Research Article
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Inverse Nodal Problem for a Conformable Fractional Diffusion Operator With Parameter-Dependent Nonlocal Boundary Condition

Year 2023, Volume: 44 Issue: 2, 356 - 363, 30.06.2023

Yaşar Çakmak

https://doi.org/10.17776/csj.1243136
Cited By: 2

Abstract

In this paper, we consider the inverse nodal problem for the conformable fractional diffusion operator with parameter-dependent Bitsadze–Samarskii type nonlocal boundary condition. We obtain the asymptotics for the eigenvalues, the eigenfunctions, and the zeros of the eigenfunctions (called nodal points or nodes) of the considered operator, and provide a constructive procedure for solving the inverse nodal problem, i.e., we reconstruct the potential functions p(x) and q(x) by using a dense subset of the nodal points.

Keywords

Diffusion operator Inverse nodal problem Conformable fractional derivative Nonlocal boundary condition

References

  • [1] McLaughlin J.R., Inverse Spectral Theory Using Nodal Points as Data—a Uniqueness Result, J. Differential Equations, 73 (2) (1988) 354-362.
  • [2] Hald O.H., McLaughlin J.R., Solutions of Inverse Nodal Problems, Inverse Problems, 5 (1989) 307-347.
  • [3] Yang X.F., A Solution of the Nodal Problem, Inverse Problems, 13 (1997) 203-213.
  • [4] Browne P.J., Sleeman B.D., Inverse Nodal Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions, Inverse Problems, 12 (1996) 377-381.
  • [5] Hald O.H., McLaughlin J.R., Inverse Problems: Recovery of BV Coefficients from Nodes, Inverse Problems, 14 (1998) 245-273.
  • [6] Law C.K., Yang C.F., Reconstructing the Potential Function and Its Derivatives Using Nodal Data, Inverse Problems, 14 (1998) 299-312.
  • [7] Shen C.L., Shieh C.T., An Inverse Nodal Problem for Vectorial Sturm–Liouville Equation, Inverse Problems, 16 (2000) 349-356.
  • [8] Yang X.F., A New Inverse Nodal Problem, J. Differential Equations, 169 (2001) 633-653.
  • [9] Law C.K., Shen C.L., Yang C.F., The Inverse Nodal Problem on the Smoothness of the Potential Function, Inverse Problems, 15 (1999) 253-263. Errata: Inverse Problems, 17 (2) (2001) 361-363.
  • [10] Freiling G., Yurko V.A., Inverse Sturm–Liouville Problems and Their Applications, New York: Nova Science Publishers, (2001).
  • [11] Shieh C.T., Yurko V.A., Inverse Nodal and Inverse Spectral Problems for Discontinuous Boundary Value Problems, J. Math. Anal. Appl., 347 (2008) 266-272.
  • [12] Keskin B., Ozkan A.S., Inverse Nodal Problems for Impulsive Sturm–Liouville Equation with Boundary Conditions Depending on the Parameter, Advances in Analysis, 2 (3) (2017) 151-156.
  • [13] Koyunbakan H., A New Inverse Problem for the Diffusion Operator, Appl. Math. Lett., 19 (2006) 995-999.
  • [14] Buterin S.A., Shieh C.T., Inverse Nodal Problem for Differential Pencils, Applied Mathematics Letters, 22 (8) (2009) 1240-1247.
  • [15] Yang C.F., Reconstruction of the Diffusion Operator with Nodal Data, Z. Naturforsch A., 65 (2010) 100-106.
  • [16] Buterin S.A., Shieh C.T., Incomplete Inverse Spectral and Nodal Problems for Differential Pencils, Results in Mathematics, 62 (2012) 167-179.
  • [17] Yang C.F., An Inverse Problem for a Differential Pencil Using Nodal Points as Data, Israel Journal of Mathematics, 204 (2014) 431-446.
  • [18] Gordeziani N., On Some Non-local Problems of the Theory of Elasticity, Bulletin of TICMI, 4 (2000) 43-46.
  • [19] Yin Y.F., On Nonlinear Parabolic Equations with Nonlocal Boundary Conditions, Journal of Mathematical Analysis and Applications, 185 (1) (1994) 161-174.
  • [20] Bitsadze A.V., Samarskii A.A., Some Elementary Generalizations of Linear Elliptic Boundary Value Problems, Doklady Akademii Nauk SSSR, 185 (4) (1969) 739-740.
  • [21] Hu Y.T., Yang C.F., Xu X.C., Inverse Nodal Problems for the Sturm-Liouville Operator with Nonlocal Integral Conditions, Journal of Inverse and Ill-Posed Problems, 25 (6) (2017) 799-806.
  • [22] Keskin B., Inverse Nodal Problems for Dirac Type Integro Differential System with a Nonlocal Boundary Condition, Turkish Journal of Mathematics, 46 (6) (2022) 2430-2439.
  • [23] Ozkan A.S., Adalar İ., Inverse Nodal Problems for Sturm-Liouville Equation with Nonlocal Boundary Conditions, Journal of Mathematical Analysis and Applications, 520 (1) (2023) 126907.
  • [24] Qin X., Gao Y., Yang C., Inverse Nodal Problems for the Sturm-Liouville Operator with Some Nonlocal Integral Conditions, Journal of Applied Mathematics and Physics, 7 (1) (2019) 111-122.
  • [25] Xu X.J., Yang C.F., Inverse Nodal Problem for Nonlocal Differential Operators, Tamkang Journal of Mathematics, 50 (3) (2019) 337-347.
  • [26] Yang C.F., Inverse Nodal Problem for a Class of Nonlocal Sturm-Liouville Operator, Mathematical Modelling and Analysis, 15 (3) (2010) 383-392.
  • [27] Çakmak Y., Keskin B., Inverse Nodal Problem for the Quadratic Pencil of the Sturm-Liouville Equations with Parameter-Dependent Nonlocal Boundary Condition, Turkish Journal of Mathematics, 47 (2023) 397–404.
  • [28] Khalil R., Al Horania M., Yousefa A., et al., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [29] Abdeljawad T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [30] Atangana A., Baleanu D., Alsaedi A., New Properties of Conformable Derivative, Open Math., 13 (2015) 889-898.
  • [31] Mortazaasl H., Jodayree Akbarfam A., Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering, 28 (4) (2020) 524–555.
  • [32] Allahverdiev B.P., Tuna H., Yalçinkaya Y., Conformable Fractional Sturm-Liouville Equation, Mathematical Methods in the Applied Sciences, 42 (10) (2019) 3508-3526.
  • [33] Keskin B., Inverse Problems for one Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems, 36 (6) (2020) 065001.
  • [34] Adalar I., Ozkan A.S., Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems, 28 (6) (2020) 775-782.
  • [35] Çakmak Y., Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering, 29 (9) (2021) 1308-1322.
  • [36] Çakmak Y., Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat, 36 (14) (2022) 4665–4674.
  • [37] Wang Y., Zhou J., Li Y., Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Adv. Math. Phys., 2016 (2016) 1-21.
  • [38] [38] Buterin S.A., On Half Inverse Problem for Differential Pencils with the Spectral Parameter in the Boundary Conditions, Tamkang Journal of Mathematics, 42 (3) (2011) 355-364.

