolu "Liu, Hongjun" https://zbmath.org/authors/?q=ai:liu.hongjun Summary: The coupled nonlinear Schrödinger (NLS) equations describing power and phase of the optical waves are used to model phase-sensitive (PS) parametric amplification in a width-modulated silicon-on-insulator (SOI) channel waveguide. Through solving the coupled NLS equations by the split-step Fourier and Runge-Kutta integration methods, the numerical results show that the coupled NLS equations can perfectly describe and character the PS amplification process in silicon waveguides. A septic B-spline collocation method for solving the generalized equal width wave equation https://zbmath.org/1463.41018 2021-07-26T21:45:41.944397Z "Karakoç, Seydi B. G." https://zbmath.org/authors/?q=ai:karakoc.seydi-battal-gazi "Zeybek, Halil" https://zbmath.org/authors/?q=ai:zeybek.halil Summary: In this work, a septic B-spline collocation method is implemented to find the numerical solution of the generalized equal width (GEW) wave equation by using two different linearization techniques. Test problems including single soliton, interaction of solitons and Maxwellian initial condition are solved to verify the proposed method by calculating the error norms $$L_2$$ and $$L_\infty$$ and the invariants $$I_1$$, $$I_2$$ and $$I_3$$. Applying the von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. As a result, the obtained results are found in good agreement with the some recent results. Analysis of Volterra integrodifferential equations with nonlocal and boundary conditions via Picard operator https://zbmath.org/1463.45039 2021-07-26T21:45:41.944397Z "Shikhare, Pallavi U." https://zbmath.org/authors/?q=ai:shikhare.pallavi-u "Kucche, Kishor D." https://zbmath.org/authors/?q=ai:kucche.kishor-d "Vanterler da Costa Sousa, José" https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose Summary: This article investigates the existence and uniqueness of solutions to the second-order Volterra integrodifferential equations with nonlocal and boundary conditions through its equivalent integral equations and fixed point of Banach. Furthermore, utilizing the Picard operator theory, we obtain the dependence of solutions on the initial nonlocal data and on functions involved on the right-hand side of the equations. Numerical study on multi-order multi-dimensional fractional optimal control problem in general form https://zbmath.org/1463.49042 2021-07-26T21:45:41.944397Z "Alipour, Mohsen" https://zbmath.org/authors/?q=ai:alipour.mohsen Summary: The aim of this work is application of Bernstein polynomials (BPs) for solving multi-order multidimensional fractional optimal control problem (MOMDFOCP). Firstly, by the Bernstein basis, we introduce operational matrices for Riemann-Liouville fractional integral and product in the arbitrary interval $$[a,b]$$. Then, via these matrices, we reduce the problem to the optimization problem. For solving this problem, we apply Lagrangian multipliers method. So, we can obtain approximate solution for MOMDFOCP. Results of some examples show that the obtained solutions are very accurate and in good agreement with exact solutions. Entropy solutions to parabolic equations in Musielak framework involving non coercivity term in divergence form. https://zbmath.org/1463.58014 2021-07-26T21:45:41.944397Z "Elemine Vall, Mohamed Saad Bouh" https://zbmath.org/authors/?q=ai:vall.mohamed-saad-bouh-elemine "Ahmed, Ahmed" https://zbmath.org/authors/?q=ai:ahmed.ahmed "Touzani, Abdelfattah" https://zbmath.org/authors/?q=ai:touzani.abdelfattah "Benkirane, Abdelmoujib" https://zbmath.org/authors/?q=ai:benkirane.abdelmoujib Summary: We prove the existence of solutions to nonlinear parabolic problems of the following type: $\begin{cases}\dfrac{\partial b(u)}{\partial t}+A(u)=f+\operatorname{div}(\Theta(x;t;u))&\text{in}\;Q,\\ u(x;t)=0&\text{on}\;\partial\Omega\times[0;T],\\ b(u)(t=0)=b(u_0)&\text{on}\;\Omega,\end{cases}$ where $$b\colon\mathbb{R}\to\mathbb{R}$$ is a strictly increasing function of class $$\mathcal{C}^1$$, the term $A(u)=-\operatorname{div}(a(x,t,u,\nabla u))$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $$\Theta\colon\Omega\times[0;T]\times\mathbb{R}\to\mathbb{R}$$ is a Carathéodory, noncoercive function which satisfies the following condition: $$\sup_{|s|\leq k}|\Theta({\cdot},{\cdot},s)|\in E_{\psi}(Q)$$ for all $$k>0$$, where $$\psi$$ is the Musielak complementary function of $$\Theta$$, and the second term $$f$$ belongs to $$L^1(Q)$$. Strong convergence of the split-step $$\theta$$ method for neutral stochastic delay differential equations https://zbmath.org/1463.60091 2021-07-26T21:45:41.944397Z "Peng, Jie" https://zbmath.org/authors/?q=ai:peng.jie "Dai, Xinjie" https://zbmath.org/authors/?q=ai:dai.xinjie "Xiao, Aiguo" https://zbmath.org/authors/?q=ai:xiao.aiguo "Bu, Weiping" https://zbmath.org/authors/?q=ai:bu.weiping Summary: Neutral stochastic delay differential equations often appear in some fields of science and engineering. The aim of this article is to investigate the strong convergence of the split-step $$\theta$$ method for neutral stochastic delay differential equations. When the drift and diffusion coefficients satisfy global Lipschitz condition with respect to the present state and the polynomial growth condition about the delay term respectively, and the neutral term may also be polynomial growth, this method is shown to be strongly convergent with order $$\frac{1}{2}$$. Some numerical results are presented to confirm the obtained theoretical results. Time-dependent reliability analysis based on probability density of first-passage time point https://zbmath.org/1463.62315 2021-07-26T21:45:41.944397Z "Yu, Shui" https://zbmath.org/authors/?q=ai:yu.shui "Wang, Zhonglai" https://zbmath.org/authors/?q=ai:wang.zhonglai Summary: Based on the first-passage time probability density (F-PTPD) for the time-dependent limit state function, an approach for estimating the cumulative probability density function of mechanical products over the whole lifecycle is proposed, which provides an effective analytical method for reliability analysis and design of mechanical products. Firstly, the sparse grid based stochastic collocation method is employed to obtain the mean value function of time-dependent limit state function, and the first-passage time point of the mean value function is obtained while the mean value function equals zero. The limit state function is then decomposed into a second-order Taylor series about time. Meanwhile, the first-passage time point function about input random variables is established via using the property of the quadratic function. The fourth origin moments of the first-passage time point function are estimated by employing the sparse grid based stochastic collocation method. Finally, the probability model of the first-passage time point for the limit state function can be obtained by the combination of the fourth origin moments and the maximum entropy method, so the cumulative probability density function can be estimated. Convergent stability for waveform relaxation methods of functional-differential equation https://zbmath.org/1463.65168 2021-07-26T21:45:41.944397Z "Fan, Zhencheng" https://zbmath.org/authors/?q=ai:fan.zhencheng Summary: The current research on wave relaxation methods for functional differential equations (WRMsFDEs) focuses on convergence. It is well known that unstable approximating methods are unmeaning. However, there are few studies on the stability of WRMsFDEs. Firstly, a definition of convergent stability for WRMsFDEs is given. Secondly, by virtue of estimating the difference between the approximating solutions of the waveform relaxation method and its disturbed systems, an estimate is derived under some common conditions. Lastly, the