Application of Laplace Transform in Solving Partial Differential Equation in the Second Derivative

Application of Laplace Transform in Solving Partial Differential Equation in the Second Derivative

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

The Laplace transform is a widely used integral transform with many applications in physics and engineering. Denoted ℓ {f(t)}= dt, it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms 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 tables. 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 images F(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 used for solving differential and integral equations. In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In such analyses, 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.

On the other hand Partial differential equations (PDEs) provide a quantitative description for many central models in physical, biological, and social sciences. The description is furnished in terms of unknown functions of two or more independent variables, and the relation between partial derivatives with respect to those variables. A PDE is said to be nonlinear if the relations between the unknown functions and their partial derivatives involved in the equation are nonlinear. Despite the apparent simplicity of the underlying differential relations, nonlinear PDEs govern a vast array of complex phenomena of motion, reaction, diffusion, equilibrium, conservation, and more. Due to their pivotal role in science and engineering, PDEs are studied extensively by specialists and practitioners. Indeed, these studies found their way into many entries throughout the scientific literature. They reflect a rich development of mathematical theories and analytical techniques to solve PDEs and illuminate the phenomena they govern. Yet, analytical theories provide only a limited account for the array of complex phenomena governed by nonlinear PDEs.

Over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments. Modern numerical methods, in particular those for solving nonlinear PDEs, are at the heart of many of these advanced scientific computations. Indeed, numerical computations have not only joined experiment and theory as one of the fundamental tools of investigation, but they have also altered the kind of experiments performed and have expanded the scope of theory. This interplay between computation, theory, and experiments was envisioned by John von Neumann, who in 1949 wrote “the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both” (Perthame, B., 2007).

1.2 STATEMENT OF THE PROBLEM

The study will try to apply Laplace transform in solving the partial differential equation = sinxsiny; with initial conditions U(x,0) = 1 + cosx, Uy(0,y) = -2siny and also the PDE + + u = 6y with initial conditions U(x,0) = 1, u(0,y) = y, Uy(0,y) = 0 in the second derivatives.

1.3 AIM AND OBJECTIVES OF STUDY

The main aim of the research work is assess the application of Laplace transform in solving partial differential equation in the second derivative. Other specific objectives of the study are:

1. To determine the exact solution of the problems stated above.

2. To determine whether PDEs can be verified using substitutions.

3. To determine whether any particular solution of PDEs can solve a non-homogenous problem.

4. To investigate on the factors affecting the use of Laplace transform in solving differential equation.

1.4 RESEARCH QUESTIONS

The study came up with research questions so as to ascertain the above stated objectives of the study. The research questions for the study are:

What is the exact solution of the problems stated above?

Can PDEs be verified using substitutions?

Can any particular solution of PDEs solve a non-homogenous problem?

What are the factors affecting the use of Laplace transform in solving differential equation?

1.5 SIGNIFICANCE OF STUDY

The study on the application of Laplace transform in solving partial differential equation in the second derivative will be of immense benefit to the mathematics department as the study will serve as a repository of information to other researchers and students that wishes to carry out similar research on the above topic because the study will educate the students and researchers on how to apply Laplace transforms to PDEs in the second derivatives. Finally the study will contribute to the body of existing literature and knowledge in this field of study and provide a basis for further research

1.6 SCOPE OF STUDY

The study on the application of Laplace transform in solving partial differential equation in the second derivative will be limited to second order PDEs. The study will cover on how to apply Laplace transforms to PDEs in the second derivatives.

1.7 DEFINITION OF TERMS

PDEs: Is a differential equation that contains unknown multivariable functions and their partial derivatives