Recent zbMATH articles in MSC 45https://zbmath.org/atom/cc/452022-11-17T18:59:28.764376ZWerkzeugCommutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebrashttps://zbmath.org/1496.170152022-11-17T18:59:28.764376Z"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.5|zhang.yi.2|zhang.yi.14|zhang.yi.3|zhang.yi|zhang.yi.1|zhang.yi.10|zhang.yi.6|zhang.yi.8|zhang.yi.4|zhang.yi.12Fix a unitary commutative associative ring \(\mathbf{k}\). A Rota-Baxter algebra is a commutative associative algebra \(R\) over \(\mathbf{k}\) together with a \(\mathbf{k}\)-linear operator \(P: R \longrightarrow R\) satisfying the so-called \textit{Rota-Baxter} identity for \(f, g\) in \(R\):
\[
P(f)P(g) = P(fP(g)) + P(P(f)g).
\]
This is a special case of a more general definition in the paper. We make this simplification in the review because the main results of the paper deal with the special case. As a pioneering example, the ring \(\mathrm{Cont}(\mathbb{R})\) of continuous functions on \(\mathbb{R}\) is a Rota-Baxter algebra over \(\mathbb{R}\), with the operator \(P\) defined by the Riemann integral for \(f \in \mathrm{Cont}(\mathbb{R})\) and \(x \in \mathbb{R}\):
\[
(P(f))(x) := \int_0^x f(t)\, dt.
\]
Let \(\Omega\) be an index set. A matching Rota-Baxter algebra with respect to \(\Omega\) is a commutative associative algebra \(R\) together with a family of linear operators \(P_{\alpha}: R \longrightarrow R\) indexed by \(\alpha \in \Omega\) such that for \(x, y\) in \(R\) and \(\alpha, \beta\) in \(\Omega\) we have:
\[
P_{\alpha}(x) P_{\beta}(y) = P_{\alpha}(xP_{\beta}(y)) + P_{\beta}(P_{\alpha}(x)y).
\]
Each pair \((R, P_{\alpha})\) for a fixed \(\alpha\) forms an ordinary Rota-Baxter algebra. If \((R, P)\) is a Rota-Baxter algebra and \((g_{\alpha})_{\alpha \in \Omega}\) is a family of elements of \(R\), then we can equip \(R\) with a structure of matching Rota-Baxter algebra by setting \(P_{\alpha}(f) := P(g_{\alpha} f)\) for \(\alpha \in \Omega\) and \(f \in R\).
Let \(\mathbf{CAlg}_{\mathbf{k}}\) denote the category of commutative associative algebras over \(\mathbf{k}\) and \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) denote the category of matching Rota-Baxter algebras. By definition we have a forgetful functor \(\mathcal{F}: \mathbf{MRBA}_{\mathbf{k},\Omega} \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\).
One of the main results of this paper is an explicit construction, via \textit{shuffle product}, of a functor \(\mathcal{G}: \mathbf{CAlg}_{\mathbf{k}} \longrightarrow \mathbf{MRBA}_{\mathbf{k},\Omega}\) left adjoint to \(\mathcal{F}\). In more details, for \(A\) a commutative associative algebra, \(\mathcal{G}(A)\) is the tensor product algebra of \(A\) with the shuffle algebra associated to the \(\mathbf{k}\)-module \(\mathbf{k}\Omega \otimes A\), which is naturally a matching Rota-Baxter algebra. The authors also extend this result to the relative setting. Fix an object \(X\) of \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) and let \(\mathcal{C}_X\) denote the category of morphisms \(X \longrightarrow Y\) in \(\mathbf{MRBA}_{\mathbf{k},\Omega}\). Then the forgetful functor \(\mathcal{C}_X \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\) sending a morphism \(X \longrightarrow Y\) to \(\mathcal{F}(Y)\) is shown to admit an explicit left adjoint.
