Fist of all the integration term has to be reduced by taking the differentiation in the entire equation w.r.t. z once. Finally, we obtain a third order differential equation which is free from the integration term.
This equation can be solved using iteration. Firstly, by solving without integral term we can obtain solution for first iteration. For second iteration solution we can use the first iteration solution for evaluation the integral term in this equation and so on.
Problems like this (integro-differential equations) often arise in EE/circuit analysis. The conventional treatment is to apply a suitable integral transform (e.g. Laplace transform), whereby the integral disappears, then solve the resulting equation in "transformed" space, and finally use the inverse transform to recover the physical solution. I suspect this will be tricky since your equation is non-linear. I think a good place to start would be the text by Lakshmikantham and Rao (1995) Theory of Integro-Differential Equations or something similar.
One complication in using the finite difference method for solving the boundary value problem is that the integral (say approximated by Simpson's rule) makes the matrix to be inverted will become non-sparse (since the "stencil" at every point spans the whole domain now) rather than three points. The other issue is that the nonlinear term BP^3....going by what you said I am assuming you have found a way to handle the BP^3 term iteratively. The integral being of P and [not any f(P)] does not seem to add nonlinearity to the problem. So, hopefully u can find an iterative matrix inversion scheme that accomodates both issues.
If you want to get the numerical solution you have not any problem.
I think you can follow the following steps:
As we see, the definite integral term gives a function independent of the integration variable (z), so that we can consider this term as a constant value.
Use any numerical method (as Trapezoidal, Newton Cotes,...etc ) to evaluate the integral term separately.
Use your favorite numerical method to solve the differential equation in the presence of the value of integral term which given by step 2. If you want to find the exact solution, just take a look to the attachement file
Thank you for all professors. Dear Dr Zakaria, Thank you for your contribution. I have looked at you answer. The parameter P is a a function of Z. You have changed the 2nd ODE to 1st ODE and this is nice. But the integration still in the equation which is P(z)dz. even in the numerical method how to fine the integration P(z)dz if I am seeking P(z).
As I said before the integral term is a constant value C0, but we seek to evaluate it by using an appropriate numerical method, so I suggest the following procedure:
Divide the range -L/2 < z < L/2 to n+1 points, and put your tolerance.
Let the value of C0=0 or any chosen value,
Use the numerical method to find the solution of ODE
After evaluating all points, Use any numerical method (as Trapezoidal, Newton Cotes,...etc ) to evaluate the integral term separately.
Let the value of C0=the new value you got in step 4,
Repeat steps 3 ,4 and 5
Stop, If you got two successive values of C0 satisfy your tolerance.
Note :
The derivation of the exact solution not just reduction the order of ODE.