Numerical Analysis for Science, Engineering and Technology

In this chapter, we will focus on the meaning of error and try to define the errors. Different and major types of errors, sources of errors, and efforts to reduce the errors will be introduced. Also, we will show how a small error in the beginning of computation will increase at the end.

Function approximations, interpolation, numerical differentiation and numerical integrations are very important topics in the numerical analysis. These topics are widely used and found in science and engineering applications. In the present chapter, we will present the main methods and techniques that had been developed for such topics. Differentiation and integration related together by so many relations as we will see later on. Finally, one can say that these topics are very important in our daily life, and therefore everyone should be aware of them.

The present chapter concerns mainly on major topics related to linear system of algebraic equations. These topics include the basic information about their constitution, Algebra, major types of numerical methods of solution, convergency and stability.

In the present chapter, we will focus on the methods of solution of nonlinear single and system of nonlinear equations. The methods presented herein are explained in brief details with error analysis and how the convergence and the stability occurred.

In science, engineering, and technology most of these applications contain one or two systems of ordinary differential equations. Many of these differential equations can't be solved using the traditional analytical method called symbolic methods. In these cases numerical or approximate methods will be used instead. In the present chapter, some elementary basic numerical methods for solving one, two and higher ordinary differential equations as well as system of ordinary differential equations will be presented with illustrative examples.

Partial differential equations have gained great interest due to their efficiency in describing the mathematical model of several physical phenomena and engineering applications. This chapter starts by a general definition for partial differential equations followed by some of the famous analytical methods of solutions and a brief description of major numerical methods for solving partial differential equations.