Reviewer: Huafeng Zhang (Villeneuve d'Ascq)A study on controllability of impulsive fractional evolution equations via resolvent operatorshttps://zbmath.org/1496.341162022-11-17T18:59:28.764376Z"Gou, Haide"https://zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the \((\alpha ,\beta)\)-resolvent operator, we concern with the term \(u^\prime(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u^\prime b)=u^\prime_b\). Finally, we present an application to support the validity study.On stability for semilinear generalized Rayleigh-Stokes equation involving delayshttps://zbmath.org/1496.350842022-11-17T18:59:28.764376Z"Lan, Do"https://zbmath.org/authors/?q=ai:lan.do"Tuan, Pham Thanh"https://zbmath.org/authors/?q=ai:tuan.pham-thanhSummary: We consider a functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability, and asymptotic behavior of solutions are addressed. By establishing a Halanay type inequality, we show the dissipativity and asymptotic stability of solutions to our problem. In addition, we prove the existence of a compact set of decay solutions by using local estimates and fixed point arguments.An application of semigroup theory to the coagulation-fragmentation modelshttps://zbmath.org/1496.351652022-11-17T18:59:28.764376Z"Das, Arijit"https://zbmath.org/authors/?q=ai:das.arijit"Das, Nilima"https://zbmath.org/authors/?q=ai:das.nilima"Saha, Jitraj"https://zbmath.org/authors/?q=ai:saha.jitrajSummary: We present the existence and uniqueness of strong solutions for the continuous coagulation-fragmentation equation with singular fragmentation and essentially bounded coagulation kernel using semigroup theory of operators. Initially, we reformulate the coupled coagulation-fragmentation problem into the semilinear abstract Cauchy problem (ACP) and consider it as the nonlinear perturbation of the linear fragmentation operator. The existence of the substochastic semigroup is proved for the pure fragmentation equation. Using the substochastic semigroup and some related results for the pure fragmentation equation, we prove the existence of global nonnegative, strong solution for the coagulation-fragmentation equation.The initial-boundary value problems of the new two-component generalized Sasa-Satsuma equation with a \(4\times 4\) matrix Lax pairhttps://zbmath.org/1496.351702022-11-17T18:59:28.764376Z"Hu, Beibei"https://zbmath.org/authors/?q=ai:hu.beibei"Zhang, Ling"https://zbmath.org/authors/?q=ai:zhang.ling"Lin, Ji"https://zbmath.org/authors/?q=ai:lin.jiSummary: In this paper, we consider a new two-component Sasa-Satsuma equation, which can simulate the propagation and interaction of ultrashort pulses and describe the propagation of femtosecond pulses in optical fibers. The unified transformation method is used to construct a \(4\times 4\) matrix Riemann-Hilbert problem. Then, the solution of the initial-boundary value problems for the new two-component generalized Sasa-Satsuma equation well can be obtained by solving this matrix Riemann-Hilbert problem. In addition, we obtain that the spectral functions satisfy an important global relation.Selection-mutation dynamics with asymmetrical reproduction kernelshttps://zbmath.org/1496.353982022-11-17T18:59:28.764376Z"Perthame, Benoît"https://zbmath.org/authors/?q=ai:perthame.benoit"Strugarek, Martin"https://zbmath.org/authors/?q=ai:strugarek.martin"Taing, Cécile"https://zbmath.org/authors/?q=ai:taing.cecileThis work mathematically studies some selection-mutation models of nonlocal type, describing the distribution of a sexual population structured by a phenotypical trait. The novelty consists in considering an asymmetry in the trait heredity or in the fecundity between the parents, situations that occur, for example, in certain mosquitoes species.
The main results are related to the asymptotic behavior of the population distribution. In particular, some non-extinction conditions and BV estimates on the total population are provided. Moreover, concentration phenomena are considered: some general estimates are given for the Hamilton-Jacobi equations that arise from this study, and concentration is obtained in some special situations.
