INVERSE SOURCE PROBLEM FOR A DEGENERATE SUBDIFFUSION EQUATION

Azizbek Mamanazarov; Shohsanam Mahmudjonova

Mathematics

No. 1 (2025): FarDU ilmiy xabarlari jurnali (ANIQ FANLAR)

INVERSE SOURCE PROBLEM FOR A DEGENERATE SUBDIFFUSION EQUATION

Mamanazarov Azizbek Otajon ugli^▸^▾
Mahmudjonova Shohsanam Bahromjon qizi^▸^▾

PDF (Uzbek)

Submitted: February 13, 2025
Published: 2025-02-25

Abstract

In the present paper, the inverse source problem is formulated and studied for a degenerate sub-diffusion equation in a rectangular domain. By employing the method of separation of variables, a spectral problem was obtained for an ordinary differential equation concerning the spatial variable. The existence of the eigenvalues and eigenfunctions of this spectral problem was established by equivalently reducing it to a homogeneous second-kind Fredholm integral equation with symmetric kernels. Using the theory of integral equations, the existence of the eigenvalues and eigenfunctions was further confirmed. The solution to the inverse source problem is expressed as the sum of a Fourier series over the system of eigenfunctions of the spectral problem. The uniform convergence of the obtained series of solution were proved.

References

Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal 1, 167–191 (1998)
Freed, A., Diethelm, K., Luchko, Y.: Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus. NASA’s Glenn Research Center, Ohio (2002)
Mainardi, F.: Fractional Calculus and Waves in Linear Visco-elasticity an Introduction to Mathematical Models. Imperial College Press, London (2010)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Frac tional Differential Equations. Elsevier, Boston (2006)
Ruzhansky, M., Tokmagambetov, N., Torebek, B.T.: On a non-local problem for a multi-term fractional diffusion-wave equation. Fract. Calc. Appl. Anal 23(2), 324–355 (2020)
Karimov, E., Mamchuev, M., Ruzhansky, M.: Non-local initial problem for second order time-fractional and space-singular equation. Hokkaido Math. J 49(2), 349 361 (2020)
Al-Salti, N., Karimov, E.: Inverse source problems for degenerate time-fractional pde. Progress in Fractional Differentiation and Applications 8(1), 39–52 (2022) https://doi.org/10.18576/pfda/080102
Ali, M., Aziz, S., Malik, S.A.: Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions. J. Pseudo-Differ. Oper. Appl 13(3) (2022) https://doi.org/10.1007/s11868-021-00434-7
Lopushanska, H., Lopushansky, A., Myaus, O.: Inverse problems of periodic spatial distributions for a time-fractional diffusion equation. Electronic Journal of Differential Equations 2016(14), 1–9 (2016)
Mikhlin, S.G.: Lektsii po Lineynym Integral’nym Uravneniyam [Lectures on Linear Integral Equations]. Fizmatlit, Moscow (1959)

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[1] Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal 1, 167–191 (1998)

[2] Freed, A., Diethelm, K., Luchko, Y.: Fractional-Order Viscoelasticity (FOV): Constitutive Development Using the Fractional Calculus. NASA’s Glenn Research Center, Ohio (2002)

[3] Mainardi, F.: Fractional Calculus and Waves in Linear Visco-elasticity an Introduction to Mathematical Models. Imperial College Press, London (2010)

[4] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Frac tional Differential Equations. Elsevier, Boston (2006)

[5] Ruzhansky, M., Tokmagambetov, N., Torebek, B.T.: On a non-local problem for a multi-term fractional diffusion-wave equation. Fract. Calc. Appl. Anal 23(2), 324–355 (2020)

[6] Karimov, E., Mamchuev, M., Ruzhansky, M.: Non-local initial problem for second order time-fractional and space-singular equation. Hokkaido Math. J 49(2), 349 361 (2020)

[7] Al-Salti, N., Karimov, E.: Inverse source problems for degenerate time-fractional pde. Progress in Fractional Differentiation and Applications 8(1), 39–52 (2022) https://doi.org/10.18576/pfda/080102

[8] Ali, M., Aziz, S., Malik, S.A.: Inverse problem for a multi-parameters space-time fractional diffusion equation with nonlocal boundary conditions. J. Pseudo-Differ. Oper. Appl 13(3) (2022) https://doi.org/10.1007/s11868-021-00434-7

[9] Lopushanska, H., Lopushansky, A., Myaus, O.: Inverse problems of periodic spatial distributions for a time-fractional diffusion equation. Electronic Journal of Differential Equations 2016(14), 1–9 (2016)

[10] Mikhlin, S.G.: Lektsii po Lineynym Integral’nym Uravneniyam [Lectures on Linear Integral Equations]. Fizmatlit, Moscow (1959)