Variable step Fehlberg’s Method –example

Problem B2 from W.H. Enright & J.D. Pryce (Two Fortran Packages For Assessing Initial Value Methods, ACM Trans. On Mathematical Software, Vol. 13, No. 1, 1987, pp 1-27) was solved using the Variable-Step Fehlberg’s method. The equations are:













The initial conditions are 1 for all equations, with α= 3. A TOL of 1E-4 was used for the calculations, with Maxbound of 1.1 and Minbound of 0.9. The results are shown below:

2-Step Runge Kutta - Example

The 2-step RK method is pretty much the same as the Euler’s method performed with two time steps. The 2-step RK method was used to solve the problem E2 from W.H. Enright & J.D. Pryce (Two Fortran packages for assessing initial value methods, ACM Trans. On Mathematical Software, Vol. 13, No. 1, 1987, pp 1-27). This is the series of equations defined by

 

 

 with y1(0)=2, y2(0)=0. The result is shown below for a step size 0.005.