finding boundary conditions when transforming a higher order ode to system of first order ode

Question

given the following ODE:

$$\frac{d^{4}w}{dx^{4}} + B\frac{d^{2}w}{dx^{2}} = 1$$

with boundary conditions $w(0) =0 , w(1) = 0,w'(0) = 0,w'(1) = 0$

its possible to solve analytically but I am attempting to solve it numerically so I can plot and see how the graph changes as I vary the parameter $B$.

in order to solve it numerically i transformed the ODE into a system of first order ODE and i obtained this system:

$$w_{1}' = w_{2}$$

$$w_{2}' = w_{3}$$

$$w_{3}' = w_{4}$$

$$w_{4}' = 1 - B \cdot w_{3}$$

where $w_{1} = w, w_{2} = w',w_{3} = w'',w_{4} = w'''$

given those boundary conditions is there a way to determine calculate the boundary conditions for $w_{3}$ and $w_{4}$ so that can can solve the system numerically?

Your original problem isn't an initial value problem becomes of the conditions at times t=0 and t=1. Rather, it's a boundary value problem. Thus you can't solve the problem with an IVP solver. — Brian Borchers, Mar 23 '21 at 17:22
so in order to solve i need to find w1,w2,w3,w4 for t=0 and t=1? — Gideon Ilung, Mar 23 '21 at 17:36
No, four is the correct number of conditions, but you need to be using a method that can solve boundary value problems rather than initial value problems. — Brian Borchers, Mar 23 '21 at 17:44
@BrianBorchers do you have any material I could use to solve this? specifically in python — Gideon Ilung, Mar 23 '21 at 17:51
You can use initial value solvers via a shooting method. Just need to formulate it that way. — A rural reader, Mar 23 '21 at 19:09
... however, I agree with @Nicoguaro and Brian that a simpler and more direct, and maybe less frustrating, approach would be via finite differences or finite elements. — A rural reader, Mar 23 '21 at 19:57

score 3 · Accepted Answer · answered Mar 23 '21 at 19:13

As mentioned in the comments, you are interested in solving a boundary value problem and not an initial value problem. Some of the most used methods are:

For this particular problem the Ritz method might be a good choice, but in general you are better using the Finite Element Method.

finding boundary conditions when transforming a higher order ode to system of first order ode

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