The Solution of Algebraic Equations

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... 

The application of the reverse Euler method of solution to a system of coupled differential equations yields a system of coupled algebraic equations that can be solved by the method of Gaussian elimination and back substitution. In this chapter I demonstrated the solution of simultaneous algebraic equations by means of this method and showed how the solution of algebraic equations can be used to solve the related differential equations. In the process, I presented subroutine GAUSS, the computational engine of all of the programs discussed in the chapters that follow. 

In physics and chemistry it is not possible to develop any useful model of matter without a basic knowledge of some elementary mathematics. This involves use of some elements of linear algebra, such as the solution of algebraic equations (at least quadratic), the solution of systems of linear equations, and a few elements on matrices and determinants. 

Of course, the complete integral of a differential equation applies to any physical process represented by the differential equation. This solution, however, may be so general as to be of little practical use. To represent any particular process, certain limitations called, limiting conditions have to be introduced. These exclude certain forms of the general solution as impossible.1 We met this idea in connection with the solution of algebraic equations, page 363. 

For general case, the IAS equations must be solved numerically and this is quite effectively done with standard numerical tools, such as the Newton-Raphson method for the solution of algebraic equations and the quadrature method for the evaluation of integral. We shall develop below a procedure and then an algorithm for solving the equilibria problem when the gas phase conditions (P, yj are given... 

Modelling is useful only if it can provide information which is not readily available, or difficult to obtain. This information can usually be obtained from the solution of the modelling equations or from the specific points which characterize these equations. It is worth remembering that the solutions of algebraic equations are numbers, whereas the solutions of differential and integral equations are functions. These specific points can be ... 

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