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Are you sure that you have the correct boundary conditions? I think the equation you state for the analytical case, $$H=x/\tanh(x)-1$$ is incompatible with the boundary condition $H(\bar{\epsilon})=\bar{\epsilon}$ where $\bar{\epsilon}$ is large. Maybe this is why you are getting incorrect behaviour for large $x$?
I tried solving your equation for the $\...
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You could use a discretization method such as the finite element or finite difference method with a linearization technique such as Picard or Newton method to solve this problem. A similar question is this.
The main idea of these solution techniques is as follows:
Pick an initial guess for the solution.
Linearize your equation and write an updated solution ...
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