Coupled Linear Differential Equations

Beside the obvious trivial solution to the above equation, there exists some value of A such that the solution is nonzero. In such a case, the value of A is called the eigenvalue and the solution vector x corresponding to that eigenvalue is called the eigenvector. The problem stated by Eq. B.78 is then called the eigenproblem. 

The eigenproblem of Section B.7 is useful in the solution of coupled linear differential equations. Let these equations be represented by the following set written in compact matrix notation 

The general solution of linear differential equations is a linear combination of a homogeneous solution and a particular solution. For Eq. B.79, the particular solution is simply the steady-state solution that is 

The general solution is the sum of the homogeneous solution and the particular solution, that is, 

Fadeev, D. K., and V. N. Fadeeva, Computational Methods of Linear Algebra, Freeman, San Francisco (1%3). 

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The coefficients of the coupled linear differential equations of Box 21.6 become ... 

This Master Equation is turned into a finite set of coupled linear differential equations by truncating the energy space at some high value that is only visited very infrequently, and graining the energy. 

SO that the diagonal components contain minus the total rate constant out of minimum i. This definition allows us to write the set of coupled linear differential equations (1,37)—the master equation —in matrix form ... 

Since the reaction involves two linear independent steps, two wavelengths of observation have to be used. One has to solve two coupled linear differential equations by formal integration ... 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

The evolution of the density matrix in time requires the solution of the equation of motion, Eq. (10) or (20) in a basis set of N states, this represents a system of coupled linear differential equations for the individual density matrix elements. It is most natural to consider the equation in Liouville space, where ordinary operators (N x N matrices) are treated as vectors (of length N ) and superoperators such as and which act on operators to create new operators, become simple matrices (of size X N ). In the Liouville space notation, Eq. (9) would... 

Substitution of equation (51) into equation (50) gives a set of four coupled linear differential equations, whose solutions can be obtained by standard procedures. We note that once we have obtained a, t) we recover... 

In this section, we will consider briefiy the eigenproblems, that is, the study of the eigenvalues and eigenvectors. Ilie study of coupled linear differential equations presented in the next section requires the analysis of the eigenproblems. 

Modern research has concentrated on elucidating the nature of L. If one has an equation representative of Eq. (3), questions abound concerning L, questions more subtle than the obvious what is it Equation (3) is irreversible in time, while any exact equation for A (t) must be reversible. This dilemma could not be resolved with irreversible thermodynamics. Also, consider that Eq. (3) may represent a set of coupled, linear differential equations. Suppose a certain set of variables A must be taken together to obtain proper dynamics. Then, if one tried to apply the principles of irreversible thermodynamics with a set smaller than the true set, an incorrect dynamical law would result no matter what was used for L. How was one to know the correct set of A Finally, the obvious question does occur Given that it is reasonable to write an irreversible equation and given that one has a proper set of variables, how does one find an expression for L ... 

Since this is a set of coupled linear differential equations, they can in principle be solved analytically. However, for more than a few spatial regions or energy groups, such a solution is too cumbersome because of the large order determinants involved. 

The set of equations (4.5), written for the various values of n, constitutes a set of coupled linear differential equations. The coupling between these equations arises solely from the existence of the perturbation 5T, which has non-zero off-diagonal matrix elements. Assume that the system is in an eigenstate of say Ej, at t=0. Thus, with the probability interpretation of the wavefunction we have... 

The 3x3 transposed Jacobian matrix = Va/ , evaluated at the stagnation point To, has real elements. It represents a nonsymmetric tensor in the absence of molecular point group symmetry. Within the linear approximation [91], only the first term in the expansion is considered and the description of the field about a stagnation point amounts to solving a system of three coupled linear differential equations whose corresponding matrix is given by the transposed Jacobian matrix. 

Some discussions of dielectric relaxation are framed in terms of normal modes of a polymer coil. The notion of normal modes of polymer chains may be traced back to the Rouse(40) and pre-averaged Zimm(41) models. These treatments are often described as bead-and-spring models, because they describe individual monomers or groups of adjoining monomers as beads, and the highly oversimplified connections between the beads as Hookean springs. Mathematically, these models have in common that they give coupled linear differential equations for the positions of the beads. 

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