Numerical Methods for Solving Partial Differential Equation

Numerical Methods for Solving Partial Differential Equation

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

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).

Numerical solutions of nonlinear PDEs were first put into use in practical problems, by von Neumann himself, in the mid-1940s as part of the war effort. Since then, the advent of powerful computers combined with the development of sophisticated numerical algorithms has revolutionized science and technology, much like the revolutions that followed the introduction of the microscope and telescope in the seventeenth century. Powered by modern numerical methods for solving for nonlinear PDEs, a whole new discipline of numerical weather prediction was formed. Simulations of nuclear explosions replaced ground experiments. Numerical methods replaced wind tunnels in the design of new airplanes. Insight into chaotic dynamics and fractal behavior was gained only by repeating “computational experiments”. Numerical solutions of nonlinear PDEs found their way from financial models on Wall Street to traffic models on Main Street.

1.2 STATEMENT OF RESEARCH PROBLEM

The study will try to solve the following problem below using numerical method:

With conditions

U(X, 0) = Sin ? X, 0≤X≤1

U (O, t) = -sin ?t, t≥0

U(X, 0) = X

(X, 0) = ∫√X

1.3 AIM AND OBJECTIVES OF STUDY

The main aim of the study is to use the numerical methods to solve the above stated problems. Other specific objectives of study are:

To provide solution to the above problems using numerical methods.

To determine the norm and error norms in the P.D.E above.

To examine various numerical methods for solving P.D.E.

To investigate on the factors affecting the use of numerical methods for solving P.D.E.

To examine the stability and convergence of partial differential equation.

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

What are the norm and error norms in the partial differential equations (PDE) above?

What are the various numerical methods for solving PDE?

What are the factors affecting the use of numerical methods for solving PDE?

What is the stability and convergence of partial differential equation?

1.5 SIGNIFICANCE OF STUDY

The study on numerical methods for solving partial differential equation will be of immense benefit to the entire mathematics department and other researchers that desire to carry out similar research on the above topic because the study will provide an explicit solution to partial differential equations using numerical methods. The study will determine the norm and error norms in the numerical solution of the PDE.

1.7 SCOPE OF STUDY

The study on numerical methods for solving partial differential equation will cover on finite difference method, stability and convergence, diagonal dominance and invertibility and convergence of the Neumann series.