Solving Linear and Nonlinear Equations

This appendix contains a brief summary of methods of solving linear and nonlinear equations. It is only a summary for details consult one of the numerous texts on numerical analysis that can be found in any library. 

Simple FORTRAN programs have been prepared for the reader s use that solve linear and nonlinear equations, retrieve the properties of water and steam, and of air-water mixtures, calculate the vapor pressure of pure substances, calculate enthalpy changes from heat capacity equations, and so on. A disk containing these codes will be found in a pocket in the back of the book. (Readers are encouraged to use library codes when available, codes that may be more accurate and robust than the simple codes provided.) As a result, the portions of the book formerly treating... 

One can solve equations in Maple using the solve and fsolve commands. The solve command is used to solve linear equations in symbolic form and the fsolve command is used to solve linear and nonlinear equations numerically. For example,... 

Under all but laminar flow conditions, the steady-state pipeline network problems are described by mixed sets of linear and nonlinear equations regardless of the choice of formulations. Since these equations cannot be solved directly, an iterative procedure is usually employed. For ease of reference let us represent the steady-state equations as... 

In the pocket in the back of this book are four simple Fortran generic codes that can be used to solve sets of linear and nonlinear equations on microcomputers or mainframes. Because these programs are simple, and thus may on occasion fell to solve your problem, you may want to use more polished computer codes that are available in your computer center software library. Such codes are more robust, but of course it takes more of your time to understand how to use them proficiently. 

In section 3.2.3, finite difference solutions were obtained for nonlinear boundary value problems. This is a straightforward and easy technique and can be used to obtain an initial guess for other sophisticated techniques. This technique is important because it forms the basis for the method of lines technique for solving linear and nonlinear partial differential equations (chapter 5 and 6). However, for stiff boundary value problems, this technique may not work and might demand prohibitively large number of node points. In addition, approximate initial guess should be provided for all the node points for stiff boundary value problems. 

The numerical method of lines was used to solve linear and nonlinear elliptic partial differential equations in section 6.1.7. This method involves using finite differences in one direction and solving the resulting system of boundary value problems in y using Maple s dsolve numeric command. This method provides a numerical solution for both the dependent variables and its derivative in the y-direction. 

Diffusion and mass transfer in multicomponent systems are described by systems of differential equations. These equations are more easily manipulated using matrix notation and concepts from linear algebra. We have chosen to include three appendices that provide the necessary background in matrix theory in order to provide the reader a convenient source of reference material. Appendix A covers linear algebra and matrix computations. Appendix B describes methods for solving systems of differential equations and Appendix C briefly reviews numerical methods for solving systems of linear and nonlinear equations. Other books cover these fields in far more depth than what follows. We have found the book by Amundson (1966) to be particularly useful as it is written with chemical engineering applications in mind. Other books we have consulted are cited at various points in the text. 

The different equations encountered in mathematical modeling can be further classified as linear and nonlinear equations. Linear equations arise in systems where the unknowns in the equations are present in the first power. Linear equations enjoy the principle of superposition, i.e., the sum of the solutions is also a solution of the equations. Linearity allows the original problem to be partitioned into simpler component problems that can be solved separately and superimposed to obtain the solution to the original problem. 

This is a particularly interesting feature that is widely used along the examples. It is the tool we can use within Excel to solve numerically a set of equations, problon optimization including fitting a set of data to a given linear and nonlinear equation and more. Solver is an add-in that needs to be activated to be used. For enabling it, we need to click on Office Button (top-left comer), then Excel Options, and in the Tab Adds-Ins, click on bottom go (Manage Excel Adds-Ins) there, look for... 

How to solve linear and nonlinear algebraic equations as well as ordinary differential equations... 

Armed with techniques for solving linear and nonlinear algebraic systems (Chapters 1 and 2) and the tools of eigenvalue analysis (Chapter 3), we are now ready to treat more complex problems of greater relevance to chemical engineering practice. We begin with the study of initial value problems (IVPs) of ordinary differential equations (ODEs), in which we compute the trajectory in time of a set of N variables Xj(t) governed by the set of first-order ODEs... 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

The Gustafson-Holden equation is a unique approach that allows both linear and nonlinear datasets to be solved since it is based on a gamma distribution. The equation is first order and has three unknowns (a, b and c) ... 

Directly following the development of the optical laser, in 1961 Frankel et al. [10] reported the first observation of optical harmonics. In these experiments, the output from a pulsed ruby laser at 6943 A was passed through crystalline quartz and the second harmonic light at 3472 A was recorded on a spectrographic plate. Interest in surface SHG arose largely from the publication of Bloembergen and Pershan [11] which laid the theoretical foundation for this field. In this publication, Maxwell s equations for a nonlinear dielectric were solved given the boundary conditions of a plane interface between a linear and nonlinear medium. Implications of the nonlinear boundary theory for experimental systems and devices was noted. Ex-... 

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 steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

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. 

Analyses of physiochemical systems often give us a set of linear alge-braie equations. Also, methods of solution of differential equations and nonlinear equations use the technique of linearizing the models. This requires repetitive solutions of sets of linear algebraic equations. Linear equations can vary from a set of two to a set having 100 or more equations. In most cases, we can employ Cramer s rule to solve a set of two or three linear algebraic equations. However, for systems of many linear... 

A variety of names are used in the literature to describe calculational procedures for solving sets of linear and nonlinear algebraic equations such as iteration, successive iteration, and successive substitution. 

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