As the name implies, Modified Euler’s Method is a modification of the original Euler’s method. It was developed by Leonhard Euler during the 1770s. It is one of the best methods to find the numerical solution of ordinary differential equation.
Unlike the analytical method, the numerical solution of ordinary differential equation by Modified Euler’s method is easy and simple to understand from both numerical and programming point of view. The analytical method is outdated, consuming a lot of time and the procedure is tedious.
Here, algorithm and flowchart modified Euler’s method have been presented in such a way that with the help of these you can write source code in any high level programming language.
For both algorithm and flowchart, the initial or the boundary conditions must be specified. The information required as the input data are the initial and final values of x i.e. x0 and xn, initial value of y i.e y0 and the value of increment i.e h.
Modified Euler’s Method Algorithm:
- Define function to calculate slope f(x,y)
- Declare variables
- Input the value of variable
- Find slope ( say m0) using initial values of x & y i.e f( x0 ,yo )
- Find new y, y = y0 + m0 * h
- Increase value of x, i.e x1 = x0 + h
- Find new slope ( say m’) using x1 and y
- Take the mean of m0 and m
- Again find ynew , and assign ynew with y
- Repeat from step 6 till two consecutive y are equal
- New, increase x and repeat from step 5 till x = xn
- Print x and corresponding y
Modified Euler’s Method Flowchart:
Analytical method often fails in case of complicated problems, but the modified Euler’s method does not fail, and gives higher degree of accuracy. It also gives improvement over the Euler’s method, though it may be somewhat long in coding. For higher degree of accuracy, you can choose the smaller values for h.
You can find complete source code, algorithm and flowchart for other methods to find solution of ordinary differential equation as well as the whole Numerical Methods course here. If you have any questions regarding Modified Euler’s method, its algorithm or flowchart, bring them up to me from the comments section.