Reviewer: Andrea Tellini (Madrid)Fractional truncated Laplacians: representation formula, fundamental solutions and applicationshttps://zbmath.org/1496.354202022-11-17T18:59:28.764376Z"Birindelli, Isabeau"https://zbmath.org/authors/?q=ai:birindelli.isabeau"Galise, Giulio"https://zbmath.org/authors/?q=ai:galise.giulio"Topp, Erwin"https://zbmath.org/authors/?q=ai:topp.erwinSummary: We introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called ``fundamental solutions''. This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to ``understand'' what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differ significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian.On fractional Schrödinger equations with Hartree type nonlinearitieshttps://zbmath.org/1496.354222022-11-17T18:59:28.764376Z"Cingolani, Silvia"https://zbmath.org/authors/?q=ai:cingolani.silvia"Gallo, Marco"https://zbmath.org/authors/?q=ai:gallo.marco"Tanaka, Kazunaga"https://zbmath.org/authors/?q=ai:tanaka.kazunagaSummary: Goal of this paper is to study the following doubly nonlocal equation
\[
(- \Delta)^s u + \mu u = (I_\alpha*F(u))F^\prime(u) \quad\text{in } \mathbb{R}^N \eqno{(\text{P})}
\]
in the case of general nonlinearities \(F \in C^1(\mathbb{R})\) of Berestycki-Lions type, when \(N \geq 2\) and \(\mu > 0\) is fixed. Here \((-\Delta)^s\), \(s \in (0, 1) \), denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential \(I_{\alpha}\), \(\alpha \in (0, N) \). We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [\textit{P. d'Avenia} et al., Math. Models Methods Appl. Sci. 25, No. 8, 1447--1476 (2015; Zbl 1323.35205); \textit{V. Moroz} and \textit{J. van Schaftingen}, Trans. Am. Math. Soc. 367, No. 9, 6557--6579 (2015; Zbl 1325.35052)].Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equationhttps://zbmath.org/1496.390162022-11-17T18:59:28.764376Z"Thuc, Nguyen Dat"https://zbmath.org/authors/?q=ai:thuc.nguyen-dat"Ngoc, Le Thi Phuong"https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Long, Nguyen Thanh"https://zbmath.org/authors/?q=ai:nguyen-thanh-long.Summary: This paper is devoted to the study of the following nonlinear functional integral equation
\[
f(x)=\sum\limits_{i=1}^q\alpha_i(x)f(\tau_i(x)) + \int_0^{\sigma_1(x)}\Psi\left(x, t, f(\sigma_2(t)), \int_0^{\sigma_3(t)}f(s)ds\right) dt + g(x),\;\forall x\in [0,1], \tag{E}
\]
where \(\tau_i, \sigma_1, \sigma_2, \sigma_3 :[0,1]\rightarrow [0,1]\); \(\alpha_i, g: [0,1]\rightarrow \mathbb{R}\); \(\Psi: [0,1]\times [0,1]\times\mathbb{R}^2\rightarrow \mathbb{R}\) are the given continuous functions and \(f:[0,1]\,\rightarrow\mathbb{R}\) is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of \(\Psi\in C^2([0, 1]\times [0,1]\times \mathbb{R}^2; \mathbb{R})\), we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.Multi-dimensional \(c\)-almost periodic type functions and applicationshttps://zbmath.org/1496.420092022-11-17T18:59:28.764376Z"Kostic, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this article, we analyze multi-dimensional Bohr \((\mathcal{B}, c)\)-almost periodic type functions. The main structural characterizations for the introduced classes of Bohr \((\mathcal{B}, c)\)-almost periodic type functions are established. Several applications of our abstract theoretical results to the abstract Volterra integro-differential equations in Banach spaces are provided, as well.Integral resolvent for Volterra equations and Favard spaceshttps://zbmath.org/1496.450012022-11-17T18:59:28.764376Z"Fadili, A."https://zbmath.org/authors/?q=ai:fadili.ahmed"Maragh, F."https://zbmath.org/authors/?q=ai:maragh.fouadThe authors study the Volterra integral equation
\[
x(t)=x_0 + \int_0^t a(t-s) Ax(s)\, ds,\quad t\geq 0,
\]
in a Banach space \(X\), where \(a\in L^1_{\text{loc}}(\mathbb R^+)\) and \(A\) is a densely define closed operator in \(X\). The integral resolvent associated with this equation is a strongly continuous family \(R(t)\) of bounded operators such that \(R(t)\) commutes with \(A\) and
\[
R(t)x=a(t)x+ \int_0^t a(t-s)AR(s)x\, ds,
\]
for all \(x\in D(A)\).
The crucial assumption used by the authors is that there exists \(\varepsilon_a>0\) and \(t_a>0\) such that for all \(0< t\leq t_a\) one has
\[
\left | \int_0^t a(t-s)a(s)\, ds\right | \geq \varepsilon_a \int_0^t |a(s)|\, ds. \]
They show that
\[
D(A)= \left \{x\in X\,:\, \lim_{t\to 0+} \frac {R(t)x-a(t)x}{(a*a)(t)}\text{ exists}\right \},
\]
and the limit is \(Ax\).
