I would appreciate it if you could help me solve the following coupled equation numerically $$ [-\frac{1}{2} \partial_r^2 + v_0(r) -E]\psi_{\ell} + v_1(r) \psi_{1-\ell}(r) = 0, $$ where $\ell = 0 , 1$ with initial conditions $$ \partial_r \psi_0(0) = 0,\\ \psi_1(0) = 0. $$ I do not know how to determine the initial start.
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$\begingroup$ Welcome to SciComp.SE. I think that you are missing another equation for $\psi_1$ or $\psi_0$. $\endgroup$– nicoguaro ♦Commented Feb 28 at 3:23
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$\begingroup$ Thank you :). This is a 2 coupled equations for $\ell =0 $ and for $\ell=1$ $\endgroup$– GhotiCommented Feb 28 at 5:39
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2$\begingroup$ What is the domain? What are $v_0$, $v_1$, and $E$? Since this is a second order problem, you are missing one initial or boundary condition for each equation. This determines whether this is an IVP or BVP, which determines suitable discretization schemes. $\endgroup$– cos_thetaCommented Feb 28 at 12:50
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$\begingroup$ Thanks for your comment @cos_theta. $v_0$ and $v_1$ are some functions of $r$, and $E$ is a parameter and it is positive $E>0$. $\endgroup$– GhotiCommented Mar 1 at 8:52
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$\begingroup$ @Ghoti, what are the missing initial or boundary conditions for each equation? It is a second order equation, so you need two conditions for each equation in order to get a unique solution. Also, what is the domain? $\endgroup$– cos_thetaCommented Mar 1 at 16:21
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