Year 2023, Volume: 44 Issue: 2, 356 - 363, 30.06.2023

Yaşar Çakmak

https://doi.org/10.17776/csj.1243136
Cited By: 2

Abstract

Keywords

Diffusion operator Inverse nodal problem Conformable fractional derivative Nonlocal boundary condition

References

  • [1] McLaughlin J.R., Inverse Spectral Theory Using Nodal Points as Data—a Uniqueness Result, J. Differential Equations, 73 (2) (1988) 354-362.
  • [2] Hald O.H., McLaughlin J.R., Solutions of Inverse Nodal Problems, Inverse Problems, 5 (1989) 307-347.
  • [3] Yang X.F., A Solution of the Nodal Problem, Inverse Problems, 13 (1997) 203-213.
  • [4] Browne P.J., Sleeman B.D., Inverse Nodal Problems for Sturm–Liouville Equations with Eigenparameter Dependent Boundary Conditions, Inverse Problems, 12 (1996) 377-381.
  • [5] Hald O.H., McLaughlin J.R., Inverse Problems: Recovery of BV Coefficients from Nodes, Inverse Problems, 14 (1998) 245-273.
  • [6] Law C.K., Yang C.F., Reconstructing the Potential Function and Its Derivatives Using Nodal Data, Inverse Problems, 14 (1998) 299-312.
  • [7] Shen C.L., Shieh C.T., An Inverse Nodal Problem for Vectorial Sturm–Liouville Equation, Inverse Problems, 16 (2000) 349-356.
  • [8] Yang X.F., A New Inverse Nodal Problem, J. Differential Equations, 169 (2001) 633-653.
  • [9] Law C.K., Shen C.L., Yang C.F., The Inverse Nodal Problem on the Smoothness of the Potential Function, Inverse Problems, 15 (1999) 253-263. Errata: Inverse Problems, 17 (2) (2001) 361-363.
  • [10] Freiling G., Yurko V.A., Inverse Sturm–Liouville Problems and Their Applications, New York: Nova Science Publishers, (2001).
  • [11] Shieh C.T., Yurko V.A., Inverse Nodal and Inverse Spectral Problems for Discontinuous Boundary Value Problems, J. Math. Anal. Appl., 347 (2008) 266-272.
  • [12] Keskin B., Ozkan A.S., Inverse Nodal Problems for Impulsive Sturm–Liouville Equation with Boundary Conditions Depending on the Parameter, Advances in Analysis, 2 (3) (2017) 151-156.
  • [13] Koyunbakan H., A New Inverse Problem for the Diffusion Operator, Appl. Math. Lett., 19 (2006) 995-999.
  • [14] Buterin S.A., Shieh C.T., Inverse Nodal Problem for Differential Pencils, Applied Mathematics Letters, 22 (8) (2009) 1240-1247.
  • [15] Yang C.F., Reconstruction of the Diffusion Operator with Nodal Data, Z. Naturforsch A., 65 (2010) 100-106.
  • [16] Buterin S.A., Shieh C.T., Incomplete Inverse Spectral and Nodal Problems for Differential Pencils, Results in Mathematics, 62 (2012) 167-179.
  • [17] Yang C.F., An Inverse Problem for a Differential Pencil Using Nodal Points as Data, Israel Journal of Mathematics, 204 (2014) 431-446.
  • [18] Gordeziani N., On Some Non-local Problems of the Theory of Elasticity, Bulletin of TICMI, 4 (2000) 43-46.
  • [19] Yin Y.F., On Nonlinear Parabolic Equations with Nonlocal Boundary Conditions, Journal of Mathematical Analysis and Applications, 185 (1) (1994) 161-174.
  • [20] Bitsadze A.V., Samarskii A.A., Some Elementary Generalizations of Linear Elliptic Boundary Value Problems, Doklady Akademii Nauk SSSR, 185 (4) (1969) 739-740.
  • [21] Hu Y.T., Yang C.F., Xu X.C., Inverse Nodal Problems for the Sturm-Liouville Operator with Nonlocal Integral Conditions, Journal of Inverse and Ill-Posed Problems, 25 (6) (2017) 799-806.
  • [22] Keskin B., Inverse Nodal Problems for Dirac Type Integro Differential System with a Nonlocal Boundary Condition, Turkish Journal of Mathematics, 46 (6) (2022) 2430-2439.
