ABSTRACT
Numerical methods is a branch of mathematics that uses numerical approximation for problems in mathematical analysis. Since this book is concerned with scientific computing and modeling real-world problems, the focus is on the numerical solution of ordinary differential equations (ODEs) and initial value problems (IVPs), partial differential equations (PDEs) and boundary value problems (BVPs). The first section is concerned with Euler's method, which uses linear approximation and the Taylor series, taking small tangent lines over short distances, to approximate a solution to an initial value problem. Euler's method is a first-order, single-step method which fails for certain systems. A more robust, and arguably most useful, multi-step solver is the fourth-order Runge-Kutta method, also known as RK4. The second section lists the algorithm for this solver and a simple example is presented. The third and fourth sections are concerned with PDEs, involving functions of several variables, which have a huge range of applications throughout science. Finite difference methods are used in the third and fourth sections when modeling heat in an insulated rod and the vibration of a string, respectively. More examples are covered in the exercises at the end of the chapter.
