On a Partial Differential Equation with Piecewise Constant Mixed Arguments

Abstract

So far, although there have been several articles on partial differential equations with piecewise constant arguments, as far as we know, there is no article on neither a heat equation with piecewise constant mixed arguments that includes three extra diffusion terms, delayed arguments $$[t-1], [t]$$ and an advanced argument $$[t+1],$$ or exploring qualitative properties of the equation. With the motivation to investigate elaborate and well-established qualitative properties of such an equation, in this paper, we deal with a problem involving a heat equation with piecewise constant mixed arguments and initial, boundary conditions. By using the separation of variables method, we obtain the formal solution of this problem. Because of the piecewise constant arguments, we get a differential equation and then a difference equation. With the help of qualitative properties of the solutions of the differential equation and with the behavior of the solutions of the difference equation, we investigate the existence of solutions and qualitative properties of the solutions of the problem such as the convergence of the solutions to zero, the unboundedness of the solutions and oscillations of them. In addition, two examples are given to illustrate the application of the results in particular cases.