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I have an ODE with the form

[tex]\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}+a\exp(b\phi)=0[/tex]

B.C. [tex]\frac{\mathrm{d}\phi}{\mathrm{d}{x}}\right|_{x=0}=-c,\frac{\mathrm{d}\phi}{\mathrm{d}{x}}\right|_{x=L}=0[/tex]

where [tex]a,b,c,L[/tex] are all positive.

I solved the problem with Maple using the dsolve routine:

dsolve({bc, equ}, type = numeric, method = bvp[midrich], range = 0 .. L)

but it says "matrix is singular". My Fortran program using the FVM method also indicate the coefficients matrix is singular at [tex]x=L[/tex]

But if i change the B.C. at [tex]x=L[/tex] to [tex]\phi\right|_{x=L}=0[/tex] the solution can be got.

So i wonder

1 why the previous Newmann B.C. cause the singularity at [tex]x=L[/tex] but the latter Dirichlet one not.

2 how to evaluate the influence of such a boundary to the nature of the solution?

any suggestions will be appreciated

hui

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# Numerical solution of an ODE be Singular at endpoint

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