Differential Equation Solution Techniques

Physics texts often introduce spherical harmonics by applying the technique of separation of variables to a differential equation with spherical symmetry. This technique, which we will apply to Laplace s equation, is a method physicists use to hnd solutions to many differential equations. The technique is often successful, so physicists tend to keep it in the top drawer of their toolbox. In fact, for many equations, separation of variables is guaranteed to find all nice solutions, as we prove in Proposition A.3. 

The present chapter provides an overview of several numerical techniques that can be used to solve model equations of ordinary and partial differential type, both of which are frequently encountered in multiphase catalytic reactor analysis and design. Brief theories of the ordinary differential equation solution methods are provided. The techniques and software involved in the numerical solution of partial differential equation sets, which allow accurate prediction of nonreactive and reactive transport phenomena in conventional and nonconventional geometries, are explained briefly. The chapter is concluded with two case studies that demonstrate the application of numerical solution techniques in modeling and simulation of hydrocar-bon-to-hydrogen conversions in catalytic packed-bed and heat-exchange integrated microchannel reactors. 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

Differential methods - in these techniques the internal grid coordinates are found via the solution of appropriate elliptic, parabolic or hyperbolic partial differential equations. 

Dyna.micPerforma.nce, Most models do not attempt to separate the equiUbrium behavior from the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

Mathematical and Computational Implementation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Comparing the two sets of equations governing pollutant dynamics (eq. 5) and meteorology (eqs. 12—14) shows that in both cases they can be put in the form ... 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundaiy conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. 

A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitraty constants. The techniques for solving difference equations resemble techniques used for differential equations. 

D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations, J. Comput. Phys. 95 (1991). 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

We now consider the solution of differential equations by means of Laplaee transforms. We have already solved one equation, namely, the first-order rate equation, but the technique is capable of more than this. It allows us to solve simultaneous differential equations. 

The treatment in this chapter has been theoretical. For a brief, dear, and very practical description of computational details for a number of standard problems, [10] is unsurpassed, and [12] can be recommended for programming techniques for automatic computers. For information on ordinary differential equations, the reader should consult [2], and for partial differential equations, [1]. For general methods of reduction to algebraic form as well as methods of solution, see [5], [7], and [8]. 

The charging of the double layer is responsible for the background (residual) current known as the charging current, which limits die detectability of controlled-potential techniques. Such a charging process is nonfaradaic because electrons are not transferred across the electrode-solution interface. It occurs when a potential is applied across the double layer, or when die electrode area or capacitances are changing. Note that the current is the tune derivative of die charge. Hence, when such processes occur, a residual current flows based on die differential equation... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

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