Numerical Methods and Applied Mathematics for Solving Engineering Problems

Abstract

Computational engineering and science problem solving is based on numerical methods, which are the computational foundation of modern engineering and science, and which allow analysis of complex systems, which have no closed-form analytical solution. This work is aimed at providing a unified summary of basic numerical methods commonly employed in engineering and the applied sciences, particularly in the context of their derivation, computation and practical application. It uses a systematic combination of classical and applied numerical methods, such as root-finding routines, numerical differentiation and integration, solutions of algebraic equations, linear and nonlinear, and methods of numerical approximation to ordinary and partial differential equations. The algorithmic procedures are brought up in terms of accuracy, convergence behavior, and stability, and their applicability to realworld engineering problems. The representative problem case studies show how the numerical methods can be useful in the approximation of the solution when analytical method is infeasible or inefficient. The findings show that the choice of method has a great impact on the accuracy of solutions and the cost of computations, that tradeoffs between accuracy and cost have to be efficiently handled based on the nature of a problem. The paper comes up with the conclusion that numerical techniques continue to be an essential tool in all fields of engineering, especially when there exists a large scale simulation and the use of computer analysis. Good knowledge of the underlying principles, error behavior, and implementation strategies is a key to effective and confident problem-solving and further improvement of numerical algorithms will increase their application in new scientific and engineering problems.