On a Partial Integro-Differential Equation on Time Scales

In this work, we study the derivation of estimation of some integral inequality in three variables over time scales. The result of this estimation is used as a tool to investigate some properties for solving partial intergrodifferential equation with initial boundary conditions on time scales, such as estimating the difference between two approximate solutions and closeness between the solutions. This difference has many applications to various scientific fields in some branches of mathematics, physics, economics, electric and biology. In this paper we study the problem from two points review, firstly we deal with estimating the difference between two ε -approximate solutions for the given certain nonlinear integrodifferential equation with initial boundary conditions, through which a suitable estimation for the approximate solutions can be obtained, and secondly we provide the conditions for closeness of solution of the problem under the study. The integral inequality in four variables over time scales may be left as a future work.