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

Abstract

In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

“…, which points to the proof of (2.4). Now, we show the proof of the relation (2.5) (see [9,11,13,29]). From (2.6), we write…”

“…Applying interpolating quadrature rules [5] and some manipulations in [11,29] for the first term and the second term of the relation (3.1), we find…”

“…In general, solving of such problems is hard and even sometimes impossible with classical analytical techniques. Thus, reliable numerical methods have been developed such as Bernstein polynomial method [8], collocation techniques [32], Nyström discretization [24], Legendre wavelets [22], finite difference methods [11,29] and Galerkin finite element approach [18].…”

A numerical method on Bakhvalov-Shishkin mesh for Volterra integro-differential equations with a boundary layer

“…Here, by the formula of g(x) given in (11), from (3) and knowing that a, f ∈ C 1 (I), K ∈ C 1 (I × I) and from ( 4) we obtain…”

A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh

“…Therefore, some important techniques have been introduced in the literature. These involve: finite difference methods [1,4,5,8,11,12,15], fitted mesh method [23], reproducing kernel method [7], factorization method [25], Galerkin method [18,26], differential transform method [9], variational iteration methods [2,10] and so on [6,16,19,20,27].…”