Semianalytical Method for Homogeneous PDEs

Consider a general linear homogeneous parabolic partial differential equation in dimensionless form 

5 Method of Lines for Parabolic Partial Differential Equations 

The method of lines involves converting the governing equation (equation (5.1)) to a system of coupled ordinary differential equations in time by applying finite difference approximations for the spatial derivatives. The governing equation (equation (5.1)) can be converted to its finite difference form as follows  

Using the boundary conditions (equations (5.7) and (5.8)) the boundary values uo and Un+1 can be eliminated. Hence, the method of lines technique reduces the linear parabolic ODE partial differential equation (equation (5.1)) to a linear system of N coupled first order ordinary differential equations (equation (5.5)). Traditionally this linear system of ordinary differential equations is integrated numerically in time.[l] [2] [3] [4] However, since the governing equation (equation (5.5)) is linear, it can be written as a matrix differential equation (see section 2.1.2)  

Start the Maple worksheet with a restart command to clear all variables. 

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