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I have found my mistake. It was on the initial condition if we take $$\phi(zi)=0$$ and $$\phi'(zi)=0$$ the both methods will give the exact same solution.


It's from the Bessel factor: Cosh[(1 - ϵ) x] /. x -> 10.`40^16 Note that the Bessel function evaluates to Underflow[]:


I Have finally found what was causing the problem. it was with notation. I define my constant a_o but in the equation I have used a_0. Once I changed that it was working.


NDSolve returned interpolation functions which come from the FEM will evaluate to Indeterminate if queried outiside of the region. In this case, for example: RegionMember[\[CapitalOmega], {10, 50}] (*False*) sol[10, 50] (*InterpolatingFunction::dmval: "Input value {10.,50.} lies outside the range of data in the interpolating function. Extrapolation will be ...


What you have, I guess, is DAE in the variable t. If you specify the initial condition g[x, 0] == Cosh[x]/Cosh[Pi], NDSolve will compute a "solution," but warns that "an insufficient number of boundary conditions have been specified for the direction of independent variable x." It then computes different initial values, that lead g to be almost identically ...

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