## Runge-Kutta solution of first to third order ODEs

version 1.0.1 (4.02 KB) by
rk4gui.m utilizes the Runge-Kutta method to directly solve 1-st to 3-rd order non-stiff ODEs which are IVPs (Initial Value Problems).

Updated 26 Feb 2020

rk4gui.m directly solve 1-st to 3-rd order non-stiff ODEs with the Runge-Kutta method to yield the same results as Matlab’s ode45.

The advantages of this code are that
(a) It eliminates the necessity of recasting ODEs of the 2-nd and 3-rd order as a system of 1-st order ODEs before using ode45.
(b) It provides a convenient way to write the ODE and specify its solution domain, initial conditions, and the solution step size in a convenient GUI menu.

A plot of the solution and its listing are provided. Three verification examples are provided below.
1st order ODE:

Comparison between the numerical and exact solutions
ODE: y' + y = sin(t) + (1+2t)cos(t)
Exact solution: y(t) = t.*(sin(t) + cos(t))
Input: a0=1, a1=1, f(t)=sin(t) + (1+2*t)*cos(t)
t0=0, tn=5, h=0.25, y0=0, y1=1

t y-Numerical y-Exact RelativeError
------ ------------ ------------ -------------
0.00 0.00000 0.00000 0.0000e+00
0.50 0.67850 0.67850 1.5479e-06
1.00 1.38174 1.38177 2.4093e-05
1.50 1.60228 1.60235 4.2614e-05
2.00 0.98621 0.98630 9.4475e-05
2.50 -0.50676 -0.50668 1.6839e-04
3.00 -2.54666 -2.54662 1.6702e-05
3.50 -4.50531 -4.50534 6.5927e-06
4.00 -5.64167 -5.64178 2.0289e-05
4.50 -5.34729 -5.34747 3.3030e-05
5.00 -3.37612 -3.37631 5.6407e-05

2nd order ODE:

ODE: (t^2)*y'' - (2*t)*y' + 2*y = (t^3)*log(t)
Exact solution: y(t) = (7/4)*t + (1/2)*(t.^3).*log(t) - (3/4)*t.^3
Input: n=2, a0=t^2, a1=-2*t, a2=2, f(t)=(t^3)*log(t)
t0=1, tn=2, h=0.1, y(t0)=1, y'(t0)=0

t y-Numerical y-Exact RelativeError
------ ------------ ------------ -------------
0.00 0.00000 0.00000 0.0000e+00
0.50 0.67850 0.67850 1.5479e-06
1.00 1.38174 1.38177 2.4093e-05
1.50 1.60228 1.60235 4.2614e-05
2.00 0.98621 0.98630 9.4475e-05
2.50 -0.50676 -0.50668 1.6839e-04
3.00 -2.54666 -2.54662 1.6702e-05
3.50 -4.50531 -4.50534 6.5927e-06
4.00 -5.64167 -5.64178 2.0289e-05
4.50 -5.34729 -5.34747 3.3030e-05
5.00 -3.37612 -3.37631 5.6407e-05

3rd order ODE:

ODE: (t^3)*y''' - (t^2)*y'' + (3*t)*y' - 4*y = (t^3)*(5*log(t) + 9)
Exact solution: y(t) = -t.^2 + t.*(cos(log(t)) + sin(log(t))) + sin(t)+(1+2*t)*cos(t)t.^3).*log(t)
Input: n=3, a0=-4, a1=3*t, a2=-t^2, a3=t^3, f(t)=(t^3)*(5*log(t) + 9)
t0=1, tn=2, h=0.1, y(t0)=0, y'(t0)=1, y''(t0)=3

t y-Numerical y-Exact RelativeError
------ ------------ ------------ -------------
1.00 0.00000 0.00000 0.0000e+00
1.10 0.11655 0.11655 8.1801e-06
1.20 0.27271 0.27274 9.1345e-05
1.30 0.47893 0.47910 3.6031e-04
1.40 0.74631 0.74700 9.2112e-04
1.50 1.08650 1.08849 1.8306e-03
1.60 1.51153 1.51626 3.1226e-03
1.70 2.03377 2.04354 4.7792e-03
1.80 2.66582 2.68401 6.7790e-03
1.90 3.42046 3.45185 9.0926e-03
2.00 4.31063 4.36158 1.1681e-02

### Cite As

Lawrence Agbezuge (2021). Runge-Kutta solution of first to third order ODEs (https://www.mathworks.com/matlabcentral/fileexchange/74358-runge-kutta-solution-of-first-to-third-order-odes), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2019b
Compatible with any release
##### Platform Compatibility
Windows macOS Linux