Substitution Equation-solving methods

Two quadratic equations in two variables can in general be solved only by numerical methods (see Numerical Analysis and Approximate Methods ). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. 

The technique consists of substituting iai for s in the characteristic equation and solving for the values of co and other parameters (e.g., controller gain) that satisfy the resulting equations. The method is best understood by looking at the example below. 

Serghides compared nine explicit approximation formulas. Various formulas gave good results but none exactly compatible with the Colebrook equation. Here is a new strategy to solve this equation. The method uses direct substitution repeatedly the results are quite accurate. 

In Chapter 3, we discussed the use of the method of substitution and the method of elimination to solve the linear inhomogeneous set of two simultaneous equations ... 

In the previous subsection, the successive substitution and Wegstein methods were introduced as the two methods most commonly implemented in recycle convergence units. Other methods, such as the Newton-Raphson method, Broyden s quasi-Newton method, and the dominant-eigenvalue method, are candidates as well, especially when the equations being solved are highly nonlinear and interdependent. In this subsection, the principal features of all five methods are compared. 

The elimination method basically involves the elimination of variables in such a way that the final equation will involve only one variable. The procedure for a set of N equations is as follows. First, from one equation solve for x, as a function of other variables, X2, X3,..., Substitute this x, into the remaining N - I equations to obtain a new set of N - 1 equations with N - I unknowns, X2, X3,..., x,s[- Next, using one of the equations in the new set, solve for X2 as a function of other variables, X3, X4,..., and then substitute this X2 into the remaining N - 2 equations to obtain a new set of N - 2 equations in terms of N - 2 unknown variables. Repeat the procedure until you end up with only one equation with one unknown, x, from which we can readily solve for x, . Knowing x, we can use it in the last equation in which X j was written in terms of x y. Repeat the same procedure to find x,. The process of going backward to find solutions is called back substitution. 

Substituting equation (5) into equation (2), multiplication by the internal basis functions and integration over (Vy, a, 3,0y, (l)y) gives a set of close-coupled equations in the translational f functions [23]. These can be solved by a variety of methods. We prefer to use the R-matrix method of Light and Walker [22]. 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

The method of lines lies midway between analytical and grid methods, and can also be used to solve the conservation equations. This method involves substituting finite differences for the derivatives with respect to one independent variable and retaining the derivatives with respect to the remaining variables. This approach replaces a given differential equation by a system of differential equations with a smaller number of independent variables, typically reducing a partial differential equation to a set of ordinary differential equations. 

The first illustrative problem comes from quantum mechanics. An equation in radiation density can be set up but not solved by conventional means. We shall guess a solution, substitute it into the equation, and apply a test to see whether the guess was right. Of course it isn t on the first try, but a second guess can be made and tested to see whether it is closer to the solution than the first. An iterative routine can be set up to cany out very many guesses in a methodical way until the test indicates that the solution has been approximated within some narrow limit. 

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

These Uj may be solved for by the methods under Numerical Solution of Linear Equations and Associated Problems and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). 

Ashton solved this problem approximately by recognizing that the differential equation, Equation (5.32), is but one result of the equilibrium requirement of making the total potential energy of the mechanical system stationary relative to the independent variable w [5-9]. An alternative method is to express the total potential energy in terms of the deflections and their derivatives. Specifically, Ashton approximated the deflection by the Fourier expansion in Equation (5.29) and substituted it in the expression for the total potential energy, V ... 

A set of n first-degree equations in n unknowns is solved in a similar fashion by multiplication and addition to eliminate n - 1 unknowns and then back substitution. Second-degree equations in 2 unknowns may be solved in the same way when two of the following are given the product of the unknowns, their sum or difference, the sum of their squares. For further solutions, see Numerical Methods. ... 

In order to see how accurate this perturbation treatment actually is, we have substituted numerical values for the S s directly into the secular equation, and then solved it rigorously by numerical methods. The calculations are not given in detail, since they are quite straightforward and proceed along well-known lines. The results are shown in Table I. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The method of false transients cannot be used to find a metastable steady state. Instead, it is necessary to solve the algebraic equations that result from setting the derivatives equal to zero in Equations (5.29) and (5.30). This is easy in the current example since Equation (5.29) (with daout/dr = Q) can be solved for Uout- The result is substituted into Equation (5.30) (with dTout/dt = Q) to obtain a single equation in a single unknown. The three solutions are... 

Euler s method for solving the above set of ODEs uses a first-order, forward difference approximation in the -direction. Equation (8.16). Substituting this into Equation (8.21) and solving for the forward point gives... 

To determine cos one should solve the set of f integral equations for probabilities of degeneration u 0(r),...,u f 1 (r) and substitute these functions into functional 0) [u] ( q. 62). Numerical solution of these equations by means of the iteration method presents no difficulties since the integral operator is a contrac-... 

In doing numerical density problems, you may always use the equation d = m/V or the same equation rearranged into the forms V = m/d or m = dV. You are often given two of these quantities and asked for the third. You will use the equation d = m/V if you are given mass and volume, but if you are given density and either of the others, you probably should use the factor-label method. That way, you need not manipulate the equation and then substitute you can solve immediately. 

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