A Generalized Eigenfunction Expansion of the Green's Function

1965 ◽  
Vol 16 (3) ◽  
pp. 348
Author(s):  
Maurice Machover
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Eigenfunction Expansion ◽  
Generalized Eigenfunction ◽  
Generalized Eigenfunction Expansion

scholarly journals Discontinuous Sturm-Liouville Problems and Associated Sampling Theories

2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
M. M. Tharwat
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Eigenfunction Expansion ◽  
Green’S Functions ◽  
Transmission Conditions ◽  
Expansion Theorem ◽  
Sampling Theorems ◽  
Sturm Liouville ◽  
Special Case

This paper investigates the sampling analysis associated with discontinuous Sturm-Liouville problems with eigenvalue parameters in two boundary conditions and with transmission conditions at the point of discontinuity. We closely follow the analysis derived by Fulton (1977) to establish the needed relations for the derivations of the sampling theorems including the construction of Green's function as well as the eigenfunction expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green's functions. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in the work of Annaby and Tharwat (2006).

Closed-Form Transient Response of Distributed Damped Systems, Part I: Modal Analysis and Green’s Function Formula

1996 ◽  
Vol 63 (4) ◽  
pp. 997-1003 ◽  
Author(s):  
Bingen Yang
Keyword(s):  
Green’S Function ◽  
Modal Analysis ◽  
Closed Form ◽  
Distributed System ◽  
Transient Response ◽  
Green's Function ◽  
Eigenfunction Expansion ◽  
Distributed Parameter ◽  
Modal Expansion ◽  
Damped Systems

An analytical method is developed for closed-form estimation of the transient response of complex distributed parameter systems that are nonproportionally damped, and subject to arbitrary external, initial, and boundary excitations. A new modal analysis leads to the Green’s function formula for the distributed system and an eigenfunction expansion of the system Green’s function. The legitimacy of the modal expansion is also shown.

Time-dependent linear water-wave scattering in two dimensions by a generalized eigenfunction expansion

2009 ◽  
Vol 632 ◽  
pp. 447-455 ◽  
Author(s):  
MICHAEL H. MEYLAN
Keyword(s):  
Eigenfunction Expansion ◽  
Water Wave ◽  
Wave Scattering ◽  
Initial Conditions ◽  
Two Dimensions ◽  
Single Frequency ◽  
Initial Surface ◽  
Water Wave Scattering ◽  
Generalized Eigenfunction ◽  
Generalized Eigenfunction Expansion

We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations.

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