Parabolic PDE

The (ode-) method called leapfrog has been mentioned in Chap. 4, where (4.38) describes it. This was used by Richardson [468] to solve a parabolic pde, apparently with success. The computational molecule corresponding to this method is... 

Leapfrog is used with apparent success to solve hyperbolic pdes [528], but was proved unconditionally unstable for parabolic pdes in 1950 [424]. Richardson had been lucky, in that the instabilities had not made themselves felt in his (pencil and paper) calculations, in the course of the few iterations he worked. 

Lang J., Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Springer, Berlin (2001)... 

The ADI method, first used by Peaceman and Rachford (1955) for solving parabolic PDEs, can also be derived from the Crank-Nicholson algorithm. In the three-dimensional MRTM model, the governing equation can be discretized by the Crank-Nicholson algorithm as... 

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... 

Consider a general parabolic PDE with a source term where y is the source term... 

In this chapter semianalytical solutions (solutions analytical in t and numerical in x) were obtained for parabolic PDEs. In section 5.1.2, the given homogeneous parabolic PDE was converted to matrix form by applying finite differences in the spatial direction. The resulting matrix differential equation was then integrated analytically in time using Maple s matrix exponential. This methodology helps us solve the dependent variables at different node points as an analytical function of time. This is a powerful technique and is valid for all linear parabolic PDEs. This... 

Numerical Method ofidnesfor Parabolic PDEs with Linear Boundary... 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

Numerical Method of Lines for Parabolic PDEs with Nonlinear Boundary... 

In section 5.2.4, a stiff nonlinear PDE was solved using numerical method of lines. This stiff problem was handled by calling Maple s stiff solver. The temperature explodes after a certain time. The numerical method of lines (NMOL) technique was then extended to coupled nonlinear parabolic PDEs in section 5.2.5. By comparing with the analytical solution, we observed that NMOL predicts the behavior accurately. 

Separation of Variables for Parabolic PDEs with Homogeneous Boundary Conditions... 

Solve the following parabolic PDE using the separation of variables method ... 

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. 

Solve the following simple parabolic PDE using the Laplace transform technique ... 

First order hyperbolic PDEs were solved numerically in section 10.1.5. Second order hyperbolic PDEs are usually specified with boundary conditions at x = 0 and X = 1. In addition, initial conditions for both the dependent variable and its time derivative are specified. The methodology is very similar to numerical method of lines for parabolic PDEs described in chapter 5.2. The only difference is that instead of a system of first order ODEs, second order hyperbolic PDEs result in a system of second order ODEs. The resulting system of second order ODEs is solved numerically in time. The methodology is illustrated with the following examples. 

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