sufficient conditions for the convergent stability of WRMsFDEs are obtained from the above estimate. Legendre-collocation spectral solver for variable-order fractional functional differential equations https://zbmath.org/1463.65169 2021-07-26T21:45:41.944397Z "Hafez, Ramy Mahmoud" https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud "Youssri, Youssri Hassan" https://zbmath.org/authors/?q=ai:youssri.youssri-hassan Summary: A numerical method for the variable-order fractional functional differential equations (VO-FFDEs) has been developed. This method is based on approximation with shifted Legendre polynomials. The properties of the latter were stated, first. These properties, together with the shifted Gauss-Legendre nodes were then utilized to reduce the VO-FFDEs into a solution of matrix equation. Sequentially, the error estimation of the proposed method was investigated. The validity and efficiency of our method were examined and verified via numerical examples. Convergence of the Euler-Maruyama method for neutral stochastic differential equations with Lévy jumps https://zbmath.org/1463.65170 2021-07-26T21:45:41.944397Z "Ma, Li" https://zbmath.org/authors/?q=ai:ma.li.1 "Yan, Liangqing" https://zbmath.org/authors/?q=ai:yan.liangqing "Han, Xinfang" https://zbmath.org/authors/?q=ai:han.xinfang Summary: In this paper, we study the Euler-Maruyama method for neutral stochastic functional differential equations with Lévy jumps. By using Gronwall inequality, Hölder inequality and BDG inequality, we prove that the numerical solution converges to the real solution, which generalizes the EM approximation for neutral stochastic functional differential equations with Poisson jumps. Magnus series expansion method for solving nonhomogeneous stiff systems of ordinary differential equations https://zbmath.org/1463.65171 2021-07-26T21:45:41.944397Z "Atay, Mehmet T." https://zbmath.org/authors/?q=ai:atay.mehmet-tarik "Eryılmaz, Aytekin" https://zbmath.org/authors/?q=ai:eryilmaz.aytekin "Köme, Sure" https://zbmath.org/authors/?q=ai:kome.sure Summary: In this paper, Magnus series expansion method, which is based on Lie groups and Lie algebras is proposed with different orders to solve nonhomogeneous stiff systems of ordinary differential equations. Using multivariate Gaussian quadrature, fourth (MG4) and sixth (MG6) order method are presented. Then, it is applied to nonhomogeneous stiff systems using different step sizes and stiffness ratios. In addition, approximate and exact solutions are demonstrated with figures in detail. Moreover, absolute errors are illustrated with detailed tables. New stable, explicit, first order method to solve the heat conduction equation https://zbmath.org/1463.65172 2021-07-26T21:45:41.944397Z "Kovács, Endre" https://zbmath.org/authors/?q=ai:kovacs.endre Summary: In this paper a novel explicit and unconditionally stable numerical algorithm is introduced to solve the inhomogeneous non-stationary heat or diffusion equation. Spatial discretization of these problems usually yields huge and stiff ordinary differential equation systems, the solution of which is still time-consuming. The performance of the new method is compared with analytical and numerical solutions. It is proven exactly as well as demonstrated numerically that the new method is first order in time and can give approximate results for extremely large systems faster than the commonly used explicit or implicit methods. The new method can be easily parallelized and it is handy to apply regardless of space dimensions and grid irregularity. Implicit-explicit second derivative diagonally implicit multistage integration methods https://zbmath.org/1463.65173 2021-07-26T21:45:41.944397Z "Abdi, Ali" https://zbmath.org/authors/?q=ai:abdi.ali "Hojjati, Gholamreza" https://zbmath.org/authors/?q=ai:hojjati.gholamreza "Sharifi, Mohammad" https://zbmath.org/authors/?q=ai:sharifi.mohammad Summary: We introduce a class of implicit-explicit (IMEX) schemes for the numerical solution of initial value problems of differential equations with both non-stiff and stiff components in which non-stiff and stiff solvers are, respectively, based on the explicit general linear methods (GLMs) and implicit second derivative GLMs (SGLMs). The order conditions of the proposed IMEX schemes are obtained. Linear stability properties of the methods are analyzed and then methods up to order four with a large area of absolute stability region of the pair are constructed assuming that the implicit part of the methods is $$L$$-stable. Due to the high stage orders of the constructed methods, they are not marred by order reduction. This is verified by the numerical experiments which demonstrate the efficiency of the proposed methods, too. A residual method using Bézier curves for singular nonlinear equations of Lane-Emden type https://zbmath.org/1463.65174 2021-07-26T21:45:41.944397Z "Adıyaman, Meltem E." https://zbmath.org/authors/?q=ai:adiyaman.meltem-evrenosoglu "Oger, Volkan" https://zbmath.org/authors/?q=ai:oger.volkan Summary: In this article, we introduce a new method to solve a singular nonlinear equation of the Lane-Emden type by approximating the solution with Bernstein polynomials. This method is based on the minimization of a residual function using Taylor's series expansion. We also apply this method to problems that are solved by other methods and the obtained results show that our method is efficient, applicable and has great potential than others. Finite integration method via Chebyshev polynomial expansion for solving 2-D linear time-dependent and linear space-fractional differential equations https://zbmath.org/1463.65175 2021-07-26T21:45:41.944397Z "Boonklurb, Ratinan" https://zbmath.org/authors/?q=ai:boonklurb.ratinan "Duangpan, Ampol" https://zbmath.org/authors/?q=ai:duangpan.ampol "Saengsiritongchai, Arnont" https://zbmath.org/authors/?q=ai:saengsiritongchai.arnont Summary: In this paper, we modify the finite integration method (FIM) using Chebyshev polynomial, to construct a numerical algorithm for finding approximate solutions of two-dimensional linear time-dependent differential equations. Comparing with the traditional FIMs using trapezoidal and Simpson's rules, the numerical results demonstrate that our proposed algorithm give a better accuracy even for a large time step. In addition, we also devise a numerical algorithm based on the idea of FIM using Chebyshev polynomial to find approximate solutions of a linear space-fractional differential equations under Riemann-Liouville definition of fractional order derivative. Several numerical examples are given and accuracy of our numerical method is demonstrated comparing to their exact solutions. It is shown that the our proposed method can give a good accuracy as high as $$10^{-5}$$ even with a few numbers of computational node. Stability of waveform relaxation methods based on linear multistep methods https://zbmath.org/1463.65176 2021-07-26T21:45:41.944397Z "Fan, Zhencheng" https://zbmath.org/authors/?q=ai:fan.zhencheng Summary: There exist a variety of research works on waveform relaxation (WR) methods. Most of them focus on the convergence and few of them is concerned with the stability. In this paper, the linear stability of WR methods based on the linear multistep methods is studied. We obtained some sufficient conditions of the linear stability, gave some examples of the linearly stable WR methods, and presented some numerical examples for supporting the theories obtained. A posteriori error estimation in maximum norm for a system of singularly perturbed Volterra integro-differential equations https://zbmath.org/1463.65177 2021-07-26T21:45:41.944397Z "Liang, Ying" https://zbmath.org/authors/?q=ai:liang.ying "Liu, Li-Bin" https://zbmath.org/authors/?q=ai:liu.li-bin "Cen, Zhongdi" https://zbmath.org/authors/?q=ai:cen.zhongdi Summary: In this paper, a system of singularly perturbed Volterra integro-differential equations is considered. The backward Euler formula is used to discretize the differential part and the right-hand rectangle rule is applied to approximate the integral term. The stabilities of the continuous and discrete solutions are carried out using the Grönwall's inequality, respectively. The a posterior error bounds are given to design an adaptive grid generation algorithm. Numerical results complement the theoretical results. Development and implementation of a computational algorithm for solving ordinary differential equations https://zbmath.org/1463.65178 2021-07-26T21:45:41.944397Z "Ogunrinde, Roseline Bosede" https://zbmath.org/authors/?q=ai:ogunrinde.roseline-bosede Summary: In this paper, we developed a numerical algorithm which aimed to solve some first order initial value problems of ordinary differential equations. We explicitly present the breakdown derivation of the new numerical algorithm. The implementation of this new numerical algorithm is on some real life problems leading to first order initial value problem of ordinary differential equations. Results comparison is also made with some existing methods. Second derivative general linear method in Nordsieck form https://zbmath.org/1463.65179 2021-07-26T21:45:41.944397Z "Okuonghae, Robert I." https://zbmath.org/authors/?q=ai:okuonghae.robert-i "Ikhile, Monday Ndidi Oziegbe" https://zbmath.org/authors/?q=ai:ikhile.monday-ndidi-oziegbe Summary: This paper considers the construction of second derivative general linear methods'' (SD-GLM) from hybrid LMM and their transformation to Nordsieck GLM. We show how the Runge-Kutta starters for the methods can be derived. The representation of the methods in Nordsieck form has the advantage of easy implementation in variable stepsize. On the numerical Picard iterations with collocations for the initial value problem https://zbmath.org/1463.65180 2021-07-26T21:45:41.944397Z "Scheiber, Ernest" https://zbmath.org/authors/?q=ai:scheiber.ernest Summary: Some variants of the numerical Picard iterations method are presented to solve an IVP for an ordinary differential system. The term numerical emphasizes that a numerical solution is computed. The method consists in replacing the right hand side of the differential system by Lagrange interpolation polynomials followed by successive approximations. In the case when the number of interpolation point is fixed, a convergence result is given. For stiff problems, starting from a stabilization principle, it is given a variant of the Picard iteration method. Finally, some numerical experiments are reported. The convergence and stability of fractional Euler method for nonlinear fractional ordinary differential equation https://zbmath.org/1463.65181 2021-07-26T21:45:41.944397Z "Tian, Xianzhen" https://zbmath.org/authors/?q=ai:tian.xianzhen "Sun, Liqiang" https://zbmath.org/authors/?q=ai:sun.liqiang "Qin, Baiying" https://zbmath.org/authors/?q=ai:qin.baiying Summary: This paper mainly investigates the convergence and stability of fractional Euler method for initial value problems of nonlinear fractional ordinary differential equation (F-ODE) with Caputo derivative, and an implicit Euler method for the nonlinear F-ODE is proposed. Convergence and stability of the implicit Euler method are established. Finally, the numerical experiment confirms the obtained theoretical results. Long-time stability and convergence of linear multistep methods to autonomous ordinary differential equation https://zbmath.org/1463.65182 2021-07-26T21:45:41.944397Z "Zhang, Fayong" https://zbmath.org/authors/?q=ai:zhang.fayong "Li, Ning" https://zbmath.org/authors/?q=ai:li.ning Summary: The approximation of solutions of the autonomous ordinary differential equation is studied. Long-time stability and convergence of the linear multistep methods are proved under the hypothesis that the exact solution $$u (t)$$ tends to a hyperbolic equilibrium point $$\underline{u}$$ as $$t\to +\infty$$. Several numerical experiments verify the correctness of the theoretical analysis. Delay-dependent stability of Runge-Kutta methods for linear neutral systems with multiple delays. https://zbmath.org/1463.65183 2021-07-26T21:45:41.944397Z "Hu, Guang-Da" https://zbmath.org/authors/?q=ai:hu.guangda This paper studies the stability of Runge-Kutta methods in conjunction with Lagrangian interpolation when applied to linear neutral systems with fixed multiple delays. This is done through the use of characteristic equations. Two simple systems of dimension two are studied with respect to the classical Runge-Kutta method. It is not clear how this analysis can be practically applied to larger dimensional systems. Delay-dependent stability of linear multi-step methods for linear neutral systems. https://zbmath.org/1463.65184 2021-07-26T21:45:41.944397Z "Hu, Guang-Da" https://zbmath.org/authors/?q=ai:hu.guangda "Shao, Lizhen" https://zbmath.org/authors/?q=ai:shao.lizhen The paper is concerned with the delay-dependent stability of numerical methods for linear neutral systems with multiple delays. Delay-dependent stability of linear multi-step (LM) methods is investigated combined with Lagrange interpolation. It was shown that the resulting difference system from an LM method combined with Lagrange interpolation is asymptotically stable for an asymptotically stable neutral system depending on the step-size. An algorithm is provided for checking the delay-dependent stability of LM methods. Two numerical examples are given using fourth-order explicit Adams-Bashforth method to demonstrate the main results of the paper. Accuracy of implicit DIMSIMs with extrapolation https://zbmath.org/1463.65185 2021-07-26T21:45:41.944397Z "Kadhim, Ali J." https://zbmath.org/authors/?q=ai:kadhim.ali-j "Gorgey, A." https://zbmath.org/authors/?q=ai:gorgey.a "Arbin, N." https://zbmath.org/authors/?q=ai:arbin.n "Razali, N." https://zbmath.org/authors/?q=ai:razali.noorhelyna|razali.nur-isnida Summary: The main aim of this article is to present recent results concerning diagonal implicit multistage integration methods (DIMSIMs) with extrapolation in solving stiff problems. Implicit methods with extrapolation have been proven to be very useful in solving problems with stiff components. There are many articles written on extrapolation of Runge-Kutta methods however fewer articles on extrapolation were written for general linear methods. Passive extrapolation is more stable than active extrapolation as proven in many literature when solving stiff problems by the Runge-Kutta methods. This article takes the first step by investigating the performance of passive extrapolation for DIMSIMs type-2 methods. In the variable stepsize and order codes, order-2 and order-3 DIMSIMs with extrapolation are investigated for Van der Pol and HIRES problems. Comparisons are made with ode23 solver and the numerical experiments showed that implicit DIMSIMs with extrapolation has greater accuracy than the method itself without extrapolation and ode23. Numerical simulation of generalized Newtonian fluids flow in bypass geometry. https://zbmath.org/1463.65186 2021-07-26T21:45:41.944397Z "Keslerová, Radka" https://zbmath.org/authors/?q=ai:keslerova.radka "Řezníček, Hynek" https://zbmath.org/authors/?q=ai:reznicek.hynek "Padělek, Tomáš" https://zbmath.org/authors/?q=ai:padelek.tomas This paper present the numerical analysis of three dimensional non-Newtonian fluid flow in a model of bypass in dependence on a parameter characterizing the geometry of the computational domain. The considered parameter is the angle of the connection between the channel