Semianalytical Method for PDEs in Composite Domains

The semianalytical method developed earlier can be used to solve partial differential equations in composite domains also. Mass or heat transfer in composite domains involves two different diffusion coefficients or thermal conductivities in the two layers of the composite material.[6] In addition, even in case of solids with a single domain and constant physical properties, the reaction may take place mainly near the surface. This leads to the formation of boundary layer near one of the boundaries. In this section, the semianalytical method developed earlier is extended to composite domains. 

For example, in the diffusion reaction problem solved in example 5.5 for higher values of Thiele modulus (O 10), concentration depletes very close to the surface (x = 0). Since we are interested in the flux at x = 0 it makes more sense to choose more node points near x = 0. Equation (5.28) can be rewritten as 

5 Method of Lines for Parabolic Partial Differential Equations 

When mass or heat transfer in composite domains are modeled, different thermal diffusivities or diffusion coefficients enter in the governing equation for each region and the mass/heat flux is continuous at x = a. Equation (5.33) can be converted to the finite difference form as  

We are using the same dependent variable Ui at interior node points, for both uiand U2 in equation (5.38) for convenience. This satisfies the continuity of dependent variable at x = a (equation (5.36)) by default. The initial conditions are 

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