Numerical Method of Lines for Parabolic Partial Differential Equations PDEs

2 Numerical Method of Lines for Parabolic Partial Differential Equations (PDEs) 

2 Numerical Method ofidnesfor Parabolic PDEs with Linear Boundary 

Consider a general nonlinear parabolic partial differential equation in dimensionless form 

The numerical method of lines[l] [3] [4] [2] (Schiesser and Silebi, 1997 Cutlip and Shacham, 1999 Taylor 1999 Constantinides and Mostoufi, 1999) involves converting the governing equation (equation (5.48)) to a system of coupled ODEs in time by applying finite difference approximations for the spatial derivatives 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows  

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