Numerical analysis differential equation solutions

Integral Equation Solutions. As a consequence of the quasi-steady approximation for gas-phase transport processes, a rigorous simultaneous solution of the governing differential equations is not necessary. This mathematical simplification permits independent analytical solution of each of the ordinary and partial differential equations for selected boundary conditions. Matching of the remaining boundary condition can be accomplished by an iterative numerical analysis of the solutions to the governing differential equations. 

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. 

For one special case, isothermal reaction at constant density, the set of differential equations comprising the time derivatives of all species concentrations is sufficient to determine the evolution of a system described by a given reaction mechanism for any assumed starting concentrations. While this special case does apply for some experiments of interest in combustion research, it does not pertain to the conditions under which most combustion processes occur. Usually we must expand our set of differential equations so as to describe the effects of chemical reaction on the physical conditions and the effects of changes in the physical conditions on the chemistry. In either case, the evolution of the system is found by numerical integration of the appropriate set of ordinary differential equations with a computer. This procedure is known in the language of numerical analysis as the solution of an initial value problem. 

In the final analysis, of greatest importance from the viewpoint of numerical analysis is the design of algorithms permitting one to obtain a solution of a differential equation on a computer with a prescribed accuracy in a finite number of operations. The user can encounter in this connection the question of the quality of an algorithm, that is, the manner in which the accuracy of the algorithm depends on... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Stone, H.L., "Iterative Solution of Implicit Approximations of Multi-Dimensional Partial Differential Equations", SIAM J. Numerical Analysis, 5, 530-558 (1968). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

On a more positive note, multimedia enable us to respond pedagogically to the shifting paradigms of science, where, as remarked above, numerical analysis is replacing analytical solutions of equations. Through it, we can present computational chemistry, for instance, and pursue the fascinating complexity that arises from systems of coupled differential equations in chemical kinetics. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

Analysis of the algorithm requires understanding the behavior of local truncation error. We use the nomenclature y(tn) to mean the exact analytic solution evaluated at some time tn and yn to represent the numerical solution at tn. It is clear that the true analytic solution must satisfy the differential equation everywhere,... 

Tphe breakthrough curve for a fixed-bed adsorption column may be pre-dieted theoretically from the solution of the appropriate mass-transfer rate equation subject to the boundary conditions imposed by the differential fluid phase mass balance for an element of the column. For molecular sieve adsorbents this problem is complicated by the nonlinearity of the equilibrium isotherm which leads to nonlinearities both in the differential equations and in the boundary conditions. This paper summarizes the principal conclusions reached from a recent numerical solution of this problem (1). The approximations involved in the analysis are realistic for many practical systems, and the validity of the theory is confirmed by comparison with experiment. 

Thoulouze-Pratt, E., 1983, Numerical analysis of the behaviour of an almost periodic solution to a periodic differential equation, an example of successive bifurcations of invariant tori. In Rhythms in Biology and Other Fields of Application, Lect. Notes in Biomath, Vol. 49, pp. 265-271. 

The task of modeling is to obtain valid scalar, differential, or other type of equations (integral, integro-differential, etc) that describe a given physical system accurately and efficiently in mathematical terms. Numerical analysis and computations then lead us to the solution values or to the solution function(s) themselves from the model equations. 

The value of the parameter n in Eqs. (9.13)—(9.14) is different from unity, and these differential equations are nonlinear and cannot be solved analytically. Therefore, Eqs. (9.13)-(9.16) subject to the conditions of Eqs. (9.17)-(9.21) were solved using numerical analysis techniques. Selim et al. (1976a) used the explicit-implicit finite-difference approximations as the method of solution. This was successfully used by Selim et al. (1975) for steady water flow conditions and by Selim et al. (1976a) for transient... 

From a mathematical point of view the PCM models can be unified according to the approach they use to solve the linear partial differential equations determining the electrostatic interactions between solute and solvent. This analysis is presented by Cances who reviews both the mathematical and the numerical aspects of such an integral equation approach when applied to PCM models. 

Irvine, D. and Savageau, M., Efficient solution of nonlinear ordinary differential equations expressed in S-system canonical form, SIAM Journal of Numerical Analysis, Vol. 27, 1990, pp. 704-735. 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

---