Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument

Abstract

We study the solution of a class of PDE with piecewise constant argument of generalized type. Separation of variables leads to a solution formed by a series of products. In previous works, the convergence and bounds of the solution could be obtained from the study of the solution on the first constancy interval only. In the general case however, each term of the series may be unbounded at every interval, implying that the solution is not computable. We establish conditions where the convergence of the solution can be verified computing a finite number of terms of the series in each constancy interval, without requiring any regularity on the initial condition. Moreover, we combine asymptotic properties for each variable of the equation to obtain an exponential bound for the solution.