Furthermore, they show that if in addition the integral resolvent is bounded and \(\int_0^\infty \text{e}^{-\omega t} |a(t)|\, dt <\infty\) for some \(\omega >0\) then the following result on the (frequency and temporal) Favard spaces associated with \((A,a)\) holds:
\[
\left \{x\in X\,:\,\sup_{\lambda > \omega}\left \| \frac 1{\hat a(\lambda)} A\left (\frac 1{\hat a(\lambda)}I-A\right)x\right\|<\infty\right \} \]
\[
= \left \{x\in X\,:\,\sup_{0<t\leq 1}\frac{\|R(t)x-a(t)x\|}{ |(a*a)(t)|}<\infty\right \}.
\]
Reviewer: Gustaf Gripenberg (Aalto)Integral boundary-value problem with initial jumps for a singularly perturbed system of integrodifferential equationshttps://zbmath.org/1496.450022022-11-17T18:59:28.764376Z"Dauylbayev, M. K."https://zbmath.org/authors/?q=ai:dauylbayev.muratkhan-k|dauylbaev.m-k"Uaissov, B."https://zbmath.org/authors/?q=ai:uaissov.bSummary: In this work, we study an integral boundary-value problem for a singularly perturbed linear system of integrodifferential equations, which has the phenomena of initial jumps. An analytical formula and asymptotic estimations of the solution and its derivatives are obtained. It is established that the solution of the boundary-value problem at the left point of the segment has the phenomenon of an initial jump of the zero order. The convergence of a singularly perturbed integral boundary-value problem to the solution of a modified degenerate problem containing the initial jumps of the solution and integral terms is proved.A fixed point theorem using condensing operators and its applications to Erdélyi-Kober bivariate fractional integral equationshttps://zbmath.org/1496.450032022-11-17T18:59:28.764376Z"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Panda, Sumati Kumari"https://zbmath.org/authors/?q=ai:panda.sumati-kumariSummary: The primary aim of this article is to discuss and prove fixed point results using the operator type condensing map, and to obtain the existence of solution of Erdélyi-Kober bivariate fractional integral equation in a Banach space. An instance is given to explain the results obtained, and we construct an iterative algorithm by sinc interpolation to find an approximate solution of the problem with acceptable accuracy.On Hyers-Ulam-Rassias stability of a Volterra-Hammerstein functional integral equationhttps://zbmath.org/1496.450042022-11-17T18:59:28.764376Z"Ciplea, Sorina Anamaria"https://zbmath.org/authors/?q=ai:ciplea.sorina-anamaria"Lungu, Nicolaie"https://zbmath.org/authors/?q=ai:lungu.nicolaie"Marian, Daniela"https://zbmath.org/authors/?q=ai:marian.daniela"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: The aim of this paper is to study the Hyers-Ulam-Rassias stability for a Volterra-Hammerstein functional integral equation in three variables via Picard operators.
For the entire collection see [Zbl 1485.65002].Iterative algorithm and theoretical treatment of existence of solution for \((k, z)\)-Riemann-Liouville fractional integral equationshttps://zbmath.org/1496.450052022-11-17T18:59:28.764376Z"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Mohiuddine, S. A."https://zbmath.org/authors/?q=ai:mohiuddine.syed-abdul"Deuri, Bhuban Chandra"https://zbmath.org/authors/?q=ai:deuri.bhuban-chandraAfter an introduction to fractional integral equations involving Riemann-Liouville fractional integrals the authors establish a new Darbo-type fixed point theorem. This allows them to discuss the existence of solutions for certain fractional integral equations. The last part of this work is devoted to the construction of a convergent iterative algorithm based on the modified homotopy perturbation method to find the solutions of given fractional integral equations.
Reviewer: Yogesh Sharma (Sardarpura)Global attractivity, asymptotic stability and blow-up points for nonlinear functional-integral equations' solutions and applications in Banach space \(BC( R_+)\) with computational complexityhttps://zbmath.org/1496.450062022-11-17T18:59:28.764376Z"Karaca, Yeliz"https://zbmath.org/authors/?q=ai:karaca.yelizQualitative analysis of integro-differential equations with variable retardationhttps://zbmath.org/1496.450072022-11-17T18:59:28.764376Z"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-j"Tunç, Osman"https://zbmath.org/authors/?q=ai:tunc.osmanThe paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs), which reads as
\[
\frac{\mathrm{d} x}{\mathrm{~d} t}=A(t) x+B F(x(t-\tau(t)))+\int_{t-\tau(t)}^{t} \Omega(t, s) F(x(s)) \mathrm{d} s+G(t, x),
\]
where \(x \in \mathbb{R}^{n},\ t \in \mathbb{R}^{+}:=[0, \infty), \ \tau \in \mathrm{C}^{1}\left(\mathbb{R}^{+}, \ \mathbb{R}^{+}\right), \ A=\left(a_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+},\ \mathbb{R}^{n \times n}\right)\), \(\Omega=\left(\Omega_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \ \mathbb{R}^{n \times n}\right)\), \(B=\left(b_{i j}\right) \in \mathbb{R}^{n \times n}, \ F \in \mathrm{C}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right), \ F(0)=0\), and \(G\in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \mathbb{R}^{n}\right)\). The authors focus on the uniform
stability and uniform asymptotic stability of the zero solution and integrability and boundedness of solutions in the case \(G(x,t)=0\). The boundedness of solutions at infinity is discussed for \(G(x,t)\neq 0\) too. Also, the authors provide two illustrative examples. Remarkably the given theorems are not only applicable to time-varying linear RIDEs, but also to nonlinear RIDEs depending on time.
