Given following inputs,
- An ordinary differential equation that defines value of dy/dx in the form x and y.
- Initial value of y, i.e., y(0)
Thus we are given below.
The task is to find value of unknown function y at a given point x.
The Runge-Kutta method finds approximate value of y for a given x. Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method.
Below is the formula used to compute next value yn+1 from previous value yn. The value of n are 0, 1, 2, 3, ….(x – x0)/h. Here h is step height and xn+1 = x0 + h
. Lower step size means more accuracy.
The formula basically computes next value yn+1 using current yn plus weighted average of four increments.
- k1 is the increment based on the slope at the beginning of the interval, using y
- k2 is the increment based on the slope at the midpoint of the interval, using y + hk1/2.
- k3 is again the increment based on the slope at the midpoint, using using y + hk2/2.
- k4 is the increment based on the slope at the end of the interval, using y + hk3.
The method is a fourth-order method, meaning that the local truncation error is on the order of O(h5), while the total accumulated error is order O(h4).
Below is implementation for the above formula.
The value of y at x is : 1.103639
Time Complexity of above solution is O(n) where n is (x-x0)/h.
Some useful resources for detailed examples and more explanation.
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