Numerical solution of Volterra–Fredholm integral equations using Legendre collocation method

Nemati, Somayeh

doi:10.1016/j.cam.2014.09.030

Cited by 54 publications

(33 citation statements)

References 8 publications

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“…In this section, the numerical results of proposed scheme on some test problems will be presented and we compare them with the results by four other efficient methods. These four methods are LECM [24], TCM [25], TPM [26] and LACM [27]. In these examples one can observe that the error of presented method is less than the errors obtained by other methods.…”

Section: Numerical Experimentsmentioning

confidence: 85%

“…When h(x) is a first-order polynomial, Equation (1) is a functional integral equation with the proportional delay. Some numerical methods such as the Legendre collocation method [24], Taylor collocation method [25], Taylor polynomial method [26] and Lagrange collocation method [27] have been applied for solving Equation (1). In this work, we will present the shifted orthonormal Bernstein polynomials method to approximate the solution of Equation (1).…”

Section: Introductionmentioning

confidence: 99%

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A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

Hesameddini

Shahbazi

2018

Journal of Taibah University for Science

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This paper deals with the numerical solution of Volterra-Fredholm integral equations. In this work, we approximate the unknown functions based on the shifted orthonormal Bernstein polynomials, in conjunction with the least-squares approximation method. The method is using a simple computational manner to obtain a quite acceptable approximate solution. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of the method for solving these equations are high. The convergence analysis of proposed method have been discussed through some theorems. Moreover, we will obtain an estimation of error bound for this algorithm. Finally, some examples are given to show the capability of presented method in comparison with four well-known algorithms in the literature namely the Legendre collocation method, Taylor collocation method, Taylor polynomial method and Lagrange collocation method. ARTICLE HISTORY

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Section: Numerical Experimentsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

Hesameddini

Shahbazi

2018

Journal of Taibah University for Science

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“…() Table 3 Maximum absolute errors of problem (67) N = M = K α 1 = β 1 = α 2 = β 2 = α 3 = β 3 = 0 α 1 = β 1 = α 2 = β 2 = 1 2 , α 3 = β 3 = - 6 3 . 0 5 × 10 -6 3.04 × 10 -6 5.52 × 10 -6 5.61 × 10 -6 8 3 .…”

Section: Numerical Simulation and Comparisonsmentioning

confidence: 99%

New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

Bhrawy¹,

Mallawi²,

Abdelkawy³

2016

Adv Differ Equ

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A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms.

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“…As Khader proposed a Chebyshev collocation method for a space‐fractional diffusion equation, Saadatmandi et al proposed and developed an efficient numerical algorithm based on the Sinc‐Legendre collocation method for the time‐fractional convection‐diffusion equation with variable coefficients on a finite domain. The reader may refer to the literature …”

Section: Introductionmentioning

confidence: 99%

“…The reader may refer to the literature. [16][17][18][19][20][21][22][23][24][25][26][27] The principal aim of this paper is to construct an efficient spectral collocation algorithm for solving NSFPDEs subject to different types of conditions. To this end, we give a numerical algorithm based on collocation by using generalized fractional-order Jacobi functions (GFJFs) in 2 steps.…”

Section: Introductionmentioning

confidence: 99%

Generalized fractional‐order Jacobi functions for solving a nonlinear systems of fractional partial differential equations numerically

Kamali

Saeedi

2018

Math Methods in App Sciences

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Numerical solution of Volterra–Fredholm integral equations using Legendre collocation method

Cited by 54 publications

References 8 publications

A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

Generalized fractional‐order Jacobi functions for solving a nonlinear systems of fractional partial differential equations numerically

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