and the bypass connection. Furthermore, the fluid is modeled as non-Newtonian with several rheology viscosity models considered. The flow is assumed to be governed by the generalized system of incompressible Navier-Stokes equations. The numerical approximation is based on the finite volume method practically realized within the OpenFOAM project. The numerical results are presented and the influence of the rheological model is discussed. For the entire collection see [Zbl 1425.65005]. Instability of Crank-Nicolson leap-frog for nonautonomous systems https://zbmath.org/1463.65187 2021-07-26T21:45:41.944397Z "Layton, William" https://zbmath.org/authors/?q=ai:layton.william-j "Takhirov, Aziz" https://zbmath.org/authors/?q=ai:takhirov.aziz "Sussman, Myron" https://zbmath.org/authors/?q=ai:sussman.myron-m Summary: used for atmosphere, ocean and climate simulations. Its stability under a CFL condition in the autonomous case was proven by Fourier methods in 1962 and by energy methods for autonomous systems in 2012. We provide an energy estimate showing that solution energy can grow with time in the nonautonomous case, with worst case rate proportional to time step size. We present two constructions showing that this worst case growth rate is attained for a sequence of timesteps $$\Delta t\to 0$$. The construction exhibiting this growth for leapfrog is for a problem with a periodic coefficient. Numerical solution of space fractional diffusion equation by spline method combined with Richardson extrapolation https://zbmath.org/1463.65188 2021-07-26T21:45:41.944397Z "Soori, Z." https://zbmath.org/authors/?q=ai:soori.zoleikha "Aminataei, A." https://zbmath.org/authors/?q=ai:aminataei.azim Summary: In this paper, we propose a high-order method for numerical solution of space fractional diffusion equation (SFDE) in one and two dimensions. The space fractional derivative of order $$1<\alpha < 2$$ is described in Caputo's sense. The spline approximation of Caputo is considered which has second-order accuracy in space. To improve the spatial accuracy, Richardson extrapolation method is presented. As a result, the high-order method can be viewed as the modification of the existing jobs [\textit{E. Sousa}, ibid. 62, No. 3, 938--944 (2011; Zbl 1228.65153); \textit{Y. Salehi} et al., ibid. 336, 465--480 (2018; Zbl 1427.65218)]. For the two-dimensional case, an alternating direction implicit (ADI) scheme is considered to split the equation into two separate one-dimensional equations. Moreover, the proposed scheme is extended to two-sided SFDE case. Numerical results confirm the theoretical supports and the effectiveness of the proposed scheme. The numerical solution of the uncertain differential equation and its application https://zbmath.org/1463.65189 2021-07-26T21:45:41.944397Z "Wang, Zhigang" https://zbmath.org/authors/?q=ai:wang.zhigang.1 "Shen, Kang" https://zbmath.org/authors/?q=ai:shen.kang "Wang, Sizhe" https://zbmath.org/authors/?q=ai:wang.sizhe Summary: Uncertain differential equations are widely used in uncertain finance, uncertain control, uncertain differential game and so on. It is difficult to find the analytical solution of some uncertain differential equations. In this paper, two numerical solutions of uncertain differential equations, Euler method and Runge-Kutta method are studied firstly, and the errors analysis are carried out. By comparing the European option pricing formulas of the stochastic domain Black-Scholes models and the uncertain domain Liu models, the rationality and practicability of the uncertain differential equations for describing the stock market are verified. A recursive analytical algorithm for dynamics analysis of nonlinear oscillators based on Riemannian geometry https://zbmath.org/1463.65190 2021-07-26T21:45:41.944397Z "Yang, Zhe" https://zbmath.org/authors/?q=ai:yang.zhe|yang.zhe.1 "Chen, Guohai" https://zbmath.org/authors/?q=ai:chen.guohai "Yang, Dixiong" https://zbmath.org/authors/?q=ai:yang.dixiong Summary: Based on Riemannian geometry and a variational principle, this paper derives the second order differential equation of a nonlinear dissipative dynamical system on the Riemannian manifold. The concept of manifold retraction is applied to discretize the dynamic equation, and the corresponding recursive scheme is established. Three autonomous nonlinear damped oscillator systems are taken as examples, and their differential dynamic equations are solved by using the recursive analytical algorithm and the Runge-Kutta algorithm, respectively. The computational time of the two algorithms with different time steps is also compared. Numerical results indicate that in comparison with the Runge-Kutta algorithm, the Riemannian geometry-based recursive algorithm can not only achieve the analytical expression of dynamic equations in each time step, but also its running time is shorter than that of the former with higher computational efficiency. The recursive algorithm for dynamic equations based on Riemannian manifolds offers a new idea for analytically solving nonlinear dynamic systems. Dissipativity of multistep Runge-Kutta methods for a class of nonlinear functional-integro-differential equations https://zbmath.org/1463.65191 2021-07-26T21:45:41.944397Z "Zhang, Yan" https://zbmath.org/authors/?q=ai:zhang.yan.5|zhang.yan.4 Summary: In this paper, the dissipation of numerical solutions for nonlinear functional-integro-differential equations is studied. A sufficient condition of the dissipation of the multistep Runge-Kutta method is presented for the equation. Furthermore, a numerical example is given to illustrate the main result of this paper. Counted decision of marginal problem for between limiting differential equation by method of modified running https://zbmath.org/1463.65192 2021-07-26T21:45:41.944397Z "Chadaev, V. A." https://zbmath.org/authors/?q=ai:chadaev.v-a (no abstract) Gradient recovery techniques in one-dimensional goal-oriented problems https://zbmath.org/1463.65193 2021-07-26T21:45:41.944397Z "El-Agamy, M." https://zbmath.org/authors/?q=ai:el-agamy.m "Elsaid, A." https://zbmath.org/authors/?q=ai:elsaid.ahmed "Nour, H. M." https://zbmath.org/authors/?q=ai:nour.hamed-mohamed Summary: In this paper, we propose a new technique to evaluate a posteriori error estimate for goal-oriented problems using recovery techniques. In our technique, we replace the gradient in the goal-oriented error estimate by the recovered gradient obtained by the polynomial preserving recovery technique. Also, we present a new local renement algorithm suitable to the proposed technique. Finally, the validity of the proposed technique is illustrated by numerical examples. Reproducing kernel shifted Legendre basis function method for solving the fractional differential equations https://zbmath.org/1463.65194 2021-07-26T21:45:41.944397Z "Gong, Quanyi" https://zbmath.org/authors/?q=ai:gong.quanyi "Yao, Huanmin" https://zbmath.org/authors/?q=ai:yao.huanmin Summary: Based on the theory of the reproducing kernel, this paper constructed a new reproducing kernel space and the reproducing kernel function of the space is given by using the shifted Legendre polynomials as the basis function. It is different from former that this function is no longer a piecewise function, so when the fractional operator is used on the kernel function, the computation is reduced. Thus the approximate solution is more accurate. Finally, the numerical examples illustrate the validity of the method. Reproducing kernel method for nonlinear higher-order multi-point boundary value problems https://zbmath.org/1463.65195 2021-07-26T21:45:41.944397Z "He, Guangting" https://zbmath.org/authors/?q=ai:he.guangting "Wang, Xinxin" https://zbmath.org/authors/?q=ai:wang.xinxin "Lv, Xueqin" https://zbmath.org/authors/?q=ai:lu.xueqin Summary: In this paper, we present an algorithm for solving a class of nonlinear $$2n$$th-order multi-point boundary value problems. The method is based on an iterative technique and the reproducing kernel method. Numerical results are shown to illustrate the accuracy of the presented method. A numerical method for the solution of fifth order boundary value problem in ordinary differential equations https://zbmath.org/1463.65196 2021-07-26T21:45:41.944397Z "Pandey, Pramod Kumar" https://zbmath.org/authors/?q=ai:kumar-pandey.pramod Summary: In this article we have proposed a technique for solving the fifth order boundary value problem as a coupled pair of boundary value problems. We have considered fifth order boundary value problem in ordinary differential equation for the development of the numerical technique. There are many techniques for the numerical solution of the problem considered in this article. Thus we considered the application of the finite difference method for the numerical solution of the problem. In this article we transformed fifth order differential problem into system of differential equations of lower order namely one and four. We discretized the system of differential equations into considered domain of the problem. Thus we got a system of algebraic equations. For the numerical solution of the problem, we have the system of algebraic equations. The solution of the algebraic equations is an approximate solution of the problem considered. Moreover we get numerical approximation of first and second derivative as a byproduct of the proposed method. We have shown that proposed method is convergent and order of accuracy of the proposed method is at lease quadratic. The numerical results obtained in computational experiment on the test problems approve the efficiency and accuracy of the method. Modified grid method for solving linear differential equation equipped with variable coefficients based on Taylor series https://zbmath.org/1463.65197 2021-07-26T21:45:41.944397Z "Radchenko, V. P." https://zbmath.org/authors/?q=ai:radchenko.vladimir-pavlovich|radchenko.v-p.1 "Usov, A. A." https://zbmath.org/authors/?q=ai:usov.aleksandr-aleksandrovich Summary: The modified grid method for solving value boundary problems for the linear differential equations based on Taylor development is described. It was demonstrated that the accuracy of the proposed method is much greater than that of a classical grids method. The results of numerical experiments are quoted. High efficient numerical algorithm for nonlinear singular boundary value problems https://zbmath.org/1463.65198 2021-07-26T21:45:41.944397Z "Sun, Pingping" https://zbmath.org/authors/?q=ai:sun.pingping "Niu, Jing" https://zbmath.org/authors/?q=ai:niu.jing "Wu, Qi" https://zbmath.org/authors/?q=ai:wu.qi "Li, Ping" https://zbmath.org/authors/?q=ai:li.ping.2|li.ping.4|li.ping|li.ping.5|li.ping.1|li.ping.3 Summary: In this work, an efficient method based on the simplified reproducing kernel method and least squares method is proposed for solving nonlinear singular boundary value problems. Furthermore, the approximate solution of the nonlinear singular boundary value problem is given, the error estimation of the proposed algorithm is also given. Finally, the numerical examples are compared with the traditional regenerative nuclear method and the variational iteration algorithm, which shows the effectiveness and high efficiency of the algorithm. Applications of modified Mickens-type NSFD schemes to Lane-Emden equations https://zbmath.org/1463.65199 2021-07-26T21:45:41.944397Z "Verma, Amit K." https://zbmath.org/authors/?q=ai:verma.amit-kumar "Kayenat, Sheerin" https://zbmath.org/authors/?q=ai:kayenat.sheerin Summary: If there is a jump discontinuity present in the forcing term of a boundary value problem (BVP), the nonstandard finite difference (NSFD) and finite difference (FD) methods do not approximate the solutions very well. Here we use fuzzy transforms (FTs) and derive fuzzy transformed NSFD schemes that are referred to as non-standard fuzzy transform methods (NFTMs). The convergence of the derived NFTMs is established. Numerical solutions of Lane-Emden type equations are obtained using NFTMs. We show that NFTMs provide better results than NSFD and FD methods when the forcing term has a jump discontinuity. Even for large jumps, the NFTMs provide more accurate results than the other methods. An improved harmony search for linear two-point boundary value problem https://zbmath.org/1463.65200 2021-07-26T21:45:41.944397Z "Yong, Longquan" https://zbmath.org/authors/?q=ai:yong.longquan Summary: Numerical solutions of linear two-point boundary value problems are studied. By using the finite difference method, the discretization of linear two-point boundary value problem is performed. An unconstrained optimization problem is obtained and solved by heuristic method named novel global harmony search algorithm (NGHS). The NGHS algorithm utilizes position updating and mutation strategy with low probability. By repeatedly adjusting the pitch of the instruments in the band, eventually NGHS reaches a wonderful state of the process of sound. Numerical results show that this method is more effective by solving 3 linear two-point boundary value problems. The hybrid finite difference schemes on the modified Bakhvalov-Shishkin mesh for the singularly perturbed problem https://zbmath.org/1463.65201 2021-07-26T21:45:41.944397Z "Zheng, Quan" https://zbmath.org/authors/?q=ai:zheng.quan "Liu, Ying" https://zbmath.org/authors/?q=ai:liu.ying.1|liu.ying.2|liu.ying.3|liu.ying.5|liu.ying.4|liu.ying|liu.ying.6 "Liu, Zhongli" https://zbmath.org/authors/?q=ai:liu.zhongli Summary: This paper develops a new hybrid finite difference scheme combining the midpoint upwind scheme with the central difference scheme on a three-piece modified Bakhvalov-Shishkin mesh to solve the singularly perturbed two-point boundary value problem. Better $$\varepsilon$$-uniform accuracy and order of convergence are obtained by adopting truncation error, discrete comparison principle, barrier functions and so on. From the coarse mesh to the fine mesh, the error estimate of second-order convergence, first-order convergence and second-order convergence are obtained in turn. The numerical examples confirm the theoretical results and illustrate the advantage on accuracy of the method over the other three methods. Uniform numerical approximation for parameter dependent singularly perturbed problem with integral boundary condition https://zbmath.org/1463.65202 2021-07-26T21:45:41.944397Z "Kudu, Mustafa" https://zbmath.org/authors/?q=ai:kudu.mustafa "Amirali, Ilhame" https://zbmath.org/authors/?q=ai:amirali.ilhame "Amiraliyev, Gabil M." https://zbmath.org/authors/?q=ai:amiraliyev.gabil-m Summary: In this paper, a parameter-uniform numerical method for a parameterized singularly perturbed ordinary differential equation containing integral boundary condition is studied. Asymptotic estimates on the solution and its derivatives are derived. A numerical algorithm based on upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error estimate for the numerical solution is established. Numerical results are presented, which illustrate the theoretical results. On the nonlocal discretization of the simplified Anderson-May model of viral infection https://zbmath.org/1463.65203 2021-07-26T21:45:41.944397Z "Korpusik, Adam" https://zbmath.org/authors/?q=ai:korpusik.adam "Bodnar, Marek" https://zbmath.org/authors/?q=ai:bodnar.marek Summary: We present five nonstandard finite difference methods designed for numerical simulation of the simplified Anderson-May model of viral infection. The proposed methods, based solely on the principle of