I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an unknown point x0.
$(1-\phi)(x-C)+\frac{1}{2}\sigma^2x^2F''(x) + rxF'(x) - rF(x) = 0$ where $F(x_0) = F'(x_0) = 0$, $x_0$ is unknown.
I guess I should make use of the parameter in bvp5c (to solve for f(x) and x0 together). Can anyone help me? Thanks in advance.
The usual strategy for a second order equation $$ \partial _{x}^{2}f(x)+a(x)\partial _{x}f(x)+b(x)f(x)=c(x) $$ is to set \begin{eqnarray*} f_{1}(x) &=&f(x), \\ f_{2}(x) &=&\partial _{x}f(x) \end{eqnarray*} Then \begin{eqnarray*} \partial _{x}f_{1}(x) &=&f_{2}(x), \\ \partial _{x}f_{2}(x)+a(x)f_{2}(x)+b(x)f_{1}(x) &=&c(x), \end{eqnarray*} or \begin{eqnarray*} \partial _{x}f_{1}(x) &=&f_{2}(x) \\ \partial _{x}f_{2}(x) &=&-a(x)f_{2}(x)-b(x)f_{1}(x)+c(x) \end{eqnarray*} or $$ \partial _{x}\left( \begin{array}{c} f_{1}(x) \\ f_{2}(x)% \end{array}% \right) =\left( \begin{array}{cc} 0 & 1 \\ -b(x) & -a(x)% \end{array}% \right) \left( \begin{array}{c} f_{1}(x) \\ f_{2}(x)% \end{array}% \right) +\left( \begin{array}{c} 0 \\ c(x)% \end{array}% \right) $$ or $$ \partial _{x}\mathbf{f}(x)=\mathsf{A}(x)\mathbf{f}(x)+\mathbf{g}(x). $$ Now let $$ \partial _{x}\mathsf{U}(x,y)=\mathsf{A}(x)\mathsf{U}(x,y),\;\mathsf{U}% (x,x)=1. $$ Then $$ \mathsf{U}(x,u)\mathsf{U}(u,y)=\mathsf{U}(x,y), $$ and $$ \mathbf{f}(x)=\mathsf{U}(x,x_{0})\mathbf{f}(x_{0})+\int_{x_{0}}^{x}dy\mathsf{% U}(x,y)\mathbf{g}(y) $$ Thus, if you can obtain $\mathsf{U}(x,y)$ in some way, you have the solution for any $x_{0}$