Nonlinear Parabolic Partial Differential Equations

We have demonstrated the method of finite differences for solving linear parabolic partial differential equations. But the utility of the numerical method is best appreciated when we deal with nonlinear equations. In this section, we will consider a nonlinear parabolic partial differential equation, and show how to deal with the nonlinear terms. 

Let us again consider the same problem of mass transfer through a membrane of Example 12.5 but this time we account for the fact that diffusivity is a function of the intramembrane concentration. The nondimensional equation can be written as 

As in Example 12.5, if we divide the domain [0,1] into N intervals, evaluate the above equation at the ( -I- l/2)th time, and use the second order correct formula for both time and space derivatives, we have the following finite difference formulas for the derivatives 

The first equation conies from Eq. 12.149, the second from Eq. 12.153, and the last is the Crank-Nicolson analog for the first spatial derivative, similar in form to the second spatial derivative Eq. 12.153 derived earlier. 

We have dealt with the derivative terms in Eq. 12.162, let us now turn to dealing with fiy) and f iy). Evaluating these at the discrete point t and time ( + 1/2), we have 

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The model used to describe the metal deposition process during hydrodemetallisation includes a system of nonlinear parabolic partial differential equations (PDEs) in one space variable (5, 6). These equations were solved numerically with a CONVEX 3840 workstation. Subroutines to solve the set of PDEs are obtained from the NAG Fortran library (1988) (11). 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab or the material in which the experiment is performed. This process is usually time varying and eventually reaches a steady state. This process is represented by parabolic partial differential equations with known initial conditions and boundary conditions at two ends. Both linear and nonlinear parabolic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear parabolic partial differential equations and numerical solutions for nonlinear parabolic partial differential equations based on the numerical method of lines. 

Consider a general nonlinear parabolic partial differential equation in dimensionless form... 

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 ... 

The evolution of polymer composition in the spatial domain can be derived using the Cahn-Hilliard equation. In numerical simulations, the fourth-order nonlinear parabolic partial differential equations are solved using Fourier-spectral methods, while the partial differential equations are transferred by the discrete cosine transform into ordinary partial equations. The result is then transformed back with the inverse cosine transform to the ordinary space. 

Prom the mathematical point of view equations (8.5.1) and (8.5.10) with definitions (8.5.3)-(8.5.5) form a system of nonlinear parabolic partial differential equations of the Reaction -I- Diffusion t3q>e, describing a wide range of physical phenomena that are of a great importance for thin films growth dynamics. 

Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or boundary value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

In section 4.4, the given linear parabolic partial differential equation in semi-infinite domain was solved by combining the independent variables (similarity solution). This technique is capable of providing special function solutions as shown in example 4.9. In section 4.5, elliptic partial differential equations were solved using the similarity solution technique. In section 4.6, similarity solution was extended for nonlinear parabolic and elliptic partial differential equations. 

The probability density of the response state vector of a nonlinear system under the excitation of Gaussian white noises is governed by a parabolic partial differential equation, called the Fokker-Planck equation. Exact solutions to such equations are difficult especially when both parametric (multiplicative) and external (additive) random excitations are present. In this paper, methods of solution for response vectors at the stationary state are discussed under two schemes based on the concept of detailed balance and the concept of generalized stationary potential, respectively. It is shown that the second scheme is more general and includes the first scheme as a special case. Examples are given to illustrate their applications. 

Let us consider the genera class of systems described by a system of n nonlinear parabolic or hyperbolic partial differential equations. For simplicity vve assume that we have only one spatial independent variable, z. 

The circumferential ((j>) momentum equation is a partial differential equation. Identify some of its basic properties. Is it elliptic, parabolic, or hyperbolic Is it linear of nonlinear What is its order ... 

Finally, substitution of the reformulated expressions for ion fluxes in material balance Eq. (2) results, for the spherically symmetric particle, in the two nonlinear parabolic type partial differential equations that follow in reduced form ... 

Nonlinear parabolic and elliptic partial differential equations are solved using the similarity solution technique in this section. The methods described in section 4.4 and sections 4.5 are valid for nonlinear partial differential equations, also. The methodology involves converting the governing equation (PDE) to an ordinary differential equation in the combined variable (Tj). This variable transformation is very difficult to do by hand. In this section, we will show how this variable transformation can be done using Maple. The original problem becomes a nonlinear boundary value problem (ODE) in the new combined variable (Tj). This is best illustrated with the following examples. 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

Mathematical models of catalytic systems in the general form are rather sophisticated. Often, they consist of nonlinear systems of differential equations containing both conventional equations and equations with partial derivatives of parabolic, hyperbolic, and other forms. Efficient simulation is only possible if a well developed qualitative theory of differential equations (mainly, equations with partial derivatives) and high performance programs for computational experiments exist. 

The method of Vishik-Lyusternik is well developed not only for elliptic but also for parabolic and hyperbolic partial differential equations. Some nonlinear problems can be solved using this method as well. The main advantage of this method is its simplicity. Usually the boundary layer functions are defined as solutions of ordinary differential equations in which the independent variable is the stretched distance along the normal to the boundary (the variable p in our case). However, there are some problems when the boundary layer functions are described by more complicated equations, for example, by parabolic equations. A survey of results obtained by the Vishik-Lyusternik method with many references can be found in [28]. 

Such a classification can also be applied to higher order equations involving more than two independent variables. Typically elliptic equations are associated with physical systems involving equilibrium states, parabolic equations are associated with diffusion type problems and hyperbolic equations are associated with oscillating or vibrating physical systems. Analytical closed form solutions are known for some linear partial differential equations. However, numerical solutions must be obtained for most partial differential equations and for almost all nonlinear equations. 

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