Linear differential time operator

It is sometimes useful to write this equation in an operator form O, using the linear differential time operator, d=dldt, i.e.,... 

In addition, the time dependence of the solution, meaning the exponential function, arises from the left hand side of Eq. (2-2), the linear differential operator. In fact, we may recall that the left hand side of (2-2) gives rise to the so-called characteristic equation (or characteristic polynomial). 

From a mathematical point of view, a random evolution is an operator 0(t,t) that is improved at both t and T times. The linear differential equation is Eq. (4.90) ... 

Laplace transforms can be used to transform this system of linear differential equations in the time domain into a system of linear equations in the Laplace domain. From the table of Laplace operations (Appendix I) we obtain... 

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... 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

The vector Fc is a complex representation of the real field F. If all our operations on time-harmonic fields are linear (e.g., addition, differentiation, integration), it is more convenient to work with the complex representation. The reason this may be done is as follows. Let be any linear operator we can operate on the field (2.10) by operating on the complex representation (2.11) and then take the real part of the result ... 

When one looks into the basic functions of the link and indirect response models, it is clear that one of the differences resides in the input functions to the effect and the receptor protein site, respectively. For the link model a linear input operates in contrast to the indirect model, where a nonlinear function operates. For the link model the time is not directly present and the pharmacological time course is exclusively dictated by the pharmacokinetic time, whereas the indirect model has its own time expressed by the differential equation describing the dynamics of the integrated response. 

This important equation is known as the Klein-Gordon equation, and was proposed by various authors [6, 7, 8, 9] at much the same time. It is, however, an inconvenient equation to use, primarily because it involves a second-order differential operator with respect to time. Dirac therefore sought an equation linear in the momentum operator, whose solutions were also solutions of the Klein-Gordon equation. Dirac also required an equation which could more easily be generalised to take account of electromagnetic fields. The wave equation proposed by Dirac was [10]... 

A proper numbering of the equations and unknowns ensures that the matrix representing the linear part of the differential operator will be a band matrix with bandwidth proportional to the lesser of m and n. Standard methods for decomposing such matrices exist (19), which allow savings in both storage and time. 

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