ODE System - Numerical Solver

Solve Ordinary differential equations systems using Runge-Kutta

Dependences

• Codes made in Fortran 90
• gfortran Compiler
• Plot solution using Matlab/Octave

How to use

Running code

The code runs in Fortran 90, you will need a fortran compiler, such as gfortran. The problem conditions are changed in code, then you'll need to compile every change:

  gfortran ode_solver_main.f90 -o <your_exe_name>

Then, run:

• On Windows

  your_exe_name.exe

• On Linux

  ./your_exe_name.out

After this, the code will generate three .out files.

• mash_info.out : Contain the points of domain discretization.
• output_solution.out : Contain the solution for each point

Plotting solution

You will need Matlab or Octave to run the .m code. Once with Matlab/Octave opened, just run the code with execute button and watch the solution change in time.

Mathematical Model

We solve a Diferential Equation System using 4 order Runge-Kutta method

Varaibles in code

To set the interval that solution will be calculated

intr_x = 0.400

The number of points that the solution will be calculated

nodes_number = 1000

To set the system, follow the steps:

• First, you'll need set the system size

Example:

!============================ System Size ============================!
system_size = 3

• Set the initial system condition

Example:

!======================== Initial conditions ==========================!
old_sol(1) = 5.000
old_sol(2) = 1.000
old_sol(3) = -3.1415

• Setting the functions

Example:

function f(x, u) result( f_res )
     real(8), intent(in)  :: x
     real(8), intent(in), dimension(:) :: u
     real(8), dimension( size(u) ) :: f_res

     !===== function =====!
     f_res(1) = 3*x - u(1)
     f_res(2) = u(2) + 3*u(1) - u(3)
     f_res(3) = log(x) - u(3) + (u(2))**2

 end function f

Solution Examples

Population Dynamics

Vertical Spring-Mass system

System with 2 springs coupled with 2 massive bodies

With Damping