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

Maleknejad, K.; Rashidinia, Jalil; Eftekhari, Tahereh

doi:10.1016/j.amc.2018.07.021

Cited by 24 publications

(14 citation statements)

References 16 publications

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“…Therefore, finding efficient numerical methods to approximate the solutions of these equations has become the main objective of many mathematicians. 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][42][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. In this research study, some classes of two-dimensional nonlinear fractional integral equations of the second kind are considered in the following forms:…”

Section: Introductionmentioning

confidence: 99%

A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations

Maleknejad

Rashidinia

Eftekhari

2021

Numerical Methods Partial

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The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two-dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two-variable shifted fractional-order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional-order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently.

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Section: Introductionmentioning

confidence: 99%

A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations

Maleknejad

Rashidinia

Eftekhari

2021

Numerical Methods Partial

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“…Maleknejad et.al. in 20 used the Bernstein polynomials of three variable with all their properties to establish the approximate solution of the 3D Fredholm-Volterra operators of the first and second kind.…”

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Hameed

Al-Saedi

2021

Baghdad Sci.J

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In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R 3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

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“…Therefore, the development of effective and easy‐to‐use numerical schemes for solving such equations acquires an increasing interest. While several numerical techniques have been proposed to solve many different problems (see, for instance [22–49], and references therein), there have been few research studies that developed numerical methods to solve DOFDEs (see [50–58]). The development, however, for efficient numerical methods to solve DOFDEs is still an important issue [51].…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

Maleknejad

Rashidinia

Eftekhari

2020

Numerical Methods Partial

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In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz-Legendre wavelet approximation. We derive a new operational vector for the Riemann-Liouville fractional integral of the Müntz-Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well-known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.

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

Cited by 24 publications

References 16 publications

A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations

A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

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