Nonlinear, Coupled, Boundary-Value Problem

In this section a spreadsheet solution is developed to solve the problem discussed in Section 6.3.2. This sheet uses a uniform mesh of just 21 points, which provides a reasonably 

A1 - B3 These cells define the extent of the domain Z = 3, the number of mesh points JJ — 21, and the mesh spacing Az = Z/(JJ — 1). Cell B3 has the formula =Z end/ (JJ-1). The values in cells B1-B3 are defined have the names in cells A1-A3 (using the command INSERTJSfAME DEFINE). 

A5 - D5 This row simply enters labels for the columns that will contain the solutions, z, u,V,du/dz. 

A6 Enter the formula =Z.end, which places the value of the extent of the domain. Here it is chosen to be Z = 3. 

A7 Enter the formula =A6-dz, which computes the z value of the next lower mesh point. 

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Numerical Solution Equations 6.40 and 6.41 represent a nonlinear, coupled, boundary-value system. The system is coupled since u and V appear in both equations. The system is nonlinear since there are products of u and V. Numerical solutions can be accomplished with a straightforward finite-difference procedure. Note that Eq. 6.41 is a second-order boundary-value problem with values of V known at each boundary. Equation 6.40 is a first-order initial-value problem, with the initial value u known at z = 0. 

Weakly coupled systems of nonlinear elliptic boundary value problems (with K. Zygour-akis). Nonlin. Anal. 6, 555-569 (1982). 

Hence, the two sources of nonlinearity in Eq. (54) are moisture concentration c and dilatational strain Consequently, Eqs. (48) and (54) are coupled. The diffusion boundary-value problem must be solved in conjunction with the nonlinear viscoelasticity boundary-value problem by using an iterative procedure. Finite-element formulation of Eq. (54) is standard (see Reddy(47)) and is given by... 

Monotone iteration methods for solving coupled systems of nonlinear boundary value problems (with K. Zygourakis). Comput. Chem. Eng. 7, 183-193 (1983). 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

Spatial coordinates and integrating numerically in time. In this chapter, we apply finite differences in one of the directions (x), convert the governing equation and boundary conditions in x to finite difference form. The resulting system of coupled nonlinear boundary values problems (second order ordinary differential equations in y) are then solved using Maple s dsolve numeric command for boundary value problems (see chapter 3.2.8). 

This application of the finite difference method to Poisson s equation as applied to semiconductors provides only a glimpse of how the code routines can be applied to nonlinear boundary value problems. However, not all problems of interest involve only a single differential equation. Many problems of interest involve equations of higher order than two or involve systems of coupled second order differential equations. Such problems are the subject of the next section where algorithms and code segments are discussed and developed for these more general... 

In the general case, eqs 4 and 5 constitute a system of nonlinear coupled second-order partial differential equations. To specify the boundary conditions for this problem, it is necessary to include the external (interphase) heat and mass transfer, as both the concentration and the temperature at the external surface of the catalyst pellet may differ from the corresponding values in the bulk of the surrounding fluid phase. 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

The latter strongly depends on the specific reaction mechanism, the stoichiometry, and the presence or absence of parallel reaction schemes (69). The rate expressions for Rt usually represent nonlinear dependences on the mixture composition and temperature. Specifically for the coupled reaction-mass transfer problems, such as Eqs. (A10), it is always essential as to whether or not the reaction rate is comparable to that of diffusion (68,77). Equations (A10) should be completed by the boundary conditions relevant to the film model. These conditions specify the values of the mixture composition at both film boundaries. For example, for the liquid phase ... 

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