Difference equations approximating wave equation

When an equation describes a system exactly but the equation cannot be solved, there are two general approaches that are followed. First, if the exact equation cannot be solved exacdy, it may be possible to obtain approximate solutions. Second, the equation that describes the system exactly may be modified to produce a different equation that now describes the system only approximately but which can be solved exactly. These are the approaches to solving the wave equation for the helium atom. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

A procedure for obtaining an approximate wave function of a molecular orbital by treating the molecule as different canonical forms. The overall wave expression is thus the weighted sum of the wave equation for each of these canonical forms. 

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

Before turning to the applications of the Debye approximation, we should elaborate more fully on a point that was glossed over. This is the assumption —made at the outset, but explicated in going from Equation (58) to Equation (59) —that the scattering behavior of each scattering element is independent of what happens elsewhere in the particle. The approximation that the phase difference between scattered waves depends only on their location in the particle and is independent of any material property of the particle is valid as long as... 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

It is interesting to compare the digital waveguide simulation technique to the recursion produced by the finite difference approximation (FDA) applied to the wave equation. Recall from (10.10) that the time update recursion for the ideal string digitized via the... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

The Eulerian finite difference scheme aims to replace the wave equations which describe the acoustic response of anechoic structures with a numerical analogue. The response functions are typically approximated by series of parabolas. Material discontinuities are similarly treated unless special boundary conditions are considered. This will introduce some smearing of the solution ( ). Propagation of acoustic excitation across water-air, water-steel and elastomer-air have been computed to accuracies better than two percent error ( ). In two-dimensional calculations, errors below five percent are practicable. The position of the boundaries are in general considered to be fixed. These constraints limit the Eulerian scheme to the calculation of acoustic responses of anechoic structures without, simultaneously, considering non-acoustic pressure deformations. However, Eulerian schemes may lead to relatively simple algorithms, as evident from Equation (20), which enable multi-dimensional computations to be carried out in a reasonable time. 

It is usually found that where there is a large difference between the calculated and the observed values of heats of formation, the calculated value of - J Hf is less than the observed that is, the actual molecule is more stable than the hypothetical molecule consisting of normal bonds. This difference has been ascribed to stabilization by resonance between a variety of valence bond structures, and is sometimes known as the observed or empirical resonance energs The concept of resonance is derived from a procedure used for obtaining approximate solutions of the wave equation. Its use is therefore a matter of convenience rather than theoretical necessity. Moreover it is often applied to systems of such complexity that no question of even an approximate solution of the wave equation arises in these cases its status is therefore that of an empirically used concept similar to the earlier notion of mesomerism . 

The essential property of all these variances is that they can only be zero if the wave equation is satisfied at all (Rj). In practice, the valuable use for these is to assess two approximate wavefunctions such as Ca<1>a and CbOb- These could be results obtained either by different workers or in one calculation at different stages of the iteration. The comparison of two such results by an independent method would be very informative. If... 

Although simple pairing of the AO s on different atoms suggests MO forms which account surprisingly well for the general properties of many diatomic molecules, its limitations soon become apparent. It must be remembered that the best MO s are solutions of a wave equation and that simple LCAO forms are rather rough approximations only. However, by building an MO out of a number of AO s instead of just a pair, a better approximation can be obtained. It will appear later that this refinement is often quite indispensible, even in qualitative descriptions. 

The Schrodinger equation provides a way to obtain the A-electron wave function of the system, and the approximate methods described in the previous section permit reasonable approaches to this wave function. From the approximate wave function the total energy can be obtained as an expectation value and the different density matrices, in particular the one-particle density matrix, can be obtained in a straighforward way as... 

Despite its strong resemblance to the Schroedinger equation, this equation encompasses veiy different physics the wave function P(x) does not describe a single-particle state in the usual sense, but is rather a building block for the (approximate) ground-state A -particle wave function... 

R. L. Higdon, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Math. Comput., vol. 47, pp. 437-459, 1986. doi 10.2307/2008166... 

In discussing many problems which cannot be directly solved, a solution can be obtained of a wave equation which differs from the true one only in the omission of certain terms whose effect on the system is small. Perturbation theory provides a method of treating such problems, whereby the approximate equation is first solved and then the small additional terms are introduced as corrections ... 

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