Reviewer: Gaston Vergara-Hermosilla (Dublin)Existence results of fractional neutral integrodifferential equations with deviating argumentshttps://zbmath.org/1496.450082022-11-17T18:59:28.764376Z"Kamalapriya, B."https://zbmath.org/authors/?q=ai:kamalapriya.b"Balachandran, K."https://zbmath.org/authors/?q=ai:balachandran.krishnan"Annapoorani, N."https://zbmath.org/authors/?q=ai:annapoorani.natarajanSummary: In this paper we prove the existence of solutions of fractional neutral integrodifferential equations with deviating arguments by using the resolvent operators and fixed point theorem. Examples are discussed to illustrate the theory.Fractional integro-differential equation with a weakly singular kernel by using block pulse functionshttps://zbmath.org/1496.450092022-11-17T18:59:28.764376Z"Mohammadi, Fakhrodin"https://zbmath.org/authors/?q=ai:mohammadi.fakhrodinSummary: In this paper, a numerical method based on block pulse functions (BPFs) is proposed for fractional integro-differential equations with a weakly singular kernel. The BPFs expansion and its fractional operational matrix along with collocation method are utilized to reduce fractional integro-differential equations with weakly singular kernel into a system of algebraic equations. The error estimate and convergence analysis of the proposed method is investigated. In order to show the effectiveness and accuracy of the proposed method, it is applied to some benchmark problems. The numerical results are compared with other methods existing in the recent literature.On one class of multidimensional integral equations of convolution type with convex nonlinearityhttps://zbmath.org/1496.450102022-11-17T18:59:28.764376Z"Khachatryan, Kh. A."https://zbmath.org/authors/?q=ai:khachatryan.khachatur-a"Petrosyan, H. S."https://zbmath.org/authors/?q=ai:petrosyan.aikanush-samvelovnaSummary: We study a class of multidimensional convolution-type integral equations with monotone and convex nonlinearity. This class of equations has applications in the theory of \(p \)-adic open-closed strings and in the mathematical theory of spatiotemporal propagation of a pandemic. A theorem on the existence of a nonnegative, nontrivial, bounded, and continuous solution is proved. The integral asymptotics of the solution is established. A uniqueness theorem is proved for a special class of nonnegative bounded functions, and applied examples of these equations are given.On the properties of a semigroup of operators generated by a Volterra integro-differential equation arising in the theory of viscoelasticityhttps://zbmath.org/1496.450112022-11-17T18:59:28.764376Z"Tikhonov, Yu. A."https://zbmath.org/authors/?q=ai:tikhonov.yu-aSummary: Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane.Study of Hyers-Ulam stability for a class of multi-singular fractional integro-differential equation with boundary conditionshttps://zbmath.org/1496.450122022-11-17T18:59:28.764376Z"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alireza"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: One of the considerable strategies for the investigation of integro-differential equation is stability. The notion of this strategy shows us we can rest assured of the numerical results obtained from the computer software. Since there are usually large errors in the numerical results of singular differential equations, this strategy will help us to be able to examine singular equations more easily with computer software. In this work, we study the stability of a multi-singular fractional boundary value problem in the sense of Hyers-Ulam stability. We also present three examples and three figures to illustrate our main result.A new aspect of generalized integral operator and an estimation in a generalized function theoryhttps://zbmath.org/1496.450132022-11-17T18:59:28.764376Z"Al-Omari, Shrideh"https://zbmath.org/authors/?q=ai:al-omari.shrideh-khalaf-qasem"Almusawa, Hassan"https://zbmath.org/authors/?q=ai:almusawa.hassan"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this paper we investigate certain integral operator involving Jacobi-Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi-Dunkl integral operator [\textit{N. B. Salem} and \textit{A. O. A. Salem}, Ramanujan J. 12, No. 3, 359--378 (2006; Zbl 1122.44002)] is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi-Dunkl operator are discussed. An inversion formula and certain convergence with respect to \(\delta\) and \(\Delta\) convergences are also introduced.On a class of operator equations in locally convex spaceshttps://zbmath.org/1496.470282022-11-17T18:59:28.764376Z"Mishin, Sergeĭ N."https://zbmath.org/authors/?q=ai:mishin.sergey-nSummary: We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.Multiplicative operator functions and abstract Cauchy problemshttps://zbmath.org/1496.470692022-11-17T18:59:28.764376Z"Früchtl, Felix"https://zbmath.org/authors/?q=ai:fruchtl.felixSummary: We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including \(C_{0}\)-groups and cosine operator functions, and more generally, Sturm-Liouville operator functions.On generalized \((\alpha,\beta)\)-nonexpansive mappings in Banach spaces with applicationshttps://zbmath.org/1496.470792022-11-17T18:59:28.764376Z"Akutsah, F."https://zbmath.org/authors/?q=ai:akutsah.francis"Narain, O. K."https://zbmath.org/authors/?q=ai:narain.ojen-kumarSummary: In this paper, we present some fixed point results for a general class of nonexpansive mappings in the framework of Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Furthermore, we establish some basic properties and convergence results for our new class of mappings in uniformly convex Banach spaces. Finally, we present an application to nonlinear integral equation and also, a numerical example to illustrate our main result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature with different choices of parameters and initial guesses. The results obtained in this paper improve, extend and unify some related results in the literature.Approximation of fixed points and the solution of a nonlinear integral equationhttps://zbmath.org/1496.471112022-11-17T18:59:28.764376Z"Ali, Faeem"https://zbmath.org/authors/?q=ai:ali.faeem"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Rodríguez-López, Rosana"https://zbmath.org/authors/?q=ai:rodriguez-lopez.rosanaSummary: In this article, we define Picard's three-step iteration process for the approximation of fixed points of Zamfirescu operators in an arbitrary Banach space. We prove a convergence result for Zamfirescu operator using the proposed iteration process. Further, we prove that Picard's three-step iteration process is almost \(T\)-stable and converges faster than all the known and leading iteration processes. To support our results, we furnish an illustrative numerical example. Finally, we apply the proposed iteration process to approximate the solution of a mixed Volterra-Fredholm functional nonlinear integral equation.Fixed point theorems in \(b\)-multiplicative metric spaceshttps://zbmath.org/1496.540262022-11-17T18:59:28.764376Z"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Kurdi, Alia"https://zbmath.org/authors/?q=ai:kurdi.aliaSummary: In this paper, we introduce the new notion of \(b\)-multiplicative metric space. We prove fixed point theorems for single and multivalued mappings on \(b\)-multiplicative metric spaces, endowed with a graph. We construct examples to illustrate our notions and results. As illustrative application, we give an existence theorem for the solution of a class of Fredholm multiplicative integral equations.A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra typehttps://zbmath.org/1496.651282022-11-17T18:59:28.764376Z"Santra, S."https://zbmath.org/authors/?q=ai:santra.somnath|santra.sanchayan|santra.sutapa|santra.sudarshan|santra.sitangshu-bikas|santra.sourav|santra.santanu|santra.srimanta|santra.sitangsu-bikas|santra.siddhartha|santra.shyam-sundar|santra.sanjiban"Mohapatra, J."https://zbmath.org/authors/?q=ai:mohapatra.jugal|mohapatra.jeetSummary: The main purpose of this work is to study the numerical solution of a time fractional partial integro-differential equation of Volterra type, where the time derivative is defined in Caputo sense. Our method is a combination of the classical L1 scheme for temporal derivative, the general second order central difference approximation for spatial derivative and the repeated quadrature rule for integral part. The error analysis is carried out and it is shown that the approximate solution converges to the exact solution. Several examples are given in support of the theoretical findings. In addition, we have shown that the order of convergence is more high on any subdomain away from the origin compared to the entire domain.A two-grid method for finite element solution of parabolic integro-differential equationshttps://zbmath.org/1496.651722022-11-17T18:59:28.764376Z"Wang, Keyan"https://zbmath.org/authors/?q=ai:wang.keyanSummary: In this paper, we study the unconditional convergence and error estimates of a two-grid finite element method for the semilinear parabolic integro-differential equations. By using a temporal-spatial error splitting technique, optimal \(L^p\) and \(H^1\) error estimates of the finite element method can be obtained. Moreover, to deal with the semilinearity of the model, we use the two-grid method. Optimal error estimates in \(L^2\) and \(H^1\)-norms of two-grid solution are derived without any time-step size conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.Effective numerical technique for solving variable order integro-differential equationshttps://zbmath.org/1496.651772022-11-17T18:59:28.764376Z"El-Gindy, Taha M."https://zbmath.org/authors/?q=ai:el-gindy.taha-m"Ahmed, Hoda F."https://zbmath.org/authors/?q=ai:ahmed.hoda-f"Melad, Marina B."https://zbmath.org/authors/?q=ai:melad.marina-bSummary: In this article, an effective numerical technique for solving the variable order Fredholm-Volterra integro-differential equations (VO-FV-IDEs), systems of VO-FV-IDEs and variable order Volterra partial integro-differential equations (VO-V-PIDEs) is given. The suggested technique is built on the combination of the spectral collocation method with some types of operational matrices of the variable order fractional differentiation and integration of the shifted fractional Gegenbauer polynomials (SFGPs). The proposed technique reduces the considered problems to systems of algebraic equations that are straightforward to solve. The error bound estimation of using SFGPs is discussed. Finally, the suggested technique's authenticity and efficacy are tested via presenting several numerical applications. Comparisons between the outcomes achieved by implementing the proposed method with other numerical methods in the existing literature are held, the obtained numerical results of these applications reveal the high precision and performance of the proposed method.A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equationhttps://zbmath.org/1496.651842022-11-17T18:59:28.764376Z"Qiao, Leijie"https://zbmath.org/authors/?q=ai:qiao.leijie"Xu, Da"https://zbmath.org/authors/?q=ai:xu.daSummary: We propose and analyze a time-stepping Crank-Nicolson (CN) alternating direction implicit (ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous mediahttps://zbmath.org/1496.652292022-11-17T18:59:28.764376Z"Domínguez, V."https://zbmath.org/authors/?q=ai:dominguez.victor"Ganesh, M."https://zbmath.org/authors/?q=ai:ganesh.mahadevanThe authors are concerned with an adaptive coupling FEM-BEM in order to solve a Helmholtz acoustic/electromagnetic 2D wave propagation problem with a bounded heterogeneous region. First they present the Helmholtz model and an equivalent decomposition formulation. Then they accomplish a numerical analysis of the FEM-BEM algorithm establishing optimal order convergence of the hybridized numerical solution. On the bounded part of the domain they approximate the solution by a FEM with classical continuous piecewise polynomials on triangular meshes. On the other hand a high-order Nyström BEM is used to compute the scattered wave in the unbounded part of the domain. The solutions are coupled by requiring the coinciding in the two artificial boundaries that ensures the matching of FEM and BEM solutions in the common region of the partition of domain. Three distinct sets of experiments are carried out, the most challenging being the algorithm for multiple-particle Janus-type configurations with non-smooth solutions.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)A vector algorithm for computing the maximal eigenvalue of an integral operatorhttps://zbmath.org/1496.652372022-11-17T18:59:28.764376Z"Rasulov, A. S."https://zbmath.org/authors/?q=ai:rasulov.a-s"Rasulov, S. I."https://zbmath.org/authors/?q=ai:rasulov.s-i"Narmanov, A. Zh."https://zbmath.org/authors/?q=ai:narmanov.a-zh(no abstract)Chebyshev spectral projection methods for Fredholm integral equations of the second kindhttps://zbmath.org/1496.652382022-11-17T18:59:28.764376Z"Laxmi Panigrahi, Bijaya"https://zbmath.org/authors/?q=ai:panigrahi.bijaya-laxmi"Kumar Malik, Jitendra"https://zbmath.org/authors/?q=ai:malik.jitendra-kumarSummary: In this paper, we will propose the Chebyshev spectral Galerkin and collocation methods for the Fredholm integral equations (FIEs) of the second kind with smooth kernel and its associated eigenvalue problem (EVPs). The convergence rates of approximated solutions, iterated solutions with exact solution in \(L^2_\omega\) norm have been investigated. We will evaluate the errors between exact eigen-elements and approximated eigen-elements both in \(L^2_\omega\) and \(L^\infty_\omega\) norms. We will show that eigenvalues and iterated eigenvectors have super-convergence rate in Chebyshev spectral Galerkin methods.