nonlocal discretization, are able to preserve all of the essential qualitative features of the original model: the non-negativity of the solution and local stability of the equilibrium points, along with their stability conditions. One of the proposed methods preserves the types of the equilibrium points (i.e. the presence and absence of oscillations) as well. All of these results are independent of the chosen step-size of simulation. Generalized Mittag-Leffler quadrature methods for fractional differential equations https://zbmath.org/1463.65205 2021-07-26T21:45:41.944397Z "Li, Yu" https://zbmath.org/authors/?q=ai:li.yu "Cao, Yang" https://zbmath.org/authors/?q=ai:cao.yang "Fan, Yan" https://zbmath.org/authors/?q=ai:fan.yan Summary: In this paper, we propose a generalized Mittag-Leffler quadrature method for solving linear fractional differential equations with a forcing term. The construction of such a scheme is based on the variation-of-constants formula which has a generalized Mittag-Leffler function in the kernel, and incorporates the idea of collocation methods and the local Fourier expansion of the system. The properties of the methods are analyzed. The numerical experiments show the high effectiveness of the new methods. Stability and convergence of Runge-Kutta methods for stiff impulsive differential equations https://zbmath.org/1463.65206 2021-07-26T21:45:41.944397Z "Yu, Yuexin" https://zbmath.org/authors/?q=ai:yu.yuexin "Wen, Haiyang" https://zbmath.org/authors/?q=ai:wen.haiyang "Xiao, Rong" https://zbmath.org/authors/?q=ai:xiao.rong "Yang, Chuanying" https://zbmath.org/authors/?q=ai:yang.chuanying Summary: Runge-Kutta methods are adopted for solving stiff impulsive differential equations. The numerical stability and asymptotic stability conditions of $$(k, l)$$-algebraically stable Runge-Kutta methods are derived. Meanwhile, it is proved that if a Runge-Kutta method for solving stiff ordinary differential equations is B-convergent of order $$r$$, then it is also B-convergent of order $$r$$ for solving stiff impulsive differential equations. A hybrid collocation method based on combining the third kind Chebyshev polynomials and block-pulse functions for solving higher-order initial value problems https://zbmath.org/1463.65207 2021-07-26T21:45:41.944397Z "Jahangiri, Saeid" https://zbmath.org/authors/?q=ai:jahangiri.saeid "Maleknejad, Khosrow" https://zbmath.org/authors/?q=ai:maleknejad.khosrow "Tavassoli Kajani, Majid" https://zbmath.org/authors/?q=ai:tavassoli-kajani.majid Summary: The purpose of this paper is to propose a new collocation method for solving linear and nonlinear differential equations of high order as well as solving the differential equation on a very large interval. The new collocation method is based on a hybrid method combining the functions of the third kind Chebyshev polynomials and block-pulse functions uses. In the proposed method, the large interval of the problems is divided to small sub-intervals and in each sub-interval, collocation method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution of the differential equation on each sub-intervals. The proposed method is more accurate than the previous methods. Numerical examples show the capability and effciency of the presented method compared to existing methods. Numerical calculation method for a class of fractional pantograph delay differential equations https://zbmath.org/1463.65208 2021-07-26T21:45:41.944397Z "Wang, Linjun" https://zbmath.org/authors/?q=ai:wang.linjun "Zhang, Lu" https://zbmath.org/authors/?q=ai:zhang.lu Summary: Based on a class of orthogonal polynomials, alternative Legendre polynomials (ALPs), we proposed a numerical calculation method for fractional pantograph delay differential equations. Firstly, we obtained the numerical approximation results of fractional calculus by using the properties of ALPs. Secondly, we transformed the fractional pantograph delay differential equations into the algebraic system for obtaining solution. Thirdly, we analyzed the error of the method and obtained the convergence results of the method. Finally, some numerical examples were given to verify the effectiveness and accuracy of the proposed method. A Galerkin-like approach to solve continuous population models for single and interacting species https://zbmath.org/1463.65209 2021-07-26T21:45:41.944397Z "Yüzbaşı, Şuayip" https://zbmath.org/authors/?q=ai:yuzbasi.suayip "Karaçayır, Murat" https://zbmath.org/authors/?q=ai:karacayir.murat Summary: In this paper, we present a Galerkin-like approach to numerically solve continuous population models for single and interacting species. After taking inner product of a set of monomials with a vector obtained from the problem under consideration, the problem is transformed to a nonlinear system of algebraic equations. The solution of this system gives the coefficients of the approximate solutions. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed in some detail. The method and the residual correction technique are illustrated with two examples. The results are also compared with numerous existing methods from the literature. The combined RKM and ADM for solving Riccati differential equations https://zbmath.org/1463.65210 2021-07-26T21:45:41.944397Z "Li, Meiyi" https://zbmath.org/authors/?q=ai:li.meiyi "Yan, Dandan" https://zbmath.org/authors/?q=ai:yan.dandan "Lv, Xueqin" https://zbmath.org/authors/?q=ai:lu.xueqin Summary: In this paper, a computational method is proposed for solving Riccati differential equations. This method is based on the Adomian decomposition method (ADM) and the reproducing kernel method (RKM). The approximate solutions are given in the form of series. Numerical results are shown to illustrate the accuracy of the present method. Solving higher order nonlinear ordinary differential equations with least squares support vector machines https://zbmath.org/1463.65211 2021-07-26T21:45:41.944397Z "Lu, Yanfei" https://zbmath.org/authors/?q=ai:lu.yanfei "Yin, Qingfei" https://zbmath.org/authors/?q=ai:yin.qingfei "Li, Hongyi" https://zbmath.org/authors/?q=ai:li.hongyi "Sun, Hongli" https://zbmath.org/authors/?q=ai:sun.hongli "Yang, Yunlei" https://zbmath.org/authors/?q=ai:yang.yunlei "Hou, Muzhou" https://zbmath.org/authors/?q=ai:hou.muzhou Summary: In this paper, a numerical method based on least squares support vector machines has been developed to solve the initial and boundary value problems of higher order nonlinear ordinary differential equations. The numerical experiments have been performed on some nonlinear ordinary differential equations to validate the accuracy and reliability of our proposed LS-SVM model. Compared with the exact solution, the results obtained by our proposed LS-SVM model can achieve a very high accuracy. The proposed LS-SVM model could be a good tool for solving higher order nonlinear ordinary differential equations. Model order reduction for sine-Gordon equation using POD and DEIM https://zbmath.org/1463.65212 2021-07-26T21:45:41.944397Z "Sukuntee, Norapon" https://zbmath.org/authors/?q=ai:sukuntee.norapon "Chaturantabut, Saifon" https://zbmath.org/authors/?q=ai:chaturantabut.saifon Summary: This work applies model reduction techniques for efficiently approximating the solution of the sin-Gordon equation. The proper orthogonal decomposition (POD) is employed to construct a low dimensional basis that can accurately capture the dynamics of the solution space. This POD basis is used with the Galerkin projection to obtain a reduced-order model that is much smaller than the original discretized system. However, the effective dimension reduction of the POD-Galerkin approach is limited to the linearity of the system. The discrete empirical interpolation method (DEIM) is then applied to