Abstract
For a fractional diffusion equation with reaction coefficient depending only on the first two components of the spatial variable \(x=(x_1,x_2,x_3)\in \mathbb {R}^3 \) and on time \(t\geq 0 \), we consider the inverse problem of determining this coefficient under the assumption that the initial value at \(t=0 \) is known for the solution of the equation and the boundary value at \( x_3=0\) is given as an additional condition. This inverse problem is reduced to equivalent integral equations, and we apply the contraction mapping principle to prove the existence of solutions of these equations. Local existence and global uniqueness theorems are proved. We also obtain a stability estimate for the solution of the inverse problem.
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REFERENCES
Caputo, M. and Mainardi, F., Linear models of dissipation in an elastic solid, La Rivista del Nuovo Cimento, 1971, vol. 1, no. 2, pp. 161–198.
Babenko, Yu.I., Teplomassoobmen. Metod rascheta teplovykh i diffuzionnykh potokov (Heat and Mass Transfer. Method for Calculating Heat and Diffusion Fluxes), Leningrad: Khimiya, 1986.
Gorenflo, R. and Mainardi, F., Fractional calculus: integral and differential equations of fractional order, in Fractals Fractional Calculus in Continuum Mechanics, Carpinteri, A. and Mainar, F., Eds., New York: Springer, 1997, pp. 223–276.
Gorenflo, R. and Rutman, R., On ultraslow and intermediate processes, in Transform Methods and Special Functions, Rusev, P., Dimovski, I., and Kiryakova, V., Eds., Sofia, 1994. Sci. Culture Technol. Singapore, 1995, pp. 61–81.
Mainardi, F., Fractional relaxation and fractional diffusion equations, mathematical aspects, in Proc. 12th IMACS World Congr., Ames, W.F., Ed., Georgia Tech Atlanta, 1994, vol. 1, pp. 329–332.
Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri, A. and Mainar, F., Eds., New York: Springer, 1997, pp. 291–348.
Kochubei, A.N., The Cauchy problem for fractional-order evolution equations, Differ. Uravn., 1986, vol. 25, no. 8, pp. 1359–1368.
Kochubei, A.N., Diffusion of fractional order, Differ. Uravn., 1990, vol. 26, no. 4, pp. 485–492.
Eidelman, S.D. and Kochubei, A.N., Cauchy problem for fractional diffusion equations, J. Differ. Equat., 2004, vol. 199, pp. 211–255.
Miller, L. and Yamamoto, M., Coefficient inverse problem for a fractional diffusion equation, Inverse Probl., 2013, vol. 29, no. 7, p. 075013.
Bondarenko, A.N. and Ivaschenko, D.S., Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient, J. Inverse Ill-Posed Probl., 2009, vol. 17, pp. 419–440.
Xiong, T.X., Zhou, Q., and Hon, C.Y., An inverse problem for fractional diffusion equation in 2-dimensional case: stability analysis and regularization, J. Math. Anal. Appl., 2012, vol. 393, pp. 185–199.
Xiong, X., Guo, H., and