Difference equations

If we select a fixed place x, then a particle moves within a time step from a rear place to forward, with the probability p. Exactly in the same way a particle moves from X + Ax with the probability q backward, thus to the grid point at x. This phenomenon is described by the difference equation 

We expand now Eq. (21.9) into a Taylor series in space and time and obtain 

Equation (21.11) is known as Fokker - Planck equation [5]. We observe the similarity with the equation of continuity, if the variance approaches zero. Or else, we realize the similarity with the equation of diffusion. In fact, the motion is a dispersive process. Also the Schrodinger equation belongs to this type of equations. The Schrodinger equation deals with the energy, p, has the dimension of a velocity. 

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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. 

J lie starting point is the standard expression for the free energy difference. Equation (11.6)... 

The rate of decay of " k is 5.543 x 10" ° yr. For any time difference ((), equation (1) relates the amount of existing now to that existing some time ago (in the present case, when the rocks were formed). 

The formulation step may result in algebraic equations, difference equations, differential equations, integr equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. 

A difference equation is a relation between the differences and the independent variable, A y, A " y,. . . , Ay, y, x) = 0, where ( ) is some given function. The general case in which the interval between the successive points is any real number h, instead of I, can be reduced to that with interval size I by the substitution x = hx. Hence all further difference-equation work will assume the interval size between successive points is I. 

The order of the difference equation is the difference between the largest and smallest arguments when written in the form of the second example. The first and second examples are both of order 2, while the third example is of order I. A hnear difference equation involves no... 

A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitraty constants. The techniques for solving difference equations resemble techniques used for differential equations. 

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. 

Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx 2 + A y -1- = ( )(x). If there... 

Substitution If it is possible to rearrange a difference equation so that it takes the form af oy, o + hf 1 -1- cfy, = ( )(x) with a, b, c constants, then the substitution =fxyx reduces the equation to one with constant coefficients. 

Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation. 

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. 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Since this equation introduces a new variable, Cq, another equation is needed and is obtained by writing the finite difference equation for = 1, too. 

If average diffusion coefficients are used, then the finite difference equation is as follows. 

We denote by C the value of c(x , t) at any time. Thus, C is a function of time, and differential equations in C are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2,. . . , n (see Fig. 3-52). 

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. 

For large temperature differences different equations are necessary ana usually are specifically applicable to either gases or liquids. Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed review of high-flux heat transfers to gases. He recommends... 

Later work by Hashem and Shepcevich [Chem. E/ig. Prog., 63, Symp. Sen 79, 35, 42 (1967)] offers more accurate second-order finite-difference equations. 

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial different equation of the second order Ficldan model, requires two boundaiy conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundaiy condition cf 0 as oo. Breakthrough behavior presumes the existence of... 

GD of the load, as referred to the motor speed, will be different. Equating the work done at the two speeds ... 

If the stress is at the primary time step loeation and the veloeities are at the middle of the time step, then the resulting finite-difference equation is second-order accurate in space and time for uniform time steps and elements. If all quantities are at the primary time step, then a more complicated predictor-corrector procedure must be used to achieve second-order accuracy. A typical predictor-corrector scheme predicts the stresses at the middle of the time step and uses them to calculate the divergence of the stress tensor. 

Lax, P.D., Hyperbolic Difference Equations A Review of the Courant-Friedrichs-Lewy Paper in the Light of Recent Developments, IBM J., 235-238 (1967). 

If we had taken the ease where the strength exeeeds the applied stress, this would have yielded a slightly different equation as shown below ... 

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