Numerical methods require tedious, cumbersome and repetitive arithmetic operations for large problems. It is almost impossible to do these cumbersome arithmetic operations manually. The development of information technology enhances the potential of these numerical techniques, and various software can handle algebra involved in these techniques in a very simple and sophisticated manner. Since most of the numerical techniques are algorithmic in nature and require repetitive cumbersome iterations, so it is practical to apply these algorithms to a computer. Of course, a computer must be given detailed and complete instructions for each step. During the formulation of any algorithm, we must keep in mind the following main features of the computer.

A systematic stepwise set of instructions utilizing above features of the computer enables us to solve complicated and cumbersome problems. Our efforts aim at the search for such algorithms. We will see that for a specific type of problems, there are several algorithms (For example, to solve nonlinear equations, we can choose among various algorithms like Bisection, Regula–Falsi, Newton–Raphson, etc.). We have a lot of choices depending on our requirement for speed, accuracy, and convergence, etc. A combination of two algorithms can also be used. For example, we can find a close approximation to the root of an equation by Bisection method keeping in mind the convergence; and then continue with the Newton–Raphson method from this approximation onwards keeping in mind the speed of the method. For modest size problem, we can easily implement any algorithm with high configuration computer. But in the case of large-scale problems, slow algorithms need to be rejected.

So far, many algorithms are developed for different kinds of problems. As we discussed above, there are so many reasons to select an algorithm over others.

The rapid growth of science and technology during the last few decades has made a tremendous change to the nature of various mathematical problems. It is not easy to solve these new problems for analytical solutions by conventional methods. In fact, the study of these mathematical problems for analytical solutions is not only regarded as a difficult endeavor, rather it is almost impossible to get analytical solutions in many cases. The tools for analysis and for obtaining the analytical solutions of complex and nonlinear mathematical systems are limited to very few special categories. Due to this reason, when confronted with such complex problems we usually simplify them by invoking certain restrictions on the problem and then solve it. But these solutions, however, fail to render much needed information about the system. These shortcomings of analytical solutions lead us to seek alternates, and various numerical techniques developed for different types of mathematical problems seem to be excellent options. During the last century, the numerical techniques have witnessed a veritable explosion in research, both in their application to complex mathematical systems and in the very development of these techniques. At many places in this book, we will compare numerical techniques with analytical techniques, and point out various problems which can not be solved through analytical techniques, and to which numerical techniques provide quite good approximate solutions.

Many researchers are using numerical techniques to investigate research problems. Numerical techniques are now widely used in a lot of engineering and science fields. Almost all universities now offer courses on introductory and advanced computer-oriented numerical methods to their engineering and science students, keeping in mind the utilization merits of these techniques. In addition, computer-oriented problems are part of various other courses of engineering/technology.

It gives me immense pleasure in presenting the book to our esteemed readers. This book is written keeping several goals in mind. It provides essential information on various numerical techniques to the students from various engineering and science streams.

Parabolic Equation (Heat Conduction or Diffusion Equation)Elliptic Equation (Laplace and Poisson Equations)Hyperbolic Equation (Wave Equation)

Most of the problems posed by nature, and which are of interest to physicists and mathematicians are usually governed by a single or a system of differential equations. In general, a physical system involves more than one independent variable; in that case, our mathematical model contains partial differential equations (PDEs). PDEs play a vital role in the study of many branches of applied sciences and engineering; for example, fluid dynamics, heat transfer, elasticity, electromagnetic theory, optics, plasma physics, quantum mechanics, etc. In fact, the theories of modern physics, generally involve a mathematical model, as far as possible it is a set of PDEs. We first solve the mathematical model for solutions and then come to mathematical and physical interpretations of these solutions. So it is necessary to solve the mathematical model to study the physical system. Often, it is very difficult to solve these sets of PDEs explicitly for exact solutions. Consequently, numerical methods are applied to obtain approximate solutions of these equations. In fact, there is much current interest in obtaining numerical solutions of the PDEs.

The finite difference method is a simple and most commonly used method to solve PDEs. In this method, various derivatives in the partial differential equation are replaced by their finite difference approximations, and the PDE is converted to a set of linear algebraic equations. This system of linear equations can be solved by any iterative procedure discussed in Chapter 5. Then, the solution of PDE is the solution of this system of linear equations. An important advantage of this method is that the most of the calculations can be carried out on the computer, and hence the solution is easy to obtain.