Numerical Methods: Euler and Runge-Kutta

Most real life phenomena change with time, hence dynamic. Differential equations are used in mathematical modeling of such scenarios. Linear differential equations can be solved analytically but most real life applications are nonlinear. Numerical solutions of nonlinear differential equations are approximate solutions. Euler and Runge-Kutta method of order four are derived, explained and illustrated as useful numerical methods for solving single and systems of linear and nonlinear differential equations. Accuracy of a numerical method depends on the step size used and degree of nonlinearity of the equations. Stiffness is another challenge with numerical solutions of nonlinear differential equations. Although better accuracy can be obtained with smaller step size, this takes more computational effort and time. Algorithms and codes can be written using available computer programming software to overcome this challenge and to avoid computational error. The Runge-Kutta method is more applicable and accurate for diverse classes of differential equations.