The emphasis in this course is on the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. A variety of numerical methods related to solve these problems will be presented in the corresponding simplified one-dimensional applications. So that both theoretical and numerical investigations can be understood easily, and the techniques could be extended intuitively to higher-dimensional problems. The last part of this course will give a brief introduction of finite difference methods for three types of PDEs, elliptic, parabolic, and hyperbolic problems.