Abstract
In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.
Highlights
Differential equations (DEs) are of great use in modeling different real life problems arising in science and engineering (Arora, 2019)
Problems arising in DEs are of two types based on the condition given at the endpoints namely initial value problems (IVPs) and boundary value problems (BVPs), and it was introduced in 1768 by British mathematician Leonhard Euler (Hossain et al, 2017)
The obtained result of our model examples of IVP and BVP is expressed through Table 1-3 and Table 4-6 and graphical representations are in Fig 1-3 and Fig 46, respectively
Summary
Differential equations (DEs) are of great use in modeling different real life problems arising in science and engineering (Arora, 2019). To find the exact solution of a complicated model equation a practice is to simplify the model equation and find the exact solution of the simplified equation, after they obtained result is used to approximate the original equation (Islam, 2015) In this circumstance, the approximate result differs from the real one. The approximate result differs from the real one To avoid such inconvenience researchers, find their faith in numerical techniques to find out the approximate solution of a complicated model equation. After several numerical methods developed for solving DEs namely Higher-order Taylor methods, Runge-Kutta.
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