Volterra and Functional Differential Equations

A VOLTERRA EQUATION WITH A TIME DEPENDENT OPERATOR

T. R. Kiffe

Department of Mathematics Texas A&M University College Station, Texas

DOI: 10.1201/9781003420026-19

In this paper, we will be concerned with the existence and uniqueness of solutions of the Volterra integro-differential equation

$$\frac{{\partial }^2\text{u}}{\partial {\text{c}}^2}\left(\text{x},\text{y},\text{t}\right)-\text{Δu}\left(\text{x},\text{y},\text{t}\right)-{\int }_0^{\text{t}}\text{a}\left(\text{t}-\text{s}\right)\text{Δu}\left(\text{x},\text{y},\text{s}\right)\text{ds}=\text{f}\left(\text{x},\text{y},\text{t}\right)$$ (1.1)

for x real, y > 0 and t ≥ 0, with the conditions

$$\begin{array}{ll}\hfill & \text{u}\left(\text{x},\text{y},0\right)={\text{u}}_0\left({\text{x}}_{,}\text{y}\right)\hfill \\ \hfill & \frac{\partial \text{u}}{\partial \text{t}}\left(\text{x},\text{y},0\right)={\text{u}}_1\left(\text{x},\text{y}\right)\hfill \\ \hfill & \text{u}\left(\text{x},0,\text{t}\right)=0\text{ for x}>\text{p}\left(\text{t}\right),\text{t}=0\hfill \\ \hfill & \frac{\partial \text{u}}{\partial \text{y}}\left(x,0,t\right)+{\int }_0^t\text{a}\left(\text{t}-\text{s}\right)\frac{\partial \text{u}}{\partial \text{y}}\hfill \end{array}$$