Solution of the Convolution Type Volterra Integral Equations of the First Kind by the Quadrature-Sum Method
- PDF / 811,074 Bytes
- 10 Pages / 612 x 792 pts (letter) Page_size
- 90 Downloads / 235 Views
tion of the Convolution Type Volterra Integral Equations of the First Kind by the Quadrature-Sum Method A. L. Karchevsky* Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia Received April 24, 2020; in final form, July 10, 2020; accepted July 16, 2020
Abstract—Some algorithm is presented for solving the convolution type Volterra integral equation of the first kind by the quadrature-sum method. We assume that the integral equation of the first kind cannot be reduced to an integral equation of the second kind but we do not assume that either the kernel or some of its derivatives at zero are unequal to zero. For the relations we propose there is given an estimate of the error of the calculated solution. Some examples of numerical experiments are presented to demonstrate the efficiency of the algorithm. DOI: 10.1134/S1990478920030096 Keywords: integral Volterra equation, convolution type equation, numerical solution
INTRODUCTION Solving the heat equation with data on a timelike boundary (for example, see [1]) or some inverse problems for the heat equation (see [2]), we have to solve the convolution type Volterra integral equation of the first kind t K(t − s)f (s) ds = g(t). (1) 0
The distinctive feature of applications of this type is that (1) cannot be reduced to the Volterra integral equation of the second kind since the function K(t), as well as all its derivatives, vanishes at the point t = 0. Example 1. Consider the function K(t) = t−1/2 e−a/t (see Fig. 1). In [1, 2] the cases are considered when the kernel is the same or more intricate but K(t) has similar behavior in the neighborhood of t = 0. In this case for the solution of (1) the regularization methods applied in [3-6] cannot be used since these methods require the condition K (n) (0) = 0 (n is some natural). The Denisov method requires the
Fig. 1. The graph of K(t). *
E-mail: [email protected]
503
504
KARCHEVSKY
knowledge of f (0), but this condition is not usually fulfilled in practice. The Tikhonov regularization method leads to the solution of a Fredholm equations of the first kind [7]. Basing on the quadrature method, we propose the numerical method for solving integral equation (1) and obtain an estimate of the error of the solution. Some example is presented of a numerical implementation of the method. The particularity of the solution of the convolution type Volterra integral equation of the first kind by applying the quadrature integration method is that we need to use only a fixed time meshsize. The time meshsize cannot be too small since, otherwise, a solution of the integral equation will take more time than we can afford. The second particularity that should be taken into account in our case is the behavior of K(t). Usually, this function changes significantly in a small neighborhood of t = 0, and then it has a fairly “calm” behavior. Example 2. The kernel K(t) = t−1/2 e−a/t attains the maximum value at tmax = 2a after which K(t) decreases smoothly as t−1/2 (see Fig. 1). In practice the parameter a is very small (
Data Loading...
