New Operational Matrix of Integrations and Coupled System of Fredholm Integral Equations

Abstract

We study Legendre polynomials and develop new operational matrix of integration. Based on the operational matrix, we develop a new method to solve a coupled system of Fredholm integral equations of the form ( ) + 11 ∫ . The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method, we compare our results with some of the results already studied with other available methods in the literature.

Existence Results for Block Matrix Operator of Fractional Orders in Banach Algebras

“…Coupled systems of integral and differential equations are studied in many papers [9][10][11][12][13]. Especially, the investigation for coupled systems of fractional differential equations appears in many studies (e.g., [9,11,[14][15][16][17]).…”

Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

“…They have the ability to solve fractional order problems, whose solutions are difficult or sometimes impossible to obtain with other traditional methods. For new readers, we strongly recommend studying the results obtained in [ 21 , 22 , 23 , 24 , 25 , 26 ] for a clear understanding and developing a good base. However, to the best of our knowledge, the spectral method becomes difficult and sometimes fails to handle the situation when boundary conditions are given in more complicated forms such as local conditions, nonlocal m-point terminal conditions, integral type terminal conditions, and radiation boundary conditions.…”

Numerical Solutions to Some Families of Fractional Order Differential Equations by Laguerre Polynomials

“…For delay differential and various other related equations, Laguerre spectral methods have been used [27][28][29][30][31][32]. Bernstein polynomials and various classes of other polynomials were also used to obtain operational matrices corresponding to fractional integrals and derivatives [33][34][35][36][37][38][39][40]. Apart from them, operational matrices were also developed with the collocation method (see Refs.…”