An introduction to the numerical treatment of volterra and abel-type integral equations

Abstract

An integral equation can be described as a functional equation in which the unknown function appears as part of an integrand.The inadequacies of this definition will not concern us here, since our subject is the numerical solution of a subclass of integral equations which we shall specify below. We shall frequently make comparisons with topics in the treatment of ordinary differential equations, with which we assume the reader to be more familiar.In any specific endeavour, the numerical analyst should adopt a broad perspective. Ideally, he should relate the application area to the theoretical analysis,to the mathematical analysis of numerical schemes, and to the construction of algorithms and automatic software. Figure 1 illustrates schematically the interaction between the various areas.Some expansion on the meaning of some of the headings in Figure 1 is in order.Under analytic theory we include classical and functional analysis of the equations, the theory of ill-and well-posedness in various contexts, and inherent stability ("conditioning"), the asymptotic theory, and, for example, possible reformulation of the problem. What constitutes elegant theory may be a subjective assessment, but, frequently, the theory of discrete schemes is at its best when it mimics the theory of the functional equation.Under numerical methods we include the basic numerical techniques: primitive quadrature schemes, basic collocation methods etc. The theoretical numerical analysis has as afirst objective results on convergence, order, and numerical stability followed by the effect of rounding error. An important feature is an assessment of the ability to model, via the numerical scheme, relevant qualitative behaviour of the functional equations. (These considerations give rise to such notions as A-stability, for example.)We remark, in passing, that whereas many will take as a point of doctrine the necessity of convergence in a numerical scheme, as (say) some discretization parameter tends to zero, this property is not sufficient for practical computation. The modelling of important qualitative properties can be absent in practical computations with theoretically convergence schemes. An assessment of which qualitative properties are important can only result from familiarity with the area of application.The development of reliable and robust adaptive software is one aim of the Observe the possibility of confusion between the use of the word stability in the sense of the classical analyst (which relates to a form of what numerical analysts term inherent conditioning) and "numerical stability".

Numerically solving a nonlinear integral equation when the reciprocal of the solution lies in the integrand

“…The numerical methods well-known for solving nonlinear integral equations are available in the articles by Atkinson [19], Adomian [20], Jafarian et al [21] and Cuminato et al [22]. Additionally, a detailed exposition of numerical solutions to integral equations can be found in books by Atkinson [23], Delves & Mohamed [24], Baker [25] and Goldberg [26]. In general, a unique solution is not possessed by nonlinear integral equations.…”

A novel super-convergent numerical method for solving nonlinear Volterra integral equations based on B-splines

“…Integral equations is a branch of mathematics that deals with equations in which an unknown function appears under an integral sign. These equations arise in many areas of science and engineering, such as physics, chemistry, biology, economics, signal processing, potential theory electrostatics and fluid dynamics [1][2][3][4]. The theory of integral equations is an important tool for understanding and solving complex problems in various fields of science and engineering.…”