further reduce the computational complexity of the nonlinear term. This work investigates the effect of using different amount of snapshots from coarse discretization in constructing the POD basis, as well as demonstrates the applicability of the POD-DEIM approach on predicting solution of the parametrized sine-Gordon equation. The POD-DEIM solutions are shown to be accurate for both numerical tests and can be solved with much less computational time and memory storage when compared to the high-dimensional discretized sine-Gordon equation. The embedded WENO scheme for solving the Euler equation https://zbmath.org/1463.65215 2021-07-26T21:45:41.944397Z "Bai, Xiaoya" https://zbmath.org/authors/?q=ai:bai.xiaoya "Zheng, Qiuya" https://zbmath.org/authors/?q=ai:zheng.qiuya "Liang, Yihua" https://zbmath.org/authors/?q=ai:liang.yihua Summary: In order to optimize the numerical calculation of Euler equation, a new scheme (E-CUSP-Embedded-WENO5) was proposed, which was obtained by embedded weighted essential-no-oscillation (Embedded-WENO) scheme coupling low-dissipation total energy convection upwind and partial pressure (E-CUSP) scheme. In the new scheme, the flux obtained by E-CUSP was reconstructed by Embedded-WENO scheme in spatial direction and the fourth-order strong stable Runge-Kutta method in time direction. The results showed that the solution of the new scheme was closer to the theoretical solution near the shock wave. It also had better stability, higher resolution and stronger ability to capture shock wave and contact discontinuity, especially for capturing shock wave, only two to three elements were needed. A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative https://zbmath.org/1463.65219 2021-07-26T21:45:41.944397Z "Cao, Junying" https://zbmath.org/authors/?q=ai:cao.junying "Wang, Ziqiang" https://zbmath.org/authors/?q=ai:wang.ziqiang "Xu, Chuanju" https://zbmath.org/authors/?q=ai:xu.chuanju Summary: In this paper, we consider numerical solutions of fractional ordinary differential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach. Analysis of time-stepping methods for the monodomain model https://zbmath.org/1463.65243 2021-07-26T21:45:41.944397Z "Roy, Thomas" https://zbmath.org/authors/?q=ai:roy.thomas "Bourgault, Yves" https://zbmath.org/authors/?q=ai:bourgault.yves "Pierre, Charles" https://zbmath.org/authors/?q=ai:pierre.charles Summary: To a large extent, the stiffness of the bidomain and monodomain models depends on the choice of the ionic model, which varies in terms of complexity and realism. In this paper, we compare and analyze a variety of time-stepping methods: explicit or semi-implicit, operator splitting, exponential, and deferred correction methods. We compare these methods for solving the monodomain model coupled with three ionic models of varying complexity and stiffness: the phenomenological Mitchell-Schaeffer model, the more realistic Beeler-Reuter model, and the stiff and very complex ten Tuscher-Noble-Noble-Panfilov (TNNP) model. For each method, we derive absolute stability criteria of the spatially discretized monodomain model and verify that the theoretical critical time steps obtained closely match the ones in numerical experiments. We also verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. The efficiency of the different methods is also considered by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured, respectively, by the eigenvalues of the model's Jacobian and the number of variables, and based on strict stability and accuracy criteria. A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem https://zbmath.org/1463.65249 2021-07-26T21:45:41.944397Z "Sumit" https://zbmath.org/authors/?q=ai:sumit. "Kumar, Sunil" https://zbmath.org/authors/?q=ai:kumar.sunil "Kuldeep" https://zbmath.org/authors/?q=ai:kuldeep.gajraj|kuldeep.b "Kumar, Mukesh" https://zbmath.org/authors/?q=ai:kumar.mukesh Summary: In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory. Stability analysis of second-order explicit TVD Runge-Kutta discontinuous Galerkin method for linear hyperbolic conservation laws https://zbmath.org/1463.65287 2021-07-26T21:45:41.944397Z "Bi, Hui" https://zbmath.org/authors/?q=ai:bi.hui "Xu, Ya'nan" https://zbmath.org/authors/?q=ai:xu.yanan Summary: The stability problem of the second-order explicit TVD Runge-Kutta discontinuous Galerkin method to solve the linear hyperbolic conservation law equations under the $$\alpha$$-th order divided differences is studied. When the solution is sufficiently smooth, it is shown by the finite element analysis technique that for any $$k$$-th order piecewise polynomial space on the non-uniform regular meshes, the algorithm is $${L^2}$$-norm stable under the CFL condition $$\tau \le \lambda {C^{-2}}h$$, where $$C$$ and $$\lambda$$ are constants independent of $$h$$ and $$\tau$$. Stability of ALE discontinuous Galerkin method with Radau quadrature. https://zbmath.org/1463.65309 2021-07-26T21:45:41.944397Z "Vlasák, Miloslav" https://zbmath.org/authors/?q=ai:vlasak.miloslav Summary: We assume the nonlinear parabolic problem in a time dependent domain, where the evolution of the domain is described by a regular given mapping. The problem is discretized by the discontinuous Galerkin (DG) method modified by the right Radau quadrature in time with the aid of Arbitrary Lagrangian-Eulerian (ALE) formulation. The sketch of the proof of the stability of the method is shown. For the entire collection see [Zbl 1425.65005]. A $$p$$-adaptive RKDG algorithm based on the troubled-cell indicators https://zbmath.org/1463.65315 2021-07-26T21:45:41.944397Z "Yuan, An'an" https://zbmath.org/authors/?q=ai:yuan.anan "Zhu, Hongqiang" https://zbmath.org/authors/?q=ai:zhu.hongqiang "Liu, Rui" https://zbmath.org/authors/?q=ai:liu.rui Summary: The Runge-Kutta discontinuous Galerkin (RKDG) method is one of the main numerical methods for solving equations of hyperbolic conservation laws. Many problems involving conservation laws have a very large scale, and they will cost considerable CPU time and storage. To solve this problem, a $$p$$-adaptive RKDG algorithm based on the troubled-cell indicators is designed. This new algorithm not only can maintain high order approximations in the smooth regions as the standard RKDG method, but also can effectively reduce storage. In the meanwhile, this work also builds the foundation for the upcoming research on $$hp$$ adaptive RKDG algorithm. A meshfree approach for analysis and computational modeling of non-linear Schrödinger equation https://zbmath.org/1463.65323 2021-07-26T21:45:41.944397Z "Jiwari, Ram" https://zbmath.org/authors/?q=ai:jiwari.ram "Kumar, Sanjay" https://zbmath.org/authors/?q=ai:kumar.sanjay.1|kumar.sanjay.2|kumar.sanjay-v "Mittal, R. C." https://zbmath.org/authors/?q=ai:mittal.ramesh-chand "Awrejcewicz, Jan" https://zbmath.org/authors/?q=ai:awrejcewicz.jan Summary: In this article, the authors propose a meshfree approach for simulation of nonlinear Schrödinger equation with constant and variable coefficients. Schrödinger equation is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and in quantum mechanics. First of all, spatial derivatives are discretized by using local radial basis functions based on differential quadrature method (LRBF-DQM) and, subsequently, the obtained system of nonlinear ordinary differential equations (ODEs) is solved by fourth-order Runge-Kutta (RK-4). The stability analysis of the proposed approach is discussed by the matrix method. Numerical experiments ensure that the proposed approach is accurate and computationally efficient. Local discontinuous Galerkin spectral element method for nonlinear reaction-diffusion equations https://zbmath.org/1463.65328 2021-07-26T21:45:41.944397Z "Wang, Hailu" https://zbmath.org/authors/?q=ai:wang.hailu "Wu, Hua" https://zbmath.org/authors/?q=ai:wu.hua Summary: In this paper, local discontinuous Galerkin spectral element method for nonlinear reaction-diffusion equations is considered. The Legendre-Galerkin Chebyshev collocation spectral method is used in the spatial direction, which means that the schemes are formulated with the Legendre-Galerkin method in each subdomain and the jump terms are controlled by numerical flux at cell boundaries. The nonlinear term is interpolated through the Chebyshev-Gauss-Lobatto points. Meanwhile, the fourth-order low-storage Runge-Kutta schemes are used in time discretization. Stability and the rate of convergence of the method are proved. We also give some numerical examples which coincide with the theoretical analysis. The numerical results are compared with those obtained by discontinuous Galerkin finite element method. Structure preserving computational technique for fractional order Schnakenberg model https://zbmath.org/1463.65410 2021-07-26T21:45:41.944397Z "Iqbal, Zafar" https://zbmath.org/authors/?q=ai:iqbal.zafar "Ahmed, Nauman" https://zbmath.org/authors/?q=ai:ahmed.nauman "Baleanu, Dumitru" https://zbmath.org/authors/?q=ai:baleanu.dumitru-i "Rafiq, Muhammad" https://zbmath.org/authors/?q=ai:rafiq.muhammad "Iqbal, Muhammad Sajid" https://zbmath.org/authors/?q=ai:iqbal.muhammad-sajid "Rehman, Muhammad Aziz-Ur" https://zbmath.org/authors/?q=ai:rehman.muhammad-aziz-ur Summary: The current article deals with the analysis and numerical solution of fractional order Schnakenberg (S-B) model. This model is a system of autocatalytic reactions by nature, which arises in many biological systems. This study is aiming at investigating the behavior of natural phenomena with a more realistic and practical approach. The solutions are obtained by applying the Grunwald-Letnikov (G-L) finite difference (FD) and the proposed G-L nonstandard finite difference (NSFD) computational schemes. The proposed formulation is explicit in nature, strongly structure preserving as well as it is independent of the time step size. One very important feature of our proposed scheme is that it preserves the positivity of the solution of continuous fractional order S-B model because the unknown variables involved in this system describe the chemical concentrations of different substances. The comparison of the proposed scheme with G-L FD method reflects the significance of the said method. A numerical framework for solving high-order pantograph-delay Volterra integro-differential equations https://zbmath.org/1463.65429 2021-07-26T21:45:41.944397Z "Mirzaee, Farshid" https://zbmath.org/authors/?q=ai:mirzaee.farshid "Bimesl, Saeed" https://zbmath.org/authors/?q=ai:bimesl.saeed "Tohidi, Emran" https://zbmath.org/authors/?q=ai:tohidi.emran Summary: In this paper, we present an efficient numerical method to solve Volterra integro-differential equations of pantograph-delay type. We use the Euler polynomials to approximate the solutions. The proposed method is discussed in detail and compared by solving some numerical examples. Moreover, the error estimates of the proposed method is given. Special cases of the main results are also mentioned. A Galerkin-like approach to solve high-order integro-differential equations with weakly singular kernel https://zbmath.org/1463.65435 2021-07-26T21:45:41.944397Z "Yüzbaşı, Şuayip" https://zbmath.org/authors/?q=ai:yuzbasi.suayip "Karaçayir, Murat" https://zbmath.org/authors/?q=ai:karacayir.murat Summary: In this study, a Galerkin-like approach is applied to numerically solve high-order integro-differential equations having weakly singular kernel. The method includes taking inner product of a set of monomials with a vector obtained from the equation in question. The resulting linear system is then solved, yielding a polynomial as the approximate solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution, is discussed briefly. Lastly, the method and the residual correction technique are illustrated with several examples. The results are also compared with numerous existing methods from the literature. Numerical solutions of the elliptic differential and the planetary motion equations by Haar wavelet-quasilinearization technique https://zbmath.org/1463.65444 2021-07-26T21:45:41.944397Z "Yangchareonyuanyong, Rawipa" https://zbmath.org/authors/?q=ai:yangchareonyuanyong.rawipa "Koonprasert, Sanoe" https://zbmath.org/authors/?q=ai:koonprasert.sanoe "Sirisubtawee, Sekson" https://zbmath.org/authors/?q=ai:sirisubtawee.sekson Summary: In this article, we apply the Haar-quasilinearization method (HQM) to solve the second order elliptic differential and planetary motion equations to which initial conditions and three types of boundary conditions including Dirichlet, Neumann-Robin, and Dirichlet-Neumann boundary conditions are equipped. By the HQM, both equations can be reduced to the recurrence relations which are linearized differential equations and then applied the Haar-quasilinearization method for solving these equations. Moreover, comparisons of the obtained results for the constructed problems with the exact solutions, HQM solutions, and some numerical solutions obtained using the standard methods are graphically demonstrated. In particular, the absolute errors, the $$L_2$$-norm errors and the maximum absolute errors $$L_\infty$$ among these solutions are computed. As a result, the HQM is considered as the effective and rapidly convergent scheme. On a hereditarity vibrating system with allowance for the effects stick-slip https://zbmath.org/1463.74090 2021-07-26T21:45:41.944397Z "Parovik, R. I." https://zbmath.org/authors/?q=ai:parovik.roman-ivanovich Summary: The work was a mathematical model that describes the effect of the sliding attachment (stick-slip), taking into account hereditarity. An explicit finite-difference scheme for the corresponding. Cauchy problem was constructed. Built on the basis of its waveform and phase trajectories. Unconditional stability of a Crank-Nicolson Adams-Bashforth 2 numerical methods https://zbmath.org/1463.76014 2021-07-26T21:45:41.944397Z "Jorgenson, Andrew D." https://zbmath.org/authors/?q=ai:jorgenson.andrew-d Summary: Nonlinear partial differential equations modeling turbulent fluid flow and similar processes present special challanges in numerical analysis. Regions of stability of implicit-explicit methods are reviewed, and an energy norm based on Dahlquist's concept of G-stability is developed. Using this norm, a time-stepping Crank-Nicolson Adams-Bashforth 2 implicit-explicit method for solving spatially-discretized convection-diffusion equations of this type is analyzed and shown to be unconditionally stable. Mathematical modeling of transient responses in a large scale multiconductor system https://zbmath.org/1463.94074 2021-07-26T21:45:41.944397Z "Sheng, Caiwang" https://zbmath.org/authors/?q=ai:sheng.caiwang "Zhang, Xiaoqing" https://zbmath.org/authors/?q=ai:zhang.xiaoqing Summary: This paper proposes a reduced-order model for the large scale circuits representing the multiconductor systems. It is based on the block Arnoldi algorithm for calculating congruence transformation matrix. By setting up the state equations in frequency domain for the multiconductor systems, the transfer functions are calculated and so the frequency response curves are obtained. A comparison is made between those curves obtained from the circuits with and without reduced-order treatment and a better agreement appears between them. By using inverse Laplace transform, the lightning transient responses in the multiconductor systems are given in time-domain. A better agreement is shown between calculated and experimental results, which confirm the validity of the proposed model.