Methods with Constant Coefficients

10 Numerical Illustrations for the Methods with Constant Coefficients and the Variable-Step Methods 

Method MI P-stable fourth order method Simos and Raptis.64 

Method MIV P-stable sixth order method of Simos67 with phase-lag of order eight. 

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Since we care for methods with constant coefficients, we expand the two algebraic conditions into their Taylor series over 0, where 0 = wh, w is the frequency, h is the step-length. 

Theory for Constructing Methods with Constant Coefficients for the Numerical Solution of the Schrodinger Type Equations... 

Remarks and Conclusion. - The most accurate fourth order P-stable method with constant coefficients is the P-stable method proposed by Simos66 with phase-lag of order sixteen. The most accurate sixth order method with constant coefficients is the modification of the family of sixth order methods of Simos69 with phase-lag of order twenty-two developed in this review. The most accurate eighth order method with constant coefficients is the eighth order method of Simos77 with phase-lag of order eighteen and interval of periodicity... 

Table 6 The basic characteristics for the dissipative methods with constant coefficients...

New Developments on Numerical Methods with Constant Coefficients and on the Methods with Coefficients Dependent on the Frequency of the Problem... 

Methods with Constant Coefficients (Generators of Numerical Methods). 

In this section we consider the one-dimensional heat conduction equation with constant coefficients and difference schemes in order to develop various methods for designing the appropriate difference schemes in the case of time-dependent problems. 

Compartmental analysis is the most widely used method of analysis for systems that can be modeled by means of linear differential equations with constant coefficients. The assumption of linearity can be tested in pharmaeokinetic studies, for example by comparing the plasma concentration curves obtained at different dose levels. If the curves are found to be reasonably parallel, then the assumption of linearity holds over the dose range that has been studied. The advantage of linear... 

The calculating procedure is based on sub-division of the Arctic Basin into grids (Eijk. This is realized by means of a quasi-linearization method (Nitu et al., 2000a). All differential equations of the SSMAE are substituted in each box E by easily integrable ordinary differential equations with constant coefficients. Water motion and turbulent mixing are realized in conformity with current velocity fields which are defined on the same coordinate grid as the E (Krapivin et al., 1998). 

The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),. .. However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations... 

Thus, as an MO basis for constructing the multielectron wave functions of configurations given by Eq. (33), we will use the Hartree-Fock MO for the occupied one-electron states and the Huzinaga MO [Eq. (34)] for the excited ones. An advantage of the Huzinaga MOs is the simplicity of the way they are obtained, since Eq. (34) is an equation with constant coefficients and one does not have to use the iteration method. Moreover, when one uses the Huzinaga... 

In the course of developing models for the impedance response of physical systems, differential equations are commonly encountered that have complex variables. For equations with constant coefficients, solutions may be obtained using the methods described in the previous sections. For equations with variable coefficients, a numerical solution may be required. The method for numerical solution is to separate the equations into real and imaginary parts and to solve them simultaneously. 

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. 

A linear homogeneous differential equation with constant coefficients can be solved by the following routine method ... 

A. Methods for Linear Systems with Constant Coefficients... 

A. METHODS FOR LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS... 

Although there is a large literature on identifiability for linear systems with constant coefficients, less has been done on nonlinear systems. Two general properties should be remembered. Whereas for linear systems one can substitute impulsive inputs for the experimental inputs for die analysis of identifiability, one cannot do that for nonlinear systems. One must analyze the input-output experiment for the actual inputs used. That is a drawback. On the other hand, experience shows dial frequently the introduction of nonlinearities makes a formerly nonidentifiable model identifiable for a given input-output experiment. Two methods are available. 

A very general method for obtaining solutions to second-order differential equations is to expand y(x) in a power series and then evaluate the coefficients term by term. We will illustrate the method with a trivial example that we have already solved, namely the equation with constant coefficients ... 

Classical method of the family is the method of the family with constant coefficients which has... 

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