Polynomial algebraic equations

A very valuable technique, useful in the solution of ordinary and partial differential equations as well as differential delay equations, is the use of Laplace transforms. Laplace transforms (Churchill, 1972), though less familiar and somewhat more difficult to invert than their cousins, Fourier transforms, are broadly applicable and often enable us to convert differential equations to algebraic equations. For rate equations based on mass action kinetics, taking the Laplace transform affords sets of polynomial algebraic equations. For DDEs, we obtain transcendental equations. 

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. 

Any algebraic equation may be written as a polynomial of nth degree in x of the form... 

This relation is equivalent to an algebraic equation of degree n in the unknown X and therefore has n roots, some of which may be repeated (degenerate). These roots are the characteristic values or eigenvalues of the matrix B, When the determinant of Eq. (69) is expanded, the result is the polynomial equation... 

POLYMATH. AIChE Cache Corp, P O Box 7939, Austin TX 78713-7939. Polynomial and cubic spline curvefitting, multiple linear regression, simultaneous ODEs, simultaneous linear and nonlinear algebraic equations, matrix manipulations, integration and differentiation of tabular data by way of curve fit of the data. 

The data of (C, t) are fitted by a polynomial or other algebraic equation and differentiated to obtain the derivative. The constants then are found from a linear plot of In r = In k + a In C. Figure 3.1 shows this plot also. 

We have found that dynamics can be more conveniently handled in the Russian transfer-function language than in the English ODE language. However, the manipulation of the algebraic equations becomes more and more difficult as the system becomes more complex and higher in order, if the system is th-order, an Afth-order polynomial in s must be factored into its N roots. For N greater than 2, we usually abandon analytical methods and turn to numerical... 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

Note how much simpler this problem is to solve in the CSTR than in the PFTR (Figure 4—6), where we had to solve simultaneous differential equations. The CSTR involves only simultaneous algebraic equations so we just need to find roots of polynomials.]... 

An equation of the form f(x)=0 where f is a polynomial, algebraic number... 

Thus we have derived an algebraic equation of A -th degree, which has, in accordance with the fundamental theorem of algebra, N roots. However, sometimes it is not as important to calculate the roots as to analyse their properties, e.g. to find out how many positive, negative, zero roots the equation has, and so on. In such cases we have to find the coefficients of the characteristic polynomial. But how ... 

The that satisfy this Eq. A.4.8 may be complex, since an algebraic equation with real coefficients may have complex conjugate pairs of roots, or they may be complex since the polynomial may have complex coefficients if the matrix [yl] has nonreal elements. 

A/j for the components (C//4, H O, H2, CO, CO2) at the collocation points (B.41-B.45) are found by a subroutine called FLUX. The subroutine FLUX evaluates ch4 Xcm the collocation points by solving the set of 2N linear algebraic equations (B.41-B.42) — excluding the centre of the pellet where the fluxes are known — by Gauss elimination with partial pivoting using the subroutine called GAUSL (Villadsen and Michelsen, 1978). The rest of the fluxes of the components are found from the stoichiometric equations (5.215). The roots (Uj) of the Jacobi polynomial (w) and the discretization... 

Mathematica can carry out both symbolic and numerical solutions of equations, including single algebraic equations, simultaneous algebraic equations, and differential equations, which we discuss later. Mathematica contains the rules needed for the symbolic solution of polynomial equations up to the fourth degree, and can... 

The derivative p must vanish at /O = /O3 and at /O = pi. A possible candidate for the front solution is p = b(p - p ) p - P3), where d is a constant to be determined. Substituting this solution into (4.22), we obtain a cubic polynomial in p. Setting the coefficients to 0 provides a system of four algebraic equations. The equation for the coefficient of p yields b = /rj2D. With this value, the equations for the... 

The coefficients of this polynomial are determined from the system of algebraic equations... 

Since the CSTR equation is simply a system of nonlinear algebraic equations, it is possible to obtain multiple CSTR steady states for a fixed feed concentration and residence time. For example, if there are polynomial terms in r(C), then more than one concentration will exist as roots to the equation. Even if these roots do not bear any physical significance to the system, they must still be known in order for the appropriate construction of the AR to be carried out. More complex expressions are also valid (and common) in modern-day rate expressions. Be wary of this when attempting to solve for CSTR solutions. The presence of multiple steady states presents... 

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