For the entire collection see [Zbl 1491.65006].Application of the Bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with Caputo fractional derivativeshttps://zbmath.org/1496.652392022-11-17T18:59:28.764376Z"Moussai, Miloud"https://zbmath.org/authors/?q=ai:moussai.miloudSummary: The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation
\[
\sum\limits_{k = 1}^mF_k(x)D^{(k\alpha)}y(x)+\lambda \int_0^xK(x, t)D^{(\alpha)}y(t)dt = g(x)y^2(x)+h(x)y(x)+P(x).
\]
In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.Approximate solution of Fredholm integral and integro-differential equations with non-separable kernelshttps://zbmath.org/1496.652402022-11-17T18:59:28.764376Z"Providas, E."https://zbmath.org/authors/?q=ai:providas.efthinios|providas.efthimiosSummary: This chapter deals with the approximate solution of Fredholm integral equations and a type of integro-differential equations having non-separable kernels, as they appear in many applications. The procedure proposed consists of firstly approximating the non-separable kernel by a finite partial sum of a power series and then constructing the solution of the degenerate equation explicitly by a direct matrix method. The method, which is easily programmable in a computer algebra system, is explained and tested by solving several examples from the literature.
For the entire collection see [Zbl 1485.65002].Bernstein operator method for approximate solution of singularly perturbed Volterra integral equationshttps://zbmath.org/1496.652412022-11-17T18:59:28.764376Z"Usta, Fuat"https://zbmath.org/authors/?q=ai:usta.fuat"Akyiğit, Mahmut"https://zbmath.org/authors/?q=ai:akyigit.mahmut"Say, Fatih"https://zbmath.org/authors/?q=ai:say.fatih"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamalSummary: An approximate solution of integral equations takes an active role in the numerical analysis. This paper presents and tests an algorithm for the approximate solution of singularly perturbed Volterra integral equations via the Bernstein approximation technique. The method of computing the numerical approximation of the solution is properly demonstrated and exemplified in the matrix notation. Besides, the error bound and convergence associated with the numerical scheme are constituted. Finally, particular examples indicate the dependability and numerical capability of the introduced scheme in comparison with other numerical techniques.The Van Vleck formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phaseshttps://zbmath.org/1496.810592022-11-17T18:59:28.764376Z"Blair, Matthew D."https://zbmath.org/authors/?q=ai:blair.matthew-dSummary: The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on \textit{Ehrenfest time scales}, i.e. \(\hbar\)-dependent time intervals which most commonly take the form \(|t| \le c|\log \hbar|\). Our derivation is based on an approximation to the integral kernel often called the \textit{Herman-Kluk approximation}, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by \textit{D. Robert} [Rev. Math. Phys. 22, No. 10, 1123--1145 (2010; Zbl 1206.35212)], this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.Ideal free dispersal in integrodifference modelshttps://zbmath.org/1496.921272022-11-17T18:59:28.764376Z"Cantrell, Robert Stephen"https://zbmath.org/authors/?q=ai:cantrell.robert-stephen"Cosner, Chris"https://zbmath.org/authors/?q=ai:cosner.chris"Zhou, Ying"https://zbmath.org/authors/?q=ai:zhou.yingSummary: In this paper, we use an integrodifference equation model and pairwise invasion analysis to find what dispersal strategies are evolutionarily stable strategies (also known as evolutionarily steady or ESS) when there is spatial heterogeneity and possibly seasonal variation in habitat suitability. In that case there are both advantages and disadvantages of dispersing. We begin with the case where all spatial locations can support a viable population, and then consider the case where there are non-viable regions in the habitat. If the viable regions vary seasonally, and the viable regions in summer and winter do not overlap, dispersal may really be necessary for sustaining a population. Our findings generally align with previous findings in the literature that were based on other modeling frameworks, namely that dispersal strategies associated with ideal free distributions are evolutionarily stable. In the case where only part of the habitat can sustain a population, we show that a partial occupation ideal free distribution that occupies only the viable region is associated with a dispersal strategy that is evolutionarily stable. As in some previous works, the proofs of these results make use of properties of line sum symmetric functions, which are analogous to those of line sum symmetric matrices but applied to integral operators.