This paper is concerned with studying some results by introducing a new Hadamard product operator for differential subordination and superordination for certain univalent functions in the open unit disc U. Firstly, we state some basic definitions and required theorems. Furthermore. Some sandwich theorems are derived. The differential subordination theory's features and outcomes are symmetric to those derived using the differential superordination theory.

In this paper, we derive explicit formulas, which can be used to solve Cauchy problems of wave equation in three and two dimension spaces, using d'alembert formula. Moreover, we apply those formulas to solve two examples.

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In this paper, we consider the analytical solution of the Nonhomogeneous mixed problem of the Heat equation. Namely, we use the separation of variables and Duhamel’s principle to derive the formula of the general solution to this problem. Moreover, we find the solution for a special case of this problem as an application to our result