The conditional stability and an iterative regularization method for a fractional inverse elliptic problem of Tricomi-Gellerstedt-Keldysh type
Abstract
The present paper is devoted to identifying an inaccessible boundary condition for a fractional elliptic problem of Tricomi-Gellerstedt-Keldysh-type. Using the expansion Fourier method, the considered problem can be reformulated as an operator equation of the first kind. To construct a stabilized approximate solution we employ a variant of the iterative method. We also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Finally, some numerical verifications on the efficiency and accuracy of the proposed algorithm is presented.
Keyword : fractional elliptic equations, Tricomi-Gellerstedt-Keldysh equations, ill-posed problems, inverse problems, a posteriori parameter choice rule, iterative regularization method
How to Cite
Djemoui, S., Meziani, M. S. E., & Boussetila, N. (2024). The conditional stability and an iterative regularization method for a fractional inverse elliptic problem of Tricomi-Gellerstedt-Keldysh type. Mathematical Modelling and Analysis, 29(1), 23–45. https://doi.org/10.3846/mma.2024.16783
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J.M. Rassias. Uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem. Complex Variables, 47(8):707–718, 2002. https://doi.org/10.1080/02781070290016368
M. Ruzhansky, B.T. Torebek and B. Turmetov. Well-posedness of Tricomi-Gellerstedt-Keldysh-type fractional elliptic problems. Journal of Integral Equations and Applications, 34(3):373–387, 2022. https://doi.org/10.1216/jie.2022.34.373
R. Sassane, N. Boussetila, F. Rebbani and A. Benrabah. Iterative regularization method for an abstract ill-posed generalized elliptic equation. Asian-European Journal of Mathematics, 14(05):2150069, 2021. https://doi.org/10.1142/S1793557121500698
O.N. Strand. Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind. SIAM Journal on Numerical Analysis, 11(4):798–825, 1974. https://doi.org/10.1137/0711066
F.G. Tricomi. Sulle equazioni lineari alle derivate parziali di secondo or-dine di tipo misto Memorie Accad, 1923.
A.F. Tsang. The solution of a nonlocal boundary value problem. Differential Equations, 42:766–769, 2006. https://doi.org/10.1134/S0012266106050181
B.Kh. Turmetov and B.T. Torebek. On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation. New York J. Math, 20:1237–1251, 2014.
F. Yang, X. Liu and X.X. Li. Landweber iterative regularization method for identifying the unknown source of the modified Helmholtz equation. Boundary Value Problems, 2017:388, 2017. https://doi.org/10.1186/s13662-017-1423-8
F. Zouyed and S. Djemoui. An iterative regularization method for identifying the source term in a second order differential equation. Mathematical Problems in Engineering, 2015, 2015. https://doi.org/10.1155/2015/713403
A.V. Bitsadze. On the problem of equations of mixed type. Trudy Matematicheskogo Instituta imeni VA Steklova, 41:3–59, 1953.
A.V. Bitsadze. Incorrectness of Dirichlet’s problem for the mixed type of equations in mixed regions. In Dokl. Akad. Nauk SSSR, volume 122, pp. 167–170, 1958.
A.V. Bitsadze. Equations of the mixed type. Elsevier, 2014.
L. Boudabsa and T. Simon. Some properties of the Kilbas-Saigo function. Mathematics, 9(3):217, 2021. https://doi.org/10.3390/math9030217
L. Boudabsa, T. Simon and P. Vallois. Fractional extreme distributions. Electron. J. Probab., (25):1–20, 2020. https://doi.org/10.1214/20-EJP520
Y. Deng and Z. Liu. Iteration methods on sideways parabolic equations. Inverse Problems, 25(9):095004, 2009. https://doi.org/10.1088/0266-5611/25/9/095004
S.Kh. Gekkieva. A boundary value problem for the generalized transfer equation with a fractional derivative in a semi-infinite domain. Izvestiya KabardinoBalkarskaya Nauchnoogo Tsentra RAN, 8(1):6–8, 2002.
E.T. Karimov, A.S. Berdyshev and N.A. Rakhmatullaeva. Unique solvability of a non-local problem for mixed-type equation with fractional derivative. Mathematical Methods in the Applied Sciences, 40(8):2994–2999, 2017. https://doi.org/10.1002/mma.4215
M.V. Keldysh. On some cases of degenerate elliptic equations on the boundary of a domain. In Doklady Acad. Nauk USSR, volume 77, pp. 181–183, 1951.
A. Kilbas and O. Repin. An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative. Fractional Calculus and Applied Analysis, 13(1):69–84, 2010.
A.A. Kilbas and M. Saigo. On solution of integral equations of Abel-Volterra type. Differ. Integral Equ., 8(5):993–1011, 1995. https://doi.org/10.57262/die/1369056041
V.A. Kozlov and V.G. Maz’ya. Iterative procedures for solving ill-posed boundary value problems that preserve the differential equations. Algebra i Analiz, 1(5):144–170, 1989.
V.A. Kozlov, V.G. Maz’ya and A.V. Fomin. An iterative method for solving the Cauchy problem for elliptic equations. Comput. Math. Phys, 31(1):45–52, 1991.
M.A. Krasnosel’skii, G.M. Vainikko, R.P. Zabreyko, Ya.B. Ruticki and V.Va. Stet’senko. Approximate solution of operator equations. Springer Science & Business Media, 2012.
A.G. Kuz’min. An equation of the mixed type associated with the direct Laval nozzle problem. Leningradskii Universitet Vestnik Matematika Mekhanika Astronomiia, pp. 65–70, 1986.
M A. Lavrent’ev and A.V. Bitsadze. On the problem of equations of mixed type. In Dokl. Akad. Nauk SSSR, number 3, pp. 373–376, 1950.
A.R. Manwell. The Tricomi equation with applications to the theory of plane transonic flow. NASA STI/Recon Technical Report A, 80:27617, 1979.
O.Kh. Masaeva. Dirichlet problem for the generalized Laplace equation with the Caputo derivative. Differential equations, 48:449–454, 2012. https://doi.org/10.1134/S0012266112030184
MSE. Meziani, N. Boussetila, F. Rebbani and A. Benrabah. Iterative regularization method for an abstract inverse Goursat problem. Khayyam Journal of Mathematics, 7(2):279–297, 2021.
E.I. Moiseev. On the solution of a nonlocal boundary value problem by the spectral method. Differentsial’nye Uravneniya, 35(8):1094–1100, 1999.
J.M. Rassias. Uniqueness of quasi-regular solutions for a bi-parabolic elliptic bi-hyperbolic Tricomi problem. Complex Variables, 47(8):707–718, 2002. https://doi.org/10.1080/02781070290016368
M. Ruzhansky, B.T. Torebek and B. Turmetov. Well-posedness of Tricomi-Gellerstedt-Keldysh-type fractional elliptic problems. Journal of Integral Equations and Applications, 34(3):373–387, 2022. https://doi.org/10.1216/jie.2022.34.373
R. Sassane, N. Boussetila, F. Rebbani and A. Benrabah. Iterative regularization method for an abstract ill-posed generalized elliptic equation. Asian-European Journal of Mathematics, 14(05):2150069, 2021. https://doi.org/10.1142/S1793557121500698
O.N. Strand. Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind. SIAM Journal on Numerical Analysis, 11(4):798–825, 1974. https://doi.org/10.1137/0711066
F.G. Tricomi. Sulle equazioni lineari alle derivate parziali di secondo or-dine di tipo misto Memorie Accad, 1923.
A.F. Tsang. The solution of a nonlocal boundary value problem. Differential Equations, 42:766–769, 2006. https://doi.org/10.1134/S0012266106050181
B.Kh. Turmetov and B.T. Torebek. On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation. New York J. Math, 20:1237–1251, 2014.
F. Yang, X. Liu and X.X. Li. Landweber iterative regularization method for identifying the unknown source of the modified Helmholtz equation. Boundary Value Problems, 2017:388, 2017. https://doi.org/10.1186/s13662-017-1423-8
F. Zouyed and S. Djemoui. An iterative regularization method for identifying the source term in a second order differential equation. Mathematical Problems in Engineering, 2015, 2015. https://doi.org/10.1155/2015/713403