Difference equation method

The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. 

The time-dependent Smith-Ewart differential difference equations methods available for their solution... 

The Time-Dependent Smith-Ewart Differential Difference Equations Methods Available for Their Solution. 

A reasonable approach for achieving long timesteps is to use implicit schemes [38]. These methods are designed specifically for problems with disparate timescales where explicit methods do not usually perform well, such as chemical reactions [39]. The integration formulas of implicit methods are designed to increase the range of stability for the difference equation. The experience with implicit methods in the context of biomolecular dynamics has not been extensive and rather disappointing (e.g., [40, 41]), for reasons discussed below. 

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

Variable Coejftcients The method of variation of parameters apphes equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. 

The term operational method implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a veiy powerful tool in the applications of mathematics, but it is hmited to linear problems. 

This equation must be solved for y The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL(Ref. 224). 

McAdams (Heat Transmission, 3d ed., McGraw-HiU, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availabihty of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. 

These results are the same as with the power series method, but difference equations are more suited to digital computation. 

Direct application of the differential equation is perhaps the simplest method of obtaining kinetic parameters from non-isothermal observations. However, the Freeman—Carroll difference—differential method [531] has proved reasonably easy to apply and the treatment has been expanded to cover all functions f(a). The methods are discussed in a sequence similar to that used in Sect. 6.2. 

Numerical methods have been developed by replacing the differential equation by a finite difference equation. Thus in a problem of unidirectional flow of heat ... 

The most general method of tackling the problem is the use of the finite-element technique 8 to determine the temperature distribution at any time by using the finite difference equation in the form of equation 9.40. 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

In the present section a direct method for solving the boundary-value problems associated with second-order difference equations will be the subject of special investigations. 

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

Solution estimation for difference bormdary-value problems by the elimination method. In tackling the first boundary-value problem difference equation (21) has the tridiagonal matrix of order TV — 1... 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

The difference boundary-value problem associated with the difference equation (7) of second order can be solved by the standard elimination method, whose computational algorithm is stable, since the conditions Ai 0, Ci > Ai -f Tj+i are certainly true for cr > 0. 

By means of the integro-interpolation method it is possible to construct a homogeneous difference scheme, whose design reproduces the availability of the heat source Q of this sort at the point x = /. This can be done using an equidistant grid u)j and accepting / = x -f Oh, 0 <0 < 0.5. Under such an approach the difference equation takes the standard form at all the nodes x [i n). In this line we write down the balance equation on the segment x,j. 

In this chapter the new difference schemes are constructed for the quasilin-ear heat conduction equation and equations of gas dynamics with placing a special emphasis on iterative methods available for solving nonlinear difference equations. Among other things, the convergence of Newton s method is established for implicit schemes of gas dynamics. 

In this regard, Newton s method suits us perfectly in connection with solving the nonlinear difference equation (2). It is worth recalling here its algorithm ... 

Numerical solution of difference equations by Newton s method. As... 

Generally speaking, Newton s method may be employed for nonlinear difference equations on every new layer, but the algorithm of the matrix elimination for a system of two three-point equations (see Chapter 10, Section 1) suits us perfectly for this exceptional case. We will say more about this later. 

The main idea behind this approach is to accelerate and simplify the algorithms by means of the method of. separate or successive eliminations. To that end, the difference equations (60) are divided into the following groups ... 

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