Inverse nodal problems for Dirac operators and their numerical approximations

Authors

  • Fei Song Nanjing Univ., Jiangsu, China
  • Yuping Wang Nanjing Forestry Univ., Jiangsu, China
  • Shahrbanoo Akbarpoor Islamic Azad Univ., Jouybar, Iran

DOI:

https://doi.org/10.58997/ejde.2023.81

Keywords:

Dirac operator; inverse nodal problem; Chebyshev wavelet; Bernstein method; uniqueness

Abstract

In this article, we consider an inverse nodal problem of Dirac operators and obtain approximate solution and its convergence based on the second kind Chebyshev wavelet and Bernstein methods. We establish a uniqueness theorem of this problem from parts of nodal points instead of a dense nodal set. Numerical examples are carried out to illustrate our method.

For more information see https://ejde.math.txstate.edu/Volumes/2023/81/abstr.html

References

S. Albeverio, R. Hryniv, Ya. Mykytyuk; Reconstruction of radial Dirac and Schr¨odinger operators from two spectra, J. Math. Anal. Appl., 339 (2008), 45-57.

E. Babolian, L. M. Delves; An augmented Galerkin method for first kind Fredholm equations, IMA J. Appl. Math., 24 (1979), 157-174.

P. J. Browne, B. D. Sleeman; Inverse nodal problem for Sturm-Liouville equation with eigen-parameter dependent boundary conditions, Inverse Probl., 12 (1996), 377-381.

S. A. Buterin, C.-T. Shieh; Incomplete inverse spectral and nodal problems for differential pencils, Results Math., 62 (2012), 167-179.

X. F. Chen, Y. H. Cheng, C. K. Law; Reconstructing potentials from zeros of one eigenfunction, Trans. Amer. Math. Soc., 363 (2011), 4831-4851.

Y. H. Cheng, C. K. Law, J. Tsay; Remarks on a new inverse nodal problem, J. Math. Anal. Appl., 248 (2000), 145-155.

S. Currie, B. A. Watson; Inverse nodal problems for Sturm-Liouville equations on graphs, Inverse Probl., 23 (2007), 2029-2040.

Y. X. Guo, G. S. Wei; Inverse problems: Dense nodal subset on an interior subinterval, J. Differential Equations, 255 (2013), 2002-2017.

Y. X. Guo, G. S. Wei; Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter, Results Math., 67 (2015), 95-110.

O. H. Hald, J. R. McLaughlin; Solutions of inverse nodal problems, Inverse Probl., 5 (1989), 307-347.

M. Horv´ath; On the inverse spectral theory of Schr¨odinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001), 4155-4171.

B. M. Levitan, I. S. Sargsjam; Sturm-Liouville and Dirac Operators, London: Kluwer Academic Publishers, 1991.

Q. Y. Li, N. C. Wang, D. Y. Yi; Numerical analysis (4th edition, in Chinese), Tsinghua University Press and Springer Press: Beijing, 2001.

G. G. Lorentz; Bernstein polynomials, AMS Chelsia Publishing, 1986.

J. R. McLaughlin; Inverse spectral theory using nodal points as data-a uniqueness result, J. Differential Equations, 73 (1988), 354-362.

S. Mosazadeh, H. Koyunbakan; On the stability of the solution of the inverse problem for Dirac operator, Appl. Math. Lett., 102 (2020), 106118.

A. Neamaty, Sh. Akbarpoor; Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition, Inverse Probl. Sci. Eng., 25 (2017), 978-994.

M. Rabbani, K. Maleknejad, N. Aghazadeh, R. Mollapourasl; Computational projection methods for solving Fredholm integral equation, Appl. Math. Comput., 191 (2007), 140-143.

M. T. Rashed; Numerical solutions of the integral equations of the first kind, Appl. Math. Comput., 145 (2003), 413-420.

C. L. Shen; On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math. Anal., 19 (1988), 1419-1424.

C.-T. Shieh, V. A. Yurko; Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), 266-272.

Y. P. Wang, E. Yilmaz, Sh. Akbarpoor; The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method, Int. J. Comput. Math., 100 (2023), 219-232.

Y. P. Wang, V. A. Yurko; On the inverse nodal problems for discontinuous Sturm-Liouville operators, J. Differential Equations, 260 (2016), 4086-4109.

Y. P. Wang, V. A. Yurko; On the missing eigenvalue problem for Dirac operators, Appl. Math. Lett., 80 (2018), 41-47.

Y. P. Wang, V. A. Yurko, C.-T. Shieh; The uniqueness in inverse problems for Dirac operators with the interior twin-dense nodal subset, J. Math. Anal. Appl., 479 (2019), 1383-1402.

Y. X. Wang, L. Zhu; SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 275 (2016), 72-80.

Z. Y. Wei, Y. X. Guo, G. S. Wei; Incomplete inverse spectral and nodal problems for Dirac operator, Adv. Differential Equations, 2015 (2015), 188.

C. F. Yang, Z. Y. Huang; Reconstruction of the Dirac operator from nodal data, Int. Equ. Oper. Theory, 66 (2010), 539-551.

X. F. Yang; A new inverse nodal problem, J. Differential Equations, 169 (2001), 633-653.

V. A. Yurko; Inverse spectral problems for differential systems on a finite interval, Results Math., 48 (2005), 371-386.

F. Y. Zhou, X. Y. Xu; Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets, Appl. Math. Comput., 247 (2014), 353-367.

L. Zhu, Q. B. Fan; Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1203-1213.

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Published

2023-12-06

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How to Cite

Inverse nodal problems for Dirac operators and their numerical approximations. (2023). Electronic Journal of Differential Equations, 2023(01-87), No. 81, 1-15. https://doi.org/10.58997/ejde.2023.81