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Journal ArticleDOI

Numerical solution of three-dimensional Volterra–Fredholm integral equations of the first and second kinds based on Bernstein’s approximation

TL;DR: A new and efficient method for solving three-dimensional Volterra–Fredholm integral equations of the second kind, first kind and even singular type and three-variable Bernstein polynomials and their properties is presented.
About: This article is published in Applied Mathematics and Computation.The article was published on 2018-12-15. It has received 27 citations till now. The article focuses on the topics: Bernstein polynomial & Integral equation.
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TL;DR: In this article, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs).
Abstract: Finding the optimal parameters and functions of iterative methods is among the main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied and instead of usual mathematical softwares the CADNA (Control of Accuracy and Debugging for Numerical Applications) library is used. Also, the convergence theorem of presented method is illustrated. In order to apply the CESTAC method we will prove a theorem that it will be our licence to use the new termination criterion instead of traditional absolute error. By using this theorem we can show that number of common significant digits (NCSDs) between two successive approximations are almost equal to NCSDs between exact and numerical solution. Finally, some examples are solved by using the Taylor-collocation method based on the CESTAC method. Several tables of numerical solutions based on the both arithmetics are presented. Comparison between number of iterations are demonstrated by using the floating point arithmetic (FPA) for different values of $\varepsilon$.

26 citations

Journal ArticleDOI

19 citations


Cites methods from "Numerical solution of three-dimensi..."

  • ...Some of these methods include Legendre wavelets [17], higher-order finite element method [18], generalized differential transform method [27], shifted Legendre polynomials [16, 21, 25], hybrid of block-pulse functions and shifted Legendre polynomials operational matrix method [31], Müntz–Legendre wavelets [32], fractional-order orthogonal Bernstein polynomials [38], delta functions operational matrix method [39], hybrid of block-pulse and parabolic functions [37], hat functions [35, 40], two-dimensional orthonormal Bernstein polynomials [41–43], two-dimensional block-pulse operational matrix method [44], homotopy analysis method [47], Haar wavelet [4, 49], orthonormal Bernoulli polynomials [52], shifted Jacobi polynomials [20, 54, 56], Bernstein polynomials [30, 55], the second kind Chebyshev wavelets [51], etc....

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Journal ArticleDOI
TL;DR: In this article, an efficient numerical method is presented to approximate solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations using shifted Jacobi polynomials.

16 citations

Journal ArticleDOI
TL;DR: The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff's fixed-point theorems.
Abstract: The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff’s fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use operational matrices of hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials via collocation method to find approximate solutions of the mentioned equations. In addition, a discussion on error bound and convergence analysis of the proposed method is presented. Finally, the accuracy and efficiency of the presented method are confirmed by solving three illustrative examples and comparing the results of the proposed method with other existing numerical methods in the literature.

16 citations


Cites background from "Numerical solution of three-dimensi..."

  • ...…Shahbazi 2018; Hassani et al. 2019a, b; Jabari Sabeg et al. 2017; Kılıçman and Al Zhour 2007; Li and Shah 2017; Mohammadi Rick and Rashidinia 2019; Maleknejad et al. 2018, 2020a, b, c; Mashoof and Refahi Shekhani 2017; Mirzaee and Samadyar 2019; Najafalizadeh and Ezzati 2016; Nouri et al. 2018;…...

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References
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17 Nov 1999
TL;DR: In this article, the authors introduce the Finite Difference Method (DFM) and Galerkin Methods to solve the problem of high-frequency problems with Finite Elements, and the solution of the problem is given.
Abstract: 1. Basic Principles of Electromagnetic Fields 2. Overview of Computational Methods in Electromagnetics 3. The Finite Difference Method 4. Variational and Galerkin Methods 5. Shape Functions 6. The Finite Element Method 7. Integral Equations 8. Open Boundary Problems 9. High- Frequency Problems with Finite Elements 10. Low-Frequency Applications 11. Solution of Equations A. Vector Operators B. Triangle Area in Terms of Vertex Coordinates C. Fourier Transform Mehtod D. Integrals of Area Coordinates E. Integrals of Voluume Coordinates F. Gauss-Legendre Quadrature Formulae, Abscissae and, Weight Coefficients G. Shape Functions for 1D Finite Elements H. Shape Functions for 2D Finite Elements I. Shape Functions for 3D Finite Elements References Index

216 citations

Journal ArticleDOI
TL;DR: In this paper, the authors presented a method to solve nonlinear Volterra-Fredholm-Hammerstein integral equations in terms of Bernstein polynomials and operational matrix of integration together with the product operational matrix.

120 citations

Journal ArticleDOI
TL;DR: This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis and the convergence of the method is established rigorously for each class of integral equation considered here.

119 citations

Journal ArticleDOI
TL;DR: A numerical method for solving Volterra integral equations of the second kind, first kind and even singular type of these equations using simple computation with quite acceptable approximate solution is presented.

111 citations

Journal ArticleDOI
TL;DR: A new approach implementing a collocation method in combination with operational matrices of Bernstein polynomials for the numerical solution of VFIDEs is introduced, which reduces such problems to ones of solving systems of algebraic equations.

81 citations