Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

Abstract

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems.

Numerical Solution of Linear Volterra Integral Equation of the Second Kind with Delay Using Lagrange Polynomials

“…On the other hand, a lot of researchers have been utilizing Lagrange polynomials to obtain numerical solutions to various types of problems. In 2014, The authors of [19] and [20] used Lagrange polynomials to find a solution to integral and integro-differential Volterra-Fredholm Integral equations respectively, Also, in 2020, the author [21] used the Lagrange polynomials to solve linear fractional Volterra-Fredholm integro-differential equations. In this work, we consider a linear Volterra integral equation of the 2 nd kind with a constant time delay 𝜏 > 0 of the form:…”

“…The purpose of this part is to integrate the Lagrange polynomial's notations and definitions that have been given entirely in [21]:…”

The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives

“…Caputo-Fabrizio FD was used in a formulation of time FOCPs and deriving the optimality system in terms of Volterra integrals by Yildiz, T A, et al 12 and for more about studying the FOCPs (see 13,14 ). Also, it is possible to see articles that provide a study of solving fractional order Volterra-Fredholm integral equations 15,16 .…”

Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations

“…Zhou and Xu [14] introduced numerical solution of FVFIDEs with mixed boundary conditions using the Chebyshev wavelet method; Dehestani et al [15] used a combination of Lucas wavelets and Legendre-Gauss quadrature; Salman and Mustafa [16] used Lagrange polynomials; Rajagopal et al [17] applied a new numerical method for FIDEs; Lotfi and Alipanah [18] employed the Legendre spectral element method for solving Volterraintegro differential equations. Also, Meng et…”