APPLICATION OF LAPLACE TRANSFORM IN SOLVING LINEAR SECOND ORDER PARTIAL DEFFERENTIAL EQUATION

The Laplace transform is awidely used integral transform with many Application in physics and engineering. Denoted{ f(t)}=F(s)= ∫_(t=0)^∞▒f (t) e^(-st) dt, it is a linear operator of a function f(t) with a real argument t(t> 0) that transform it to a function F(s) with a complex argument s. This transformation is essentially Bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in table. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the imagesF(s). It is named after pierre-Simon Laplace, who introduced the transform in his work on probability. Theory. The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is sued for solving differential and integral equation. In physics and engineering it is used for analysis of liner time-invariant system such as electrical circuit, harmonic oscillator, optical devices, and mechanical systems. In such analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.