Difference equation order

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 ... 

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... 

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... 

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. 

So concludes Robert M. May his now famous 1976 Nature review article [may76] of what was then known about the behavior of first-order difference equations of the form... 

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. 

It should be noted here that a second-order difference equation also may... 

Replacing in (1) u[ by fi we cancel 0 h ) and multiply the resulting equation by 2h. As a final result we get the second-order difference equation... 

The first-order difference equations and inequalities. Of our concern is the first-order difference equation... 

The second-order difference equations. The Cauchy problem. Boundary-value problems. The second-order difference equation transforms into a more transparent form... 

The latter difference equation clarifies that (6) is an analog of a second-order differential equation. 

For the second-order difference equations capable of describing the basic mathematical-physics problems, boundary-value problems with additional conditions given at different points are more typical. For example, if we know the value for z = 0 and the value for i = N, the corresponding boundary-value problem can be formulated as follows it is necessary to find the solution yi, 0 < i < N, of problem (6) satisfying the boundary conditions... 

Other ideas are connected with reduction of the original second-order difference equation (9) to three first-order ones, which may be, generally speaking, nonlinear. First of all, the recurrence relation with indeterminate coefficients a,- and f3i is supposed to be valid ... 

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... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

The second-order difference equations with constant coefficients. If... 

The complete posing of a difference problem necessitates specifying the difference analogs of those conditions in addition to the approximation of the governing differential equation. The set of difference equations approximating the differential equation in hand and the supplementary boundary and initial conditions constitute what is called a difference scheme. In order to clarify the essence of the matter, we give below several examples. 

In order to develop a difference equation from (11), we substitute linear combinations of the values of u at the grid nodes in place of w and the integral with u that can be obtained through the interpolations in some neighborhood of the node x. The simplest interpolation gives... 

Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

We now turn to a difference equation of second order. We learn from Chapter 1, Section 1 that any difference equation of second order A, — CiUi + 2/i+i = Fi can be treated as an equation of divergent type,... 

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. 

The intuition suggests that in such a setting the governing difference equation and the boundary condition at the point a = 0 have one and the same order of approximation 0(r + h ). To make sure of it, it suffices only to evaluate the residual... 

Additive schemes. The general formulations and statements. Considerable effort is devoted to a discussion of additive schemes after introducing the notion of summarized approximation. With this aim, we recall the notion of the n-layer difference scheme as a difference equation with respect to t of order n — 1 with operator coefficients ... 

All the equations reported above were derived for first-order reactions with respect to the reactant. The laws change when different reaction orders are involved. In particular, plots of i vs. will be different in shape. At zero reaction order (Fig. 6.9,... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

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