Partial differential equations the finite differences method

3 Partial differential equations the finite differences method 

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The model equations (9.10)-(9.14) are a group of non-linear partial differential equations. The finite difference method is applied to the partial derivative terms with respect to the radius in the equations, for example... 

The preliminary steps for solving partial differential equations by finite difference methods is similar to the last section. We shall now illustrate this for parabolic partial differential equations. 

Abstract Chapter 5 provides an examination of the numerical solutions of the dyeing models that can be applied to different conditions. Numerical simulation of the system involves the use of Matlab software to solve systems of highly non-linear simultaneous coupled partial differential equations. The finite difference and finite element methods are introduced The partition of the fibrous assembly geometry into small units of a simple shape, or mesh, is examined. Polygonal shapes used to define the element are briefly described. The defined geometries, boundary conditions, and mesh of the system enable solutions to the equations of flow or mass transfer models. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Mitchell, A. and Griffits, D. (1980) The Finite Difference Methods in Partial Differential Equations. Wiley New York. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

The numerical methods for partial differential equations can be classified according to the type of equation (see Partial Differential Equations ) parabolic, elliptic, and hyperbolic. This section uses the finite difference method to illustrate the ideas, and these results can be programmed for simple problems. For more complicated problems, though, it is common to rely on computer packages. Thus, some discussion is given to the issues that arise when using computer packages. 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

Most of these tools use the finite difference method (at least for one-dimensional models) in which the continuous space coordinate is divided into a number of boxes. So we are back to the box-model technique. To demonstrate the procedure, in Box 23.4 we show how the partial differential Eqs. 23-44 and 23-45 are transformed into discrete (box) equations. 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced By approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, tinite difference, finite volume, and finite element methods are the most common. The finite difference method estimates spatial derivatives in terms of the nodi values and spacing between nodes. The governing equations are then written in terms of... 

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