# eigenfunction expansion

## eigenfunction expansion

[′ī·gən‚fəŋk·shən ik′span·chən]
(mathematics)
By using spectral theory for linear operators defined on spaces composed of functions, in certain cases the operator equals an integral or series involving its eigenvectors; this is known as its eigenfunction expansion and is particularly useful in studying linear partial differential equations.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In SEM, the eigenfunction expansion method (EEM) is used as discretization method, where the eigenfunctions are those of the Laplacian -[DELTA].
Figure 12 shows a comparison between the resulting scattering amplitude for the electric field obtained by the eigenfunction expansion method and the FDTD simulation for varying values of the homogeneous plasma parameters.
, on the other hand, introduced exact solutions using the eigenfunction expansion and Laplace transform techniques.
This shows that the off-diagonal terms in the eigenfunction expansion contribute less in the infinite time asymptotic regime.
Most of the obtained results are analogous for the ones of regular Sturm-Liouville eigenvalue problems and they open the door for establishing other results such as the countability of eigenfunctions and completeness of eigenfunctions which are essential in solving fractional differential equations by fractional eigenfunction expansion.
These analytical methods include Green's function method, the Laplace transform, separation of variables, and eigenfunction expansion method.
Additionally, the solution is verified using four different techniques: the WHE with numerical Pickard's iterations, WHEP with numerical estimation, analytical eigenfunction expansion solved using Mathematica, and the Monte-Carlo simulations (MCS).
Comparing these results with literature example on two spheres and eigenfunction expansion , where N=10 instances were used; we find that the method of mirror images is still competitive and has similar convergence behaviour.
All chapters have been revised and updated for this edition, which has an expanded introduction to Green's functions, discussion of the eigenfunction expansion method and sections on the convergence speed of series solutions and the importance of alternate GF, a section on intrinsic verification, new examples and figures, a new chapter on steady-periodic heat conduction, and new appendices on the Dirac delta function, the Laplace transform, and properties of common materials.
Keywords: Time scale, delta and nabla derivatives and integrals, Green's function, completely continuous operator, eigenfunction expansion.
In turn, this provides us with the eigenfunction expansion

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