Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

“…In addition, by using the Tau method, the problem is converted to a set of algebraic equations from which the solution can be obtained iteratively. In [32], a class of two-dimensional nonlinear Volterra integral equations has been solved using operational matrices of Legendre polynomials; the operational matrices of integration and product together with the collocation points have been utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. In [33,34], the one and two dimensional fractional equations were studied by using Legendre operational matrix.…”

Section: Research Literaturementioning

confidence: 99%

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Parand

Hossayni

Rad

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

Please cite this article as: K. Parand, Sayyed A. Hossayni, J.A. Rad, An operation matrix method based on Bernstein polynomials for Riccati differential equation and Volterra population model, Applied Mathematical Modelling (2015), Highlight• We present an exact formulation for the operational matrix.• This method transforms the original problem into a algebraic equations system.• Riccati equation and Volterra population model are solved by new method. AbstractIn this paper, we present a modified configuration, including exact formulation for the operational matrix form of the integration, differentiation and product operators to be applied in the Galerkin method. Up to now, so many studies have been conducted on the methods of obtaining operational matrices (derivative, integral and product) for Fourier, Chebyshev, Legendre and Jacobi polynomials and, sometimes, the non-orthogonal bases, which almost all of them operate approximately; but, in this research, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to the previous works, this method transforms the original problem into a system of nonlinear, algebraic equations. To keep the simplicity of the procedure, samples have been considered in one dimensional contexts; however, the proposed technique can be employed for two and three-dimensional problems, as well. For confirming the accuracy of the new approach and for showing the performance of EOMs over ordinary operational matrices (OOMs), two examples are presented. The corresponding results demonstrated the increased accuracy of the new method. Also the convergence of the EOM method is studied numerically and analytically, to prove the method efficiency.

show abstract

“…As a very important application, Legendre spectral methods are successfully used to obtain numerical solutions of the various differential equations. Through Google Scholar search, there are almost 54,000 articles from 1980 to 2019 on the use of the Legendre spectral methods in the study of various problems, such as numerical solving for integrodifferential equations ( [1][2][3][4][5][6]) and ordinary differential equations with fractional order ( [7][8][9]) and integer order ( [10]). Recently, the Legendre spectral method was proved to be an effective method to solve fractional differential equations, which has been studied by many scholars ( [7,8,[11][12][13]).…”

Section: Introductionmentioning

confidence: 99%

Left- and Right-Shifted Fractional Legendre Functions with an Application for Fractional Differential Equations

Yang

She

2020

Advances in Mathematical Physics

View full text Add to dashboard Cite

Two new orthogonal functions named the left-and the right-shifted fractional-order Legendre polynomials (SFLPs) are proposed. Several useful formulas for the SFLPs are directly generalized from the classic Legendre polynomials. The left and right fractional differential expressions in Caputo sense of the SFLPs are derived. As an application, it is effective for solving the fractional-order differential equations with the initial value problem by using the SFLP tau method.

show abstract

Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

Cited by 113 publications

References 7 publications

Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions

Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Left- and Right-Shifted Fractional Legendre Functions with an Application for Fractional Differential Equations

Contact Info

Product

Resources

About