Theorems on unilateral, bilateral multidimensional Laplace transforms with partial differential equations

Abstract

<p>This dissertation is on the study of theorems on unilateral, bilateral multidimensional Laplace transforms and partial differential equations. We have formulated and proved theorems involving unilateral and bilateral Laplace transforms. In the case of the bilateral Laplace transform, we have considered original functions with argument of the form (ax + by)/(cx + dy). A theorem is proved and the result of this theorem is used to obtain new bilateral operational results in two dimensions. We have studied a rule similar to the exponential transformation rule in a new and generalized context. Two theorems are proved in connection with this rule by using one dimensional operational relations of the form: [phi](p) \doteq a[superscript] -cx f(a[superscript] -bx), a > 1, b > 0, and c ϵ R;The section on bilateral relations is concluded by proving two theorems in three dimensions on original functions with radical arguments;The theorems on unilateral Laplace transforms involve only two dimensional Laplace transform pairs. Six theorems are proved for original functions of the form: (1[over] x+1[over] y)[superscript][alpha] (xy)[superscript][beta] F((1[over] x+1[over] y)[gamma]) for various values of [alpha], [beta] and [gamma]. The results from these theorems are used to obtain several new unilateral Laplace transform pairs in two dimensions;Lastly, we apply the techniques of both unilateral and bilateral Laplace transforms to solve boundary value problems in partial differential equations. The methods of the unilateral Laplace transforms and Green's function techniques are used to solve the two-dimensional heat equation in a semi-infinite slab. The technique of bilateral relations together with convolutions is used to show how to obtain particular solutions of certain partial differential equations.</p>