N-dimensional Laplace transforms with associated transforms and boundary value problems

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1988
Authors
Debnath, Joyati
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Rajbir S. Dahiya
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Mathematics
Abstract

In this dissertation important theoretical results on n-dimensional Laplace transform, for n ≥ 2, are developed. It starts with a brief review on literature of Laplace transformation. The definitions and the concept of the region of convergence in n-dimensional Laplace transform are successfully extended from those of two dimensional Laplace transform. A number of new and useful theorems on multidimensional Laplace transforms and inverse multidimensional Laplace transform are presented. Proofs of these theorems are explicitly shown. Furthermore, in order to justify the validity of these results, several examples corresponding to each of these theorems are discussed in some detail;In certain non-linear system analysis it becomes necessary to find the inverse of n-dimensional Laplace transform and specify the inverse image at a single variable. A commonly used technique to obtain the inverse of the multidimensional Laplace transform is known as the association of variable. Several new and useful theorems on the association of variables are also developed with illustrative examples. With the concept of association of variables, a transformed function in n-dimensions is first evaluated at a single transformed variable and is then taken single dimensional inverse Laplace transform. This notion shows a technique of evaluating complicated integrals in a straight-forward manner;As a direct application of multidimensional Laplace transform, several boundary value problems characterized by partial differential equation are solved. These boundary value problems include the flow of electricity in a transmission line, semi-infinite string problem, heat transfer for a thin semi-infinite plate, electrostatic potential and a temperature distribution for the semi-infinite slab problem;Finally, a summary of the major contributions of this dissertation together with several possible directions for future research are included.

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Fri Jan 01 00:00:00 UTC 1988