Laplace Transform Technique for Parabolic PDEs

Parabolic partial differential equations are solved using the Laplace transform technique in this section. Diffusion like partial differential equations are first order 

4 Partial Differential Equations in Semi-infinite Domains 

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Laplace Transform Technique for Parabolic PDEs The dimensionless temperature profile is given by > u =convert(u,erfc) ... 

Laplace Transform Technique for Parabolic PDEs -Advanced Problems... 

Linear first order parabolic partial differential equations in finite domains are solved using the Laplace transform technique in this section. Parabolic PDEs are first order in the time variable and second order in the spatial variable. The method involves applying the Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes a boundary value problem (BVP) in the spatial direction with s, the Laplace variable as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 3.1 for solving linear boundary value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). Certain simple problems can be inverted to the time domain using Maple. This is best illustrated with the following examples. 

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