Abstract
Non-standard finite difference (NSFD) methods are widely used for the numerical integration of modelled ordinary differential equations (ODEs) or partial differential equations (PDEs). The NSFD methods are easy to use and have more stability than the standard methods. In this chapter, a brief history of NSFD methods is given along with the rules of construction and pertinence to ODEs and PDEs. The exact solution of the ODEs is discussed by standard and non-standard finite difference methods. Some of the magnificent problems are provided with excellent numerical solutions by the NSFD scheme. The solution of non-linear ODE representing the combustion model and Duffing oscillator second-order ODE is obtained. Also, the NSFD scheme for models describing the fractional-order problems is explained using the Grunwald-Letnikov and Riemann-Liouville derivatives to approximate the variable order fractional-order equations.
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