Integral Volterra Equations

Abstract

In this chapter, we conducted a thorough examination of the Volterra integral equation of the second kind for an arbitrary real parameter λ, assuming that the free term f (x) is real-valued and continuous on the interval [a, b] and that the kernel K(x, t) is real-valued, continuous, and separable on the square Q(a, b) = {(x, t): [a, b] × [a, b]}. A question naturally arises: What, if anything, can be proven if K(x, t) is a general kernel, i.e., an arbitrary kernel that is only assumed to be real-valued and continuous? In this chapter, we will answer this question completely by providing several tools of the trade which are indispensible for the comprehension of the material in this chapter. We use these tools to show that the Volterra integral equation of the second kind with a general kernel has a unique solution if the product of the parameter λ and the “size” of the kernel are small. We show how to construct the resolvent kernel that appears in the solution to the integral equation recursively and introduce numerical methods for producing an approximation to the solution of a Volterra integral equation. These methods are necessary due to the inherent computational difficulties in constructing the resolvent kernel in case of integral type creep equation. Particular attention is paid to this kind Volterra integral equation , which is most common in solving problems related to high temperatures creep. This Volterra integral equation in this case has two main features. First, the high temperatures range [a, b] in real fire scenario is so large that it has to divide into smaller subintervals to obtain a convergent solution. The solutions of the integral equation in the two adjacent subintervals are joined together on the basis of the conditions of continuity of the solution at the junction of these subintervals. This method is called strip method . Second, the kernel of the Volterra integral equation K(x, t) is on the one hand an increasing function (due to the presence of the Arrhenius law) and, on the other hand should be tending to zero as t tends to infinity. Many numerical examples show that the precision of strip method solutions are reasonable in comparison to the classical methods.