Partial differential equations orthogonal function

The set of equations (10.5-22) is quite readily handled by the numerical method of orthogonal collocation. Basically, the coupled partial differential equations (eq. 10.5-22) are discretized in the sense that the spatial domain r is discretized into N collocation points, and the governing equation is valid at these points. In this way, the coupled partial differential equations will become coupled ordinary differential equations in terms of concentrations at those points. These resulting coupled ODEs are function of time and are solved by any standard ODE solver. Details of the orthogonal collocation analysis are given in Appendix 10.5, and a computer code ADSORB3A is provided with this book for the readers to learn interactively and explore the simulation of this model. 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

Since we have not used the symmetry boundary condition that dTfdy = 0 at y = 1/2, we will solve the problem by performing an orthogonal collocation in the spatial domain using the shifted Legendre polynomials as the basis function. This will reduce the problem to a set of initial-value ordinary differential equations that can be solved using IMSL ordinary differential equation routines. As discussed by Cooper et al. (1986), seven internal collocation points accurately describe the solution to the partial differential equations. Therefore, we use equation (8.12.14) to approximate the second spatial derivative. This reduces the original partial differential equation of (8.12.15) to... 

The collocation method is one of a general class of approximate methods known as the method of weighted residuals (Ames 1965). The method involves expanding the temperature and concentration in a series 2af(z)Fj(r) of known functions of radius, Pi(r), multiplied by unknown functions of z. The trial functions are substituted into the partial differential equations that are satisfied at discrete radial collocation points, r. This gives a set of ordinary differential equations governing Ui(z). The trial functions are orthogonal polynomials, e.g.r... 

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