Fundamentals of Computational Methods for Engineers

In this chapter, fundamentals of computational methods are presented. Mathematical modeling of the physical systems is described. Then, various type of methods, error analysis and the algorithm with software packages are also discussed here.

In this chapter, computational methods for the solution of nonlinear equations, particularly the solution of transcendental equations, have been presented. Various types of computational methods are discussed with the engineering problem analysis. 

In this chapter, computational methods for the solution of linear system equations have been presented. At first, the fundamentals of a matrix with matrix operation have been discussed. Then, various types of computational methods, such as Gauss elimination, Gauss-Jordan, Gauss Seidel, etc., for the solution of linear system equations are discussed with the engineering problem analysis. 

In this chapter, we study numerical techniques that deal with given set of data points arising from experimental works. Starting from linear interpolation, different interpolating polynomials are discussed that are used to find functional value at intermediate points of the given data set. Lagrange interpolation is discussed that does not require equally spaced data points. Newton forward and backward difference interpolation formulae are derived to evaluate function near the beginning and end parts of the given data sets. Linear least-squares fit that is widely used to approximate unknown functions is presented and an algorithm is developed. We also discuss least-squares approximations for approximating an explicit function on given interval.

In this chapter, we study numerical differentiation and integration. At first, fundamental theories on differentiation and integration are discussed. Then, Newton's forward difference, Leibniz's notation and Lagrange's notation are presented. At the end of the chapter, various types of integration methods are discussed with the engineering problem analysis

Practically, the engineer’s deal with a lot of problems that can be expressed mathematically by ordinary differential equations (ODEs). These ODEs can be solved using both direct and iterative methods. The latter is popular as, in this case, the solution techniques are based only on the basic arithmetic operations. In this chapter, at first, we studied ordinary differential equations. Secondly, fundamental theories for the solutions of these differential equations are discussed. Then, various numerical solution techniques are explained in this regard. At the end of each technique, solutions to various engineering problems are discussed.

In this chapter, advanced computational methods such as FD, FDTD, MoM, and FEM have been presented. Various types of computational methods are discussed with the engineering problem analysis.