N-dimensional Laplace transformations and their applications in partial differential equations
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This dissertation focuses on the theoretical and computation aspects of N-dimensional Laplace transformation pairs, for N ≥ 2. Laplace transforms can be defined either as a unilateral or a bilateral integral. We concentrate on the unilateral integrals. We have successfully developed a number of theorems and corollaries in N-dimensional Laplace transformations and inverse Laplace transformations. We have given numerous illustrative examples on applications of these results in N and particularly in two dimensions. We believe that these results will further enhance the use of N-dimensional Laplace transformation and help further development of more theoretical results;Specifically, we derive several two-dimensional Laplace transforms and inverse Laplace transforms in two-dimension pairs. We believe most of these results are new. However, we have established some of the well-known results for the case of commonly used special functions;Several initial boundary value problems (IBVPs) characterized by non-homogenous linear partial differential equations (PDEs) are explicitly solved in Chapter 4 by means of results developed in Chapters 2 and 3. In the absence of necessitous three and N-dimensional Laplace transformation tables, we solve these IBVPs by the double Laplace transformations. These include non-homogenous linear PDEs of the first order, non-homogenous second order linear PDEs of Hyperbolic and Parabolic types;Even though multi-dimensional Laplace transformations have been studied extensively since the early 1920s, or so, still a table of three or N-dimensional Laplace transforms is not available. To fill this gap much work is left to be done. To this end, we have established several new results on N-dimensional Laplace transformations as well as inverse Laplace transformations and many more are still under our investigation. A successful completion of this task will be a significant endeavor, which will be extremely beneficial to the further research in Applied Mathematics, Engineering and Physical Sciences. Especially, by the use of multi-dimensional Laplace transformations a PDE and its associated boundary conditions can be transformed into an algebraic equation in n-independent variables. This algebraic equation can be used to obtain the solution of the original PDE.