Inverse Nodal Problem For An Integro-Differential Operator

Yıl 2015, Cilt: 12 Sayı: 1, - , 01.05.2015

Emrah Yılmaz

Öz

In this study, we consider an inverse nodal problem of recovering integro-differential operator
with the Sturm-Liouville differential part and the integral part of Volterra type. Furthermore, we obtain a
reconstruction formula for function M. So, we reconstruct the operator L with a dense subset of nodal points
provided that the function q is known. Even if not all nodes are taken as data but a dense subset of nodes,
inverse problem is determined.

Anahtar Kelimeler

Integro-differential operator, nodal points, reconstruction formula

Kaynakça

• [1] J. R. McLaughlin, Inverse Spectral Theory Using Nodal Points as Data-a Uniqueness Result, Journal of Differential Equations, 73, (1988), 354-362.
• [2] C. L. Shen, On the Nodal Sets of the Eigenfunctions of the String Equations, SIAM Journal on Mathematical Analysis, 19, (1988), 1419-1424.
• [3] V. A. Yurko, An Inverse Problem for Integro-differential Operators, Matematicheskie Zametki, 50(5), (1991), 134–146 (Russian); English Translation in Mathematical Notes, 50(5–6), (1991), 1188–1197.
• [4] Y. V. Kuryshova and C. T. Shieh, An Inverse Nodal Problem for Integro-Differential Operators, Journal of Inverse and III-posed Problems, 18, (2010), 357–369.
• [5] O. Hald and J. R. McLaughlin, Solutions of Inverse Nodal Problems, Inverse Problems, 5, (1989), 307-347.
• [6] P. J. Browne and B. D. Sleeman, Inverse Nodal Problem for Sturm-Liouville Equation with Eigenparameter Depend Boundary Conditions, Inverse Problems, 12, (1996), 377-381.
• [7] Y. T. Chen, Y. H. Cheng, C. K. Law and J. Tsay, L1 Convergence of the Reconstruction Formula for the Potential Function, Proceedings of the American Mathematical Society, 130, (2002), 2319-2324.
• [8] C. K. Law, C. L. Shen and C. F. Yang, The Inverse Nodal Problem on the Smoothness of the Potential Function, Inverse Problems, 15(1), (1999), 253-263 (Erratum, Inverse Problems, 17, (2001), 361-363.
• [9] E. Yilmaz and H. Koyunbakan, Reconstruction of Potential Function and its Derivatives for Sturm-Liouville Problem with Eigenvalues in Boundary Conditions, Inverse Problems in Science and Engineering, 18(7), (2010), 935-944.
• [10] H. Koyunbakan and E. Yilmaz, Reconstruction of the Potential Function and its Derivatives for the Diffusion Operator, Verlag der Zeitschrift f¨ur Naturforch, 63(a), (2008), 127-130.
• [11] C. T. Shieh and V. A. Yurko, Inverse Nodal and Inverse Spectral Problems for Discontinuous Boundary Value Problems, Journal of Mathematical Analysis and Applications, 347(1),(2008), 266-272.
• [12] V. S. Gerdjikov, On the Spectral Theory of the Integro-Differential Operator a Generating Nonlinear Evolution Equations, Letters in Mathematical Physics, 6(5), (1982), 315-324.
• [13] Y. V. Kuryshova, Inverse Spectral Problem for Integro-Differential Operators, Mathematical Notes, 81(6), (2007), 767-777.
• [14] B. Wu and J. Yu, Uniqueness of an Inverse Problem for an Integro-Differential Equation Related to the Basset Problem, Boundary Value Problems, 229, (2014).
• [15] C. F. Yang and X. P. Yang, Inverse Nodal Problems for Differential Pencils on a Star-Shaped Graph, Inverse Problems, 26, (2010).
• [16] A. S. Ozkan and B. Keskin, Inverse Nodal Problems for Sturm–Liouville Equation with EigenparameterDependent Boundary and Jump Conditions, Inverse Problems in Science and Engineering, (2014).
• [17] S. A. Buterin, On an Inverse Spectral Problem for a Convolution Integro-Differential Operator, Results in Mathematics, 50, (2007), 173-181.
• [18] S. A. Buterin, The Inverse Problem of Recovering the Volterra Convolution Operator from the Incomplete Spectrum of its Rank-One Perturbation, Inverse Problems, 22, (2006), 2223–2236.
• [19] G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publishers, New York, (2001).