Solving a system of ODE with BCs

Hi, I'm trying to solve a system of 2 ODE with boundary conditions.

The integration variable is r, which is the sferical coordinate (radius of a sphere). At the end of the simulation I should get the values of A and B as a functions of r, where the maximum value of r is 1.

k,c,a,b are constants.

The system is as follows:

Equations:

BCs:

I converted the system to first order one, as follows:

Equations:

BCs:

Y3(0)=Y4(0)=0

Y1(1)=a

Y2(1)=b

How can I solve it?

Thank you!

Answer 1

[Divija Aleti]

Hi Elia,

The following link gives an example of how to solve a system of two first-order differential equations with boundary conditions. Similarly, you can solve for your converted system of four first order differential equations.

Solve a System of Differential Equations

Hope this helps!

Regards,

Divija

[Elia Paini]

Thanks! However, my system is a bit different because BCs are present, while in your link the focus is on initial conditions

[Elia Paini]

I think the only way is to solve with bvp4c...or may I adapt your system?

[Divija Aleti]

You can solve in the same way shown in the example, just by replacing 't' in the example, with 'r', and giving your four boundary conditions.

[Elia Paini]

Ok, thank you!!

[Elia Paini]

I tried to use the way you suggested me, but Matlab can't find a solution. The message is as follows:

Warning: Unable to find symbolic solution.

I think that, because the complexity of the system, bvp4c must be used. In this way, results are correct.