Difference equation linear

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. 

The resulting finite difference equations constitute a set of nonho-mogeneous linear algebraic equations. Because there are three dependent variables, the number of equations in the set is three times the number of material points. Obviously, if a large number of points is required to accurately represent the continuous elastic body, a computer is essential. 

Deterministic Randomness. On the one hand, equation 4.8 is a trivial linear difference equation possessing an equally as trivial solution for each initial point Xq Xn = 2"a o (mod 1). Once an initial point is chosen, the future iterates are determined uniquely. As such, this simple system is an intrinsically deterministic one. On the other hand, look again at the binary decimal expansion of a randomly selected a o- This expansion can also be thought of as a particular semi-infinite sequence of coin tosses. 

The starting point in more a detailed exploration is the simplest systems of linear algebraic equations, namely, difference equations with special matrices in simplified form, for example, with tridiagonal matrices. 

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,... 

Difference schemes as operator equations. After replacing differential equations by difference equations on a certain grid we obtain a system of linear algebraic equations that can be written in matrix form. The outcome of this is... 

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... 

The books by Gelfand (1967), Samarskii and Nikolaev (1989) cover in full details the general theory of linear difference equations. Sometimes the elimination method available for solving various systems of algebraic equations is referred to, in the foreign literature, as Thomas algorithm and this... 

Lax, P. and Richtmyer, R. (1956) A servey of stability of linear finite difference equations. Comm. Pure Appl. Mathem., 9, 267-293. 

This equation is the equivalent of Eq. (9-12) for the induced dipole model but has one important difference. Equation (9-13), the derivative of Eq. (9-12), is linear and standard matrix methods can be used to solve for the p. because Eq. (9-12) is a quadratic function of p , while Eq. (9-54) is not a quadratic function of d and thus matrix methods are usually not used to find the Drude particle displacements that minimize the energy. 

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 1. A linear difference equation involves no products or other nonlinear functions of the dependent variable and its differences. The first and third examples are linear, while the second example is nonlinear. 

B is the magnetic field strength and, as always, A denotes an isotopic difference. Equation 8.9 shows that isotopes of different masses will move in paths with radii R /R = (M /M)1/2. For a 180° sector like that in Fig. 8.6 the two isotopes are separated linearly by... 

In the present variation-perturbation calculations the first order corrections were expanded in 600-term ECG basis defined in equations (15) and (16). The components of the polarizability were computed from equation (11) using the optimized The optimization was performed separately for each component and intemuclear distance. The values of aj, (co) are arithmetic sums of the plus and minus components (equation (12)) computed from two separate first-order corrections. For a given component v (either or ), and are expanded in the same basis but, because they are solutions to two different equations (equation (9)) they differ in the linear expansion coefficients. The computed components of the static polarizability an(/ ) and a R) are drawn in Fig. 2 and their numerical values at selected intemuclear distances are listed in Table 1. 

In our numerical model, Eq.(2.8) was transformed into a six-point finite-difference equation using the alternative direction implicit method (ADIM). At the edges of the computational grid (—X,X) radiation conditions were applied in combination with complex scaling over a region x >X2, where —X X j) denotes the transverse computational window. For numerical solution of the obtained tridiagonal system of linear equations, the sweep method" was used. 

We know from Proposition 1 that the constant term Bq C vanishes at the thermodynamic equilibrium. Some features of Equation (67) similar to the known LHHW-kinetic equation. There is a "potential term" Bq responsible for thermodynamic equilibrium, there is a "denominator" of the polynomial type. However there is also a big difference. Equation (67) includes the term D, which is generated by the non-linear steps. 

Substituting the expressions for differences Akyi, k = 1,2,..., m, one can modify it to an mth order linear difference equation related to an unknown y ... 

This partial difference equation is of the first order, due to the fact that the coefficients in (6.4) were linear in the y. The characteristics of (6.12) are determined by... 

The accuracy of this difference scheme is 0(Ar2 + Aa 2), provided the integrals entering (5.1.7) and (5.1.8) are calculated with accuracy of 0(Aa 2). The difference equations are solved first from the left to the right with the boundary condition gjq = 0 (no correlations at r > (). Similar to the discussed scheme (5.1.18), non-linear effects are taken here into account iteratively. 

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