Initial Value Problems for ODEs Single-step Methods

When it comes to the application of science in physics in real life we are often left to deal with differential equations.

It may be possible that the solution to those differential equations can be found out easily but in most cases, it is not so and we often get stuck in the problem. This is where numerical analysis comes to the rescue.

According to a research paper published in the CFM “The idea behind numerical solutions of a Differential Equation is to replace differentiation by differencing”. A computer cannot do a continuous process (i.e. differentiation) but it can easily handle a discreet process (i.e. differencing).

To have a good grasp of this topic you need to have a good practice of Taylor series expansion.

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You will get multiple methods just by truncating the Taylor series and replacing the derivative with a difference. Thus it can be said that this particular method (Euler method) has truncation error.

Always check for Lipschitz continuity to your initial value problem. It guarantees the existence and uniqueness of the differential equation. Although the Taylor series method is a one-step method it requires the explicit form of the derivates off.

To avoid the disadvantage of the Taylor series method, we can use the Runge-Kutta method. Though this is still a one-step method, it depends on the estimation of the solution at different points.

The second-order method requires two evaluations of the function, the fourth-order requires 4 evaluations.

One big advantage of this approach is that you don’t need to know the derivative of the function. One would think that they could continue to increase the evaluations to get faster results but it is to be noted that RK-4 offers the highest order.

Take up a differential equation from a spring-mass damper system and try to apply various methods and compare the accuracy of the output.

For Discussion Comment below or Go to the Next Chapter Initial Value Problems for ODEs: Multistep Methods

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