Numerical Method of Lines for Parabolic PDEs with Nonlinear Boundary

3 Numerical Method of Lines for Parabolic PDEs with Nonlinear Boundary 

When the boundary conditions are nonlinear, the procedure described in section 5.2.2 cannot be used because the boundary values cannot be eliminated because of the nonlinear boundary conditions. This is handled by differentiating the finite difference form of the boundary condition with respect to t. This yields two additional nonlinear ODEs in time (see section 2.2.6 on DAEs), which are then solved simultaneously with N nonlinear ODEs arising from the discretization of 

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