A Computational Method for Volterra Integro-Differential Equation

Abstract

In this paper, we examine the initial value problem for a linear first order Volterra integro-differential equation. In order to solve the problem computationally, we present a novel finite difference method, which is based on the method of integral identities with the use of the basis functions and interpolating quadrature rules with remainder term in integral form. Furthermore, as a consequence of error analysis the method is proved to be first-order convergent in the discrete maximum norm. Finally, an example is provided to support our theoretical results.

Approximate Solution of Singularly Perturbed Problems with Numerical Integration Method

“…Ayrıca, literatürde Volterra integro-diferansiyel denklemlerini çözmek için kullanılan farklı yaklaşımlar vardır. Bunlar, Piecewise-quasilinearization yöntemi (Ramos, 2007), exponential teknik ve implicit Runge-Kutta metodu (Ramos, 2008), coupled metodu (Tao & Zhang, 2019), sonlu fark metodu (Mbroh ve ark., 2007;Sevgin, 2014;Cimen, 2018;Yapman & Amiraliyev, 2020) ve diferansiyel dönüşüm metodu (Celik & Tabatabaei, 2013). Volterra integro-diferansiyel denklemlerin çözümlerinin varlığı ve tekliği de literatürde yer almaktadır (Ross ve ark., 1996;Jerri, 1999;Kythe & Puri, 2002;Burton, 2005;Nefedov ve ark., 2006).…”

A novel computational method for solving nonlinear Volterra integro-differential equation

“…Moreover, more different studies are seen. Finite difference method is examined for first-order VIDEs (Cimen, 2018). In addition, for high-order linear VIDEs, a new numerical method with Taylor polynomials is developed (Laib et al, 2018).…”

An Alternative Method for Numerical Solution of Fredholm Integro Differential Equation

“…Örneğin, dikdörtgen metodu, yamuk metodu, Simpson metodu gibi geleneksel metotlar sıklıkla kullanılmaktadır (Hackbusch, 1995;Jerri, 1999;Kythe ve Puri, 2002). Bununla birlikte, son yıllarda birçok yazarın; bölge ayrıştırma metodu, Taylor seri metodu, dalgacıklar (wavelets) yöntemi, doğuran çekirdekli Hilbert metodu ve sonlu farklar metodu gibi yeni metotları kullandıkları görülmektedir (Darania ve Ebadian, 2007;Lakestani vd., 2011;Maleknejad ve Attary, 2011;Wazwaz, 2011;Arqub vd., 2013;Pandey, 2015;Amiraliyev vd., 2018;Çimen, 2018;Yapman vd., 2019).…”