Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order

Abstract

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

“…where 𝒃 𝑖 = 𝑩 𝑖 (0) is called the Bernoulli number for each 𝑖 = 0,1, …. These numbers are calculated by the following identity [62]:…”

Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses

“…B(E) the Banach space of bounded linear operators from E into E. First we recall the concept of the evolution operator S(t, s) for problem (2), introduced by Kozak in [15] and recently used by Henríquez, Poblete and Pozo in [20].…”

“…The theory and application of integrodifferential equations are important subjects in applied mathematics, see, for example [1][2][3][4][5][6][7][8] and recent development of the topic, see the monographs of [9]. In recent times there have been an increasing interest in studying the abstract autonomous second order, see for example [10][11][12][13][14].…”