Developments in Runge–Kutta Method to Solve Ordinary Differential Equations
Geeta Arora, Varun Joshi, Isa Sani Garki
Abstract
Differential equations need to be solved by researchers while performing experiments with different entities involved in a process through mathematical modelling. Among the various types of existing differential equations, the most common type of differential equation is an ordinary differential equation. There are various analytical and numerical methods available to solve the ordinary differential equations (ODEs). A numerical method to solve ODE can be a single-step method like Taylor’s series method or Picard’s method. Alternatively, it can be a step-by-step method like Euler, Milne, Adams–Bashforth, or Runge–Kutta method. This chapter is aimed to study one of the most popular numerical methods famous as Runge–Kutta (R-K) method for solving ordinary differential equations. R-K method was expanded upon in stages by its contributing authors and researchers. Presently, there are many methods related to R-K that are widely applied. In this chapter, the different methods of this category are studied to discuss their applicability and to compare their relative accuracies. An ordinary differential equation is solved by running the selected methods on MATLAB displaying the exact, approximated, as well as error values. Thus, this chapter is aimed to summarize the contribution and development of R-K methods in the field of numerical analysis to solve the ordinary differential equations.