  • [23] Ozkan A.S., Adalar İ., Inverse Nodal Problems for Sturm-Liouville Equation with Nonlocal Boundary Conditions, Journal of Mathematical Analysis and Applications, 520 (1) (2023) 126907.
  • [24] Qin X., Gao Y., Yang C., Inverse Nodal Problems for the Sturm-Liouville Operator with Some Nonlocal Integral Conditions, Journal of Applied Mathematics and Physics, 7 (1) (2019) 111-122.
  • [25] Xu X.J., Yang C.F., Inverse Nodal Problem for Nonlocal Differential Operators, Tamkang Journal of Mathematics, 50 (3) (2019) 337-347.
  • [26] Yang C.F., Inverse Nodal Problem for a Class of Nonlocal Sturm-Liouville Operator, Mathematical Modelling and Analysis, 15 (3) (2010) 383-392.
  • [27] Çakmak Y., Keskin B., Inverse Nodal Problem for the Quadratic Pencil of the Sturm-Liouville Equations with Parameter-Dependent Nonlocal Boundary Condition, Turkish Journal of Mathematics, 47 (2023) 397–404.
  • [28] Khalil R., Al Horania M., Yousefa A., et al., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014) 65-70.
  • [29] Abdeljawad T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015) 57-66.
  • [30] Atangana A., Baleanu D., Alsaedi A., New Properties of Conformable Derivative, Open Math., 13 (2015) 889-898.
  • [31] Mortazaasl H., Jodayree Akbarfam A., Trace Formula and Inverse Nodal Problem for a Conformable Fractional Sturm-Liouville Problem, Inverse Problems in Science and Engineering, 28 (4) (2020) 524–555.
  • [32] Allahverdiev B.P., Tuna H., Yalçinkaya Y., Conformable Fractional Sturm-Liouville Equation, Mathematical Methods in the Applied Sciences, 42 (10) (2019) 3508-3526.
  • [33] Keskin B., Inverse Problems for one Dimentional Conformable Fractional Dirac Type Integro Differential System, Inverse Problems, 36 (6) (2020) 065001.
  • [34] Adalar I., Ozkan A.S., Inverse Problems for a Conformable Fractional Sturm-Liouville Operators, Journal of Inverse and Ill-posed Problems, 28 (6) (2020) 775-782.
  • [35] Çakmak Y., Inverse Nodal Problem for a Conformable Fractional Diffusion Operator, Inverse Problems in Science and Engineering, 29 (9) (2021) 1308-1322.
  • [36] Çakmak Y., Trace Formulae for a Conformable Fractional Diffusion Operator, Filomat, 36 (14) (2022) 4665–4674.
  • [37] Wang Y., Zhou J., Li Y., Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Adv. Math. Phys., 2016 (2016) 1-21.
  • [38] [38] Buterin S.A., On Half Inverse Problem for Differential Pencils with the Spectral Parameter in the Boundary Conditions, Tamkang Journal of Mathematics, 42 (3) (2011) 355-364.
There are 38 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Yaşar Çakmak 0000-0002-6820-1322

Submission Date January 27, 2023
Acceptance Date June 6, 2023
Publication Date June 30, 2023
Published in Issue Year 2023 Volume: 44 Issue: 2

Cite

APA Çakmak, Y. (2023). Inverse Nodal Problem for a Conformable Fractional Diffusion Operator With Parameter-Dependent Nonlocal Boundary Condition. Cumhuriyet Science Journal, 44(2), 356-363. https://doi.org/10.17776/csj.1243136

Cited By

Inverse Problems for a Conformable Fractional Diffusion Operator
Journal of New Theory
https://doi.org/10.53570/jnt.1335702
An Interior Inverse Problem with Local Derivative
Differential Equations
https://doi.org/10.1134/S0012266125050027

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