# zbMATH — the first resource for mathematics

Eigenvalues of Sturm-Liouville problems with discontinuity conditions inside a finite interval. (English) Zbl 1124.65066
The author considers an eigenvalue problem of Sturm-Liouville type, where a discontinuity of the solution is specified at one point inside the domain of the two-point boundary value problem. For a detailed analysis, the solution is split into a left and a right part with respect to the discontinuity. Bounds related to each part are proved. A regularised sampling method has been introduced for Sturm-Liouville problems with parameter-dependent boundary conditions in the previous work by the author [Math. Comput. 74, No. 252, 1793–1801 (2005; Zbl 1080.34010)]. This method represents an improvement of techniques based on Shannon’s sampling theory, since any integration is avoided. Thus high-order approximations are achieved at low costs.
In the paper at hand, this approach is applied to the Sturm-Liouville problem with discontinuity. Used functions of a Paley-Wiener space exhibit a representation by a specific series due to the Whitaker-Shannon-Kotel’nikov theorem. A bound on the truncation error of a finite series can be given. Since the exact eigenvalues represent the squares of the zeros of some characteristic function, an approximation is obtained by using the truncated series in this function. Consequently, the author shows a bound on the approximation error of the constructed method. Finally, a brief numerical simulation of a test problem is presented, which verifies that high-order approximations are achieved.

##### MSC:
 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations 34B24 Sturm-Liouville theory 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
Full Text:
##### References:
 [1] Amirov, R. Kh., On sturm – liouville operators with discontinuity conditions inside an interval, J. math. anal. appl., 317, 163-176, (2006) · Zbl 1168.34019 [2] Boumenir, A.; Chanane, B., Eigenvalues of S-L systems using sampling theory, Appl. anal., 62, 323-334, (1996) · Zbl 0864.34073 [3] Chanane, B., Computation of the eigenvalues of sturm – liouville problems with parameter dependent boundary conditions using the regularized sampling method, Math. comput., 74, 252, 1793-1801, (2005), (published electronically S 0025-5718(05)01717-5 in 2005) · Zbl 1080.34010 [4] Chanane, B., High order approximations of the eigenvalues of sturm – liouville problems with coupled self-adjoint boundary conditions, Appl. anal., 80, 317-330, (2001) · Zbl 1031.34030 [5] Chanane, B., High order approximations of the eigenvalues of regular sturm – liouville problems, J. math. anal. appl., 226, 121-129, (1998) · Zbl 0916.34029 [6] Dunford, N.; Schwartz, Linear operators, part III, (1971), Wiley Interscience New York · Zbl 0243.47001 [7] Edmunds, E.E.; Evans, W.D., Spectral theory and differential operators, (1987), Clarendon/Oxford University Press New York · Zbl 0628.47017 [8] Hinton, D.; Schaefer, P.W., Spectral theory and computational methods of sturm – liouville problems, (1997), Marcel Dekker, Inc. [9] M.A. Naimark, Linear Differential Operators, Part I, Ungar, 1968. · Zbl 0227.34020 [10] Pryce, J.D., Numerical solution of sturm – liouville problems, (1993), Oxford Science Publications, Clarendon Press · Zbl 0795.65053 [11] Zayed, A.I., Advances in shannonâ€™s sampling theory, (1993), CRC Press · Zbl 0868.94011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.