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In this paper, we introduce a new method for solving some partial differential equations called double Laplace-Shehu transform, some useful properties for the transform are presented. In addition, we use this transform to solve the Laplace, Poisson, Wave and Heat equations and find Laplace-Shehu transform for some functions. 1. Introduction In the literature, there are many different types of integral transforms such as Fourier transform, Laplace transform, Sumudu transform, Ezaki transform, Shehu transform, and so on. These kinds of integral transforms have many applications in various fields of mathematical sciences and engineering such as physics, mechanics, chemistry, acoustic, etc, [11]. They play an important role for solving integral equations or partial differential equations describing the physical phenomena [3, 4, 7, 13, 12]. Solving such equations using single transforms is more difficult than using the double transforms. In recent years, great attention has been given to deal with the double integral transform, see for example [2, 5, 8, 9]. Eltayeb and Kilicman [10], applied double Laplace transform (DLT) to solve wave, Laplace’s and heat equations with convolution terms, general linear telegraph and partial integro-differential equations. Alfaqeih and Misirli in [5] deal with double Shehu transform to get the solution of initial and boundary value problems in different areas of real life science and engineering. Analogous to [2], we applied new double Laplace-Shehu transform to solve Laplace, Poisson, Wave and Heat equations, through the derivation of general formula for solutions of these equations, or by applying the double Laplace-Shehu transform directly to the given equation. The main objective of this paper is to introduce new method for solving some partial differential equations subject to the initial and boundary conditions called double Laplace-Shehu transform, the definition of double Laplace-Shehu transform and its inverse. We also discuses some theorems, properties, elementary functions about the double Laplace- Shehu transform. And we implement the double Laplace-Shehu transform method to some examples.
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