Linear integral equations

The linear integral equation (5) is solved by a standard technique, including expansion of the unknown An z) by some basis functions and transformation of (5) into a matrix equation to... 

Cochran, J. A. The Analysis of Linear Integral Equations, McGraw-Hill, New York (1972). 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

R. Kress Linear Integral Equations (Springer, Berlin, 1989). 

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

If the anharmonic interaction is not weak, then D(t) should be found from the equation of motion, which in case k > 2 turns out to be a non-linear integral equation. To get the corresponding equation, one should start from the equation of motion for the displacement operator(s) q. From equation (5) it follows that... 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

Kress, R., 1999, Linear integral equations Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 365 pp. 

For a whole vector of N travel time observations, one obtains a system of linear integral equations which after quadrature discretization gives a linear system of equations for slowness perturbations As. For practical applications a rectangular grid is generally chosen and the wave slowness is discretized by treating it as constant in the cells determined by the grid. Thus one forms a vector s of L slowness perturbations ... 

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) ... 

This is a linear integral equation (of the first kind) that now can be solved directly, in principle, to determine the unknown surface temperature distribution 9S (x). The main advantage of this formulation, relative to solving the whole problem (11-6) with (11-108) as a boundary condition, is that (11-109) can be solved to determine 9S (x) directly, without any need to determine the temperature distribution elsewhere in the domain. This latter problem is only ID, in spite of the fact that the original problem was fully 2D. 

In this case, we again have a linear integral equation (of the second kind) for the unknown surface temperature distribution. 

The solutions to (11-109) or (11-111) can be carried out either numerically or, because the equations are linear, by an assortment of approximate analytical techniques. An attempt is not made to present these solution methods here. Instead, to preserve space, the interested reader is referred to any of the standard textbooks on linear integral equations. A classic is the book Linear Integral Equations by W. V Lovitt, which is available from Dover in paperback.3... 

W. Y. Lovitt, Linear Integral Equations (Dover, New York, 1950 originally published in 1924 by McGraw-Hill). 

Kress, R., Linear Integral Equations, Berlin Springer-Verlag, 1989. 

In analogy to the Ward Tordai (1946) equation (4.1) the following non-linear integral equation was derived on the basis of these relations (Miller, 1980) ... 

L. Fox, E.T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. R. Soc., Lond. 245A (1953) 501. 

M. A. Krasnoselskii, Topological Methods in the Theory of Non-linear Integral Equations, Pergamon, New York, 1964. 

The immediate result of a spectroscopic ensemble technique is a signal spec-tmm, i.e. the variation of the measured signal g over the spectral parameter (time, space or frequency). Each size fraction v possesses a characteristic spectrum k/s ), which in general covers the whole spectral range. Assuming that each size fraction contributes independently and linearly to the measured signal spectmm, the determination of the size distribution requires the inversion of a linear integral equation (Fredholm type) ... 

Inversion (or deconvolution) the transformation of a signal spectrum to the distribution of a quantity—frequenfly, the signal spectrum can be described as a linear integral equation of the requested distribution function inversion is achieved by numerical algorithms that find a biased guess of the distribution function and minimise its dependency on the measurement error the result is influenced in every instance by the employed inversion algorithm and its parameterisation moreover, the inversion of measurement data cannot reveal all details of the distribution function (i.e. limited information content). 

Fxmctions Adj, Bdj, Fdj and Gdj satisfy the linear integral equations with the linearized operators of elastic collisions and inelastic ones with internal energy transitions. 

The solution to the linear integral equations, Eqs. (6.94) and (6.95), are given in detail in various references, which enables an accurate determination of the functions Aj and B/. Since the solution to linear integral equations is a fairly standard mathematical exercise, we instead turn our attention to the specific forms of the property flux expressions and the final closed forms of the conservation equations. 

Note that the xc potential is still a local potential, albeit being obtained through the solution of an extremely non-local and non-linear integral equation. In fact, the solution of (4.37) poses a very difficult numerical problem. Fortunately, by performing an approximation first proposed by Krieger, Li, and lafrate (KLI) it is possible to simplify the whole procedure, and obtain an semi-anal3dic solution of (4.37) [19]. The KLI approximation turns out to be a very good approximation to the EXX potential. Note that both the EXX and the KLI potential have the correct — 1/r asymptotic behavior for neutral finite systems. 

R. icss Linear Integral Equations (Spnagei ,Jier in, 1989). 

Thus, whereas patchwise heterogeneity generates a linear integral equation, random heterogeneity results in a nonlinear integral equation. Intermediate cases between patchwise heterogeneity and random heterogeneity are also possible. 

Following Friedman [4, chap.8] one can turn the problem of seeking such solutions to the partial differential problem (2.1-6) into an equivalent existence problem for non-linear integral equations of Volterra type. Indeed one can write... 

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