Generalized Fourier Series as Green’s Function Expansion for Multi-interval Sturm–Liouville Systems

Author(s):  
K. Aydemir ◽  
O. Sh. Mukhtarov
2012 ◽  
Vol 535-537 ◽  
pp. 1354-1358
Author(s):  
Jian Xiang Gao ◽  
Fang Cheng Zhang

The integral theory for electromagnetic theory of gratings, which has complex mathematical tool and long computation cost, is needed to improve computing speed of the algorithm. A very shot-cut equation, which is called symmetrical equation of diffracted wavevectors by us, is presented. It makes the number of Green’s function expansion terms shorter and simplifies the integral kernel by taking into account its symmetry. Take the grating in Littrow mount for computing example, because of applying this equation, the computation cost of integral theory which has good convergence, is shorten to 1/2 to 2/5. This work modifies the integral theory and improves its computing speed.


2011 ◽  
Vol 2011 ◽  
pp. 1-30 ◽  
Author(s):  
M. M. Tharwat

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).


2009 ◽  
Vol 82 (6) ◽  
pp. 756-772 ◽  
Author(s):  
Omar M. Sallah ◽  
L. J. Gray ◽  
M. A. Amer ◽  
M. S. Matbuly

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
K. Aydemir ◽  
O. Sh. Mukhtarov

The purpose of this paper is to investigate some spectral properties of Sturm-Liouville type problems with interior singularities. Some of the mathematical aspects necessary for developing our own technique are presented. By applying this technique we construct some special solutions of the homogeneous equation and present a formula and the existence conditions of Green's function. Furthermore, based on these results and introducing operator treatment in adequate Hilbert space, we derive the resolvent operator and prove self-adjointness of the considered problem.


2022 ◽  
Vol 7 (4) ◽  
pp. 4887-4897
Author(s):  
Youyu Wang ◽  
◽  
Xianfei Li ◽  
Yue Huang

<abstract><p>By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.</p></abstract>


Author(s):  
S. Angel Auxzaline Mary ◽  
T. Ramesh

In this paper, we describe Green's function to determine the importance of this function, i.e. Boundary & Initial Value problem, Sturm-Liouville Problem. Along with the series representation of Green's Function.


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