Difference equations linear finite

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. 

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

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. 

Consequently, there are 4 independent variables m], mj, ipp and ipj for each icelike layer j, and a system of 4 linear equations with finite differences (eqs 7a—7d) is obtained. We try solutions of the type80... 

Cook and Moore35 studied gas absorption theoretically using a finite-rate first-order chemical reaction with a large heat effect. They assumed linear boundary conditions (i.e., interfacial temperature was assumed to be a linear function of time and the interfacial concentration was assumed to be a linear function of interfacial temperature) and a linear relationship between the kinetic constant and the temperature. They formulated the differential difference equations and solved them successively. The calculations were used to analyze absorption of C02 in NaOH solutions. They concluded that, for some reaction conditions, compensating effects of temperature on rate constant and solubility would make the absorption rate independent of heat effects. 

B. A. Luty, M. E. Davis, and J. A. McCammon,/. Comput. Chem., 13, 1114 (1992). Solving the Finite-Difference Non-Linear Poisson—Boltzmann Equation. 

A powerful tool for EM modeling and inversion is the integral equation (IE) method and the corresponding linear and nonlinear approximations, introduced in the previous chapter. One important advantage which the IE method has over the finite difference (FD) and finite element (FE) methods is its greater suitability for inversion. Integral equation formulation readily contains a sensitivity matrix, which can be recomputed at each inversion iteration at little expense. With finite differences, however, this matrix has to be established anew on each iteration at a cost at least equal to the cost of the full forward simulation. 

The system of linear equations originating from the difference equation (2.308) has to be supplemented by the difference equations for the points around the boundaries where the decisive boundary conditions are taken into account. As a simplification we will assume that the boundaries run parallel to the x- and y-directions. Curved boundaries can be replaced by a series of straight lines parallel to the x- and y-axes. However a sufficient degree of accuracy can only be reached in this case by having a very small mesh size Ax. If the boundaries are coordinate lines of a polar coordinate system (r, differential equation and its boundary conditions are formulated in polar coordinates and then the corresponding finite difference equations are derived. 

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

We use here the Neumann stability analysis [57], which is the most widely used procedure for the determination of the stabihty of a calculation scheme using a finite difference approximation. In this stability analysis, an initial error is introduced as a finite Fourier series and one studies the growth or decay of this error during the calculation. The Neumann method applies only to initial value problems with a periodical initial condition it neglects the influence of the bormd-ary condition, and it is applied only to linear finite difference approximations with constant coefficients, i.e., to linear equations. This method gives only a necessary condition for the stability of a munerical procedure. It turns out, however, that this condition is sufficient in many cases. 

Convert the governing equation to finite difference form by using central difference expression accurate to the order h for the first and second derivatives in the spatial variable, x (equation (6.11)). This gives raise to N second order linear ODEs in This system of second order equations is converted to 2N first order linear ODEs in as described in equation (6.12). The variable ui(Q, i = 0..N-I-1 corresponds to the dependent variable, ui at node point i. 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

Some engineering problems give sets of simultaneous linear equations that are diagonally dominant. This involves the computation of finite difference equations, derived from the approximation of partial differential equations. A diagonally dominant system of linear equations has coefficients on the diagonal that are larger in absolute value than the sum of the absolute values of the other coefficients. 

This is a linear finite-difference equation in the independent variable j Its solutions can be found by comparing it with the trigonometric identity,... 

In this chapter, we develop analytical solution methods, which have very close analogs with methods used for linear ODEs. A few nonlinear difference equations can be reduced to linear form (the Riccati analog) and the analogous Euler-Equidimensional finite-difference equation also exists. For linear equations, we again exploit the property of superposition. Thus, our general solutions will be composed of a linear combination of complementary and particular solutions. 

SOLUTION METHODS FOR LINEAR FINITE DIFFERENCE EQUATIONS... 

The general linear finite difference equation of kth order can be written just as we did in Section 2.5 for ODE... 

Solution Methods for Linear Finite Difference Equations 167 5.2.1 Complementary Solutions... 

In solving ODE, we assumed the existence of solutions of the form y = A exp(/%) where r is the characteristic root, obtainable from the characteristic equation. In a similar manner, we assume that linear, homogeneous finite difference equations have solutions of the form, for example, in the previous extraction problem... 

Very few nonlinear equations yield analytical solutions, so graphical or trial-error solution methods are often used. There are a few nonlinear finite difference equations, which can be reduced to linear form by elementary variable transformation. Foremost among these is the famous Riccati equation... 

Thus, our finite difference equations are Eq. 12.123, Eq. 12.120b for i = 1,7),..., N - 1, and Eq. 12.124, totalling N equations, and we have exactly N unknown discrete variables to be found (y, y2,..., y y). Since the starting equation we choose is linear, the resulting set of N equations are linear algebraic equations, which can be handled by the usual matrix methods given in Appendix B. The matrix formed by this set of linear equations has a special format, called the tridiagonal matrix, which will be considered in the next section. 

To summarize the finite difference method, all we have to do is to replace all derivatives in the equation to be solved by their appropriate approximations to yield a finite difference equation. Next, we deal with boundary conditions. If the boundary condition involves the specification of the variable y, we simply use its value in the finite difference equation. However, if the boundary condition involves a derivative, we need to use the fictitious point which is outside the domain to effect the approximation of the derivative as we did in the above example at x =. The final equations obtained will form a set of algebraic equations which are amenable to analysis by methods such as those in Appendix A. If the starting equation is linear, the finite difference equation will be in the form of tridiagonal matrix and can be solved by the Thomas algorithm presented in the next section. 

In Chapter 5, we examine the problem concerning the reducibility of a linear system of difference equations with quasiperiodic coefficients to a linear system of difference equations with constant coefficients. We consider the case when the right-hand side of the equation is not analytic but has a finitely many derivatives. 

These methods can be applied both to the simplified model with constant s, and to models with space dependent s(r )or real charges dispersed in the whole dielectric medium. They aim at solving the Poisson equation (1) expressed as a set of finite difference equations for each point of the grid. The linear system to be solved has elements depending both on e... 

All factors on the right-hand side of Eq. (10.3-20) are constant. This equation is a linear first-order difference equation and can be solved by the calculus of finite-difference methods (Gl. M1). The final derived equations are as follows. 

The application of a finite-difference method transforms the first derivatives V[u L)k point Sk into a linear combination of function values y(u L),m at contiguous points Sm around s. This yields two linear difference equations for every inner grid point s. The resulting set of linear equations can be combined into a 2m X 2m matrix equation... 

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