Elliptic PDE

In Chapters 7 and 8, we presented numerical methods for solving ODEs of initial and boundary value type. The method of orthogonal collocation discussed in Chapter 8 can be also used to solve PDEs. For elliptic PDEs with two spatial domains, the orthogonal collocation is applied on both domains to yield a set of algebraic equations, and for parabolic PDEs the collocation method is applied on the spatial domain (domains if there are more than one) resulting in a set of coupled ODEs of initial value type. This set can then be handled by the methods provided in Chapter 7. 

We will illustrate the application of orthogonal collocation to a number of examples. Elliptic PDEs will be dealt with first and typical parabolic equations occurring in chemical engineering will be considered next. 

Before we start with orthogonal collocation, it is worthwhile to list in Table 12.2 a number of key formulas developed in Chapter 8, since they will be needed in this section. Table 12.2 shows the various formula for the few basic properties of the orthogonal collocation method. 

We now show the application of the orthogonal collocation method to solve an elliptic partial differential equation. 

The level 4 of modeling the cooling of a solvent bath using cylindrical rods presented in Chapter 1 gave rise to an elliptic partial differential equation. These equations were derived in Chapter 1, and are given here for completeness. 

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Semianalytical and Numerical Method of Lines for Elliptic PDEs... 

Semianalytical Method for Elliptic PDEs in Rectangular Coordinates... 

The procedure involved in solving a linear steady state elliptic PDE is summarized as follows ... 

Transform the elliptic PDE from y coordinate to coordinate using the variable transformation = ye/h. 

Example 6.1 (heat transfer in a rectangle) is solved again using the numerical method of lines. The procedure involved in solving a linear or nonlinear steady state elliptic PDE numerically is summarized as follows ... 

Steady state linear elliptic PDEs in finite domains are solved by applying finite difference technique in both x and y coordinates in this section. When finite differences are applied, a linear elliptic PDE is converted to a system of linear algebraic equations. This resulting system of linear equations can be directly solved using Maple s solve or fsolve command. This is best illustrated with the following examples. 

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