Solution of algebraic equation

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

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. 

One can always fmd solutions of algebraic equations of the form F (a ) = F2(x) by plotting F] and versus x and determining intersections (Figure 6-2). For the situation on the left there is only one solution xj, while for the situation on the right there are three solutions X, X2, and X3... 

Newton s method is a procedure for the numerical solution of algebraic equations, applicable to any number M of such equations expressed as functions of M variables. 

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. 

Section 3.3 Numerical Solution of Algebraic Equations TABLE 3.2 > Mathematical Functions in Mathematics... 

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. 

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