The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform
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The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform Tongke Wang1
· Meng Qin1 · Huan Lian1
Received: 7 April 2019 / Accepted: 15 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly singular convolution kernel. The expansions about zero and infinity, as well as their Pad´e approximations, are used to approximate the solution when the argument is small and large, respectively, and the Pad´e approximations are more accurate. The methods are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations of the second kind with convolution kernels. Some examples confirm the correctness of the methods and the effectiveness of the asymptotic expansions. They show that numerical methods are only necessary in a small interval in practical computation when uniform high precision evaluations are needed for solving these kinds of Volterra integral equations. Keywords Linear Volterra type integral equations of the second kind · Convolution kernel · Laplace transform · Asymptotic expansion about zero or infinity · Pad´e approximation Mathematics Subject Classification (2010) 65R20 · 41A21 · 41A60 · 45D05
1 Introduction Integral equations arise in many scientific and engineering problems [11]. In this paper, we consider the following linear Volterra integral equation of the second kind x u(x) = f (x) + κ(x, t)u(t)dt, (1.1) 0 Tongke Wang
[email protected] 1
School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
Numerical Algorithms
where f (x) and κ(x, t) are known functions. If κ(x, t) = κ(x − t), it is called a convolution or difference kernel. If κ(x, t) involves singular factors such as (x − t)−α , 0 < α < 1 or log(x − t), the equation (1.1) is called a weakly singular Volterra integral equation of the second kind. Although some types of integral equations can be solved analytically [21], most of them need to be solved numerically. Hence, a lot of researches focused on exploring the properties of the equation or designing numerical methods over the past decades. In these researches, finding the asymptotic expansions for the solution about zero and infinity is of importance in mathematical and numerical analysis since they can provide a first step into exhibiting the general structure of the solution. Brunner [3, 4] systematically studied the singularity expansion of the solution of (1.1) about zero when the kernel κ(x, t) = (x − t)−α w(x, t), where w(x, t) is sufficiently differentiable. Cao et al. [5] extended the results to the case κ(x, t) = (x − t)−α w(x, t) + M(x, t). Kilbas and Saigo [14, 15] considered the asymptotic behavior of the solution near zero for some kinds of linea
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