On a weakly singular Volterra integral equation

Abstract

The need for providing reliable numerical methods for the solution of weakly singular Volterra integral equations ofIst Kind stems from the fact that they are connected to important problems in the theory and applications of stochastic processes. In the first section are briefly sketched the above problems and some peculiarities of such equations. Section 2 described the method for obtaining an approximate solution whose properties are described in section 3: such properties guarantee that our approximate solution always oscillates around the rigorous one. Section 4 discusses the applicability to our case of some classical bounds on the errors. The remaining sections are all devoted to the construction of upper bounds on the oscillating error in order to reach a high degree of reliability for our solution. All the bounds are independent on the numerical method which is employed for obtaining the numerical solution. In section 5 is derived a Volterra II Kind integral equation by subtracting to the original kernel the weak singularity, while in section 6 is given an upper bound to the error in the case of Wiener and Ornstein-Uhlenbeck kernels with constant barriers. Such a bound is generalized to other kinds of barriers in section 7 while in section 8 is suggested an approximation of the Kernel for the O. U. case with constant barriers and by means of it is given an explicit bound for the error in terms of Abel's transform of the known term in the original integral equation. A rough estimation of the error is also given under the assumption that\(y(t) - \int\limits_0^t {K(t,\tau )\tilde x(\tau )d_\tau [\tilde x(\tau )} \) denotes any approximate solution of (1a) obtained by any method] can be approximated by means of a sinusoidal function.