Recent zbMATH articles in MSC 45B05 https://zbmath.org/atom/cc/45B05 2021-05-28T16:06:00+00:00 Werkzeug A study of error estimation for second order Fredholm integro-differential equations. https://zbmath.org/1459.65244 2021-05-28T16:06:00+00:00 "Parvaz, R." https://zbmath.org/authors/?q=ai:parvaz.reza "Zarebnia, M." https://zbmath.org/authors/?q=ai:zarebnia.mohammad "Bagherzadeh, A. Saboor" https://zbmath.org/authors/?q=ai:saboor-bagherzadeh.a|bagherzadeh.amir-saboor Summary: In this work, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of second order Fredholm integro-differential equations. We use the defect correction principle to find the deviation of the error estimation and show that collocation method by using $$m$$ degree piecewise polynomial provides order $$\mathcal{O}(h^{m+2})$$ for the deviation of the error. Also, the theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical analysis. On the solution of quadratic nonlinear integral equation with different singular kernels. https://zbmath.org/1459.65240 2021-05-28T16:06:00+00:00 "Basseem, M." https://zbmath.org/authors/?q=ai:basseem.m "Alalyani, Ahmad" https://zbmath.org/authors/?q=ai:alalyani.ahmad Summary: All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton-Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software. A highly efficient and accurate finite iterative method for solving linear two-dimensional Fredholm fuzzy integral equations of the second kind using triangular functions. https://zbmath.org/1459.65245 2021-05-28T16:06:00+00:00 "Ramadan, Mohamed A." https://zbmath.org/authors/?q=ai:ramadan.mohamed-abdel-latif "Osheba, Heba S." https://zbmath.org/authors/?q=ai:osheba.heba-s "Hadhoud, Adel R." https://zbmath.org/authors/?q=ai:hadhoud.adel-rashad Summary: This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two $$m$$-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions. Iterative solution for systems of a class of abstract operator equations in Banach spaces and application. https://zbmath.org/1459.65076 2021-05-28T16:06:00+00:00 "Su, Hua" https://zbmath.org/authors/?q=ai:su.hua Summary: In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained. A unified spectral collocation method for nonlinear systems of multi-dimensional integral equations with convergence analysis. https://zbmath.org/1459.65250 2021-05-28T16:06:00+00:00 "Zaky, Mahmoud A." https://zbmath.org/authors/?q=ai:zaky.mahmoud-a "Ameen, Ibrahem G." https://zbmath.org/authors/?q=ai:ameen.ibrahem-g "Elkot, Nermeen A." https://zbmath.org/authors/?q=ai:elkot.nermeen-a "Doha, Eid H." https://zbmath.org/authors/?q=ai:doha.eid-h Summary: The main purpose of this paper is to construct and analyze a spectral collocation method for solving a general class of nonlinear systems of multi-dimensional integral equations. In order to obtain high-order accuracy for the approximation, the integral terms in the resulting equation are approximated by using the Legendre spectral quadrature rule. The spectral rate of convergence for the proposed method is established in the $$L^2$$-norm showing that the error of approximate solution decays exponentially. Numerical examples are presented to confirm this theoretical prediction.