Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE)

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Comparison of Runge Kutta Methods

The exact general solution of the example Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) is Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) . Inputting the initial value Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) gives Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) and the particular solution is Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) . From initial value, this solution only exist in the range up to about Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) because beyond that the right hand side is negative and square root of negative number give complex number.

Plot of all the four Runge-Kutta method and the exact solution is given below

Comparison Runge Kutta

Note that the numerical solutions of the four methods as well as the exact solution produce almost the same results. When there is a gap between the numerical solution and the exact solution, this gap usually can be narrowed by setting smaller value of Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) .

Though the exact solution does not exist after Comparison of Runge-Kutta methods: Solving Ordinary Differential Equation (ODE) , the numerical solutions still produce some results but the four methods produce different results. Care must be taken to ensure that you use the domain that produces solution.

Numerical results is not always correct

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See also: Numerical Excel tutorial , Dynamical System tutorial , Kardi Teknomo's Tutorial

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Preferable reference for this tutorial is

Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\tutorial\ODE\