Linear inhomogeneous

This is an inhomogeneous linear differential equation of second order with constant coefficient a, where g is its right hand side. The parameter a is very small, and it is approximately... 

In genera], Eq. (32) represents a system of inhomogeneous linear equations. It is assumed that A and B are known and the elements of the vector X are the unknowns. For simplicity, the following arguments will be limited to the case in which A is square, that is, n = m. If all elements of the vector B are equal to zero, the equations are homogeneous and Eq. (33) becomes AX=0. 

According to the rule for matrix multiplication introduced earlier, each element of y is calculated as the scalar product between c and the corresponding column of A. These linear operations are represented exactly by the following system of inhomogeneous linear equations ... 

Solving a set of inhomogeneous linear equations amounts to finding the inverse of the coefficient matrix. 

If the total number of basis functions f l for all cells is N,p, then for each global solution index X there are 2N,p equations for the 2N,p elements of the column vectors oj1"- and Thus the variational equations derived from Ea provide exactly the number of inhomogeneous linear equations needed to determine the two coefficient matrices, oj and /I These equations have not yet been implemented, but they promise to provide an internally consistent energy-linearized full-potential MST. 

This is a system of inhomogeneous linear equations for the functions (vectors) T m ) (the mixed notation for the perturbation corrections to eigenvalues and eigenvectors is used above). The 0-th order in A yields the unperturbed problem and thus is satisfied automatically. The others can be solved one by one. For this end we multiply the equation for the first order function by the zeroth-order wave function and integrate which yields ... 

Higher-order NLO coefficients are given by higher-order correction vectors, starting with It satisfies the inhomogeneous linear equation... 

Correction vectors lead to inhomogeneous linear equations of the form Ax = b that involve sparse matrices in the VB representation. An iterative small mar trix method gives rapid convergence using coordinate relaxation that resembles the Davidson algorithm for eigenvalues. We summarize the procedure here [48, 40]. We begin with a set of m orthonormal A-vectors Q,- i = 1... m and construct x the zeroth approximation to the true solution x as,... 

After the first order approximation is introduced through (2.246), (2.245) is said to be an inhomogeneous linear integral equation for The form of is then established without actually obtaining a complete solution. Instead, by functional analysis a partial solution is written in the form (e.g.. Chapman and Cowling [12], sect. 7.3 Hirschfelder et al. [39], chap. 7, sect. 3) ... 

The equilibrium equation of inhomogeneous linear elastic materials with temperature effects is expressed as... 

So far we have considered only the solutions to sets of inhomogeneous linear equations where at least one of the b,- is non-zero. If, however, we have a set of homogeneous equations, where all the bi are zero, then we may define two further possible limiting cases ... 

Method of Solution for Inhomogeneous Equations Given the following inhomogeneous linear equations ... 

S.2 Inhomogeneous Linear Differential Equations with Constant Coefficients... 

Consider an inhomogeneous linear differential equation with constant coefficients given by... 

An inhomogeneous linear differential equation can be solved if a particular solution can be found. 

The Forced Harmonic Oscillator Inhomogeneous Linear Differential Equations... 

The inhomogeneous linear differential Eq. (2.59) is solved numerically using the Runge-Kutta methode of second order with variable time steps (cf. [25]). Scenario for the calculation... 

TABLE 1. Symmetry, numeric codes and structures of selected inhomogeneous linear structure series with Q = 1... 

Grin, Yu.N. Akselrud, LG. (1990). Use of inhomogeneous linear structure series for the crystal-chemiceil description of the superconducting oxides. Acta Cryst. A46 Suppl., C-338. 

In the thin-absorber approximation, an inhomogeneous linear transformation is used to make connection between the vector of the Mossbauer parameters Pm (set up from isomer shifts, quadrupole splittings, magnetic splittings, etc.) and the vector of the peak parameters Ppeak ... 

Here, 7 . was given by the expression in the square bracket of Eq. (3.5). With removing a diagonal one, say = 0, from Eq. (3.7a), the above relations constitute inhomogeneous linear equations that determine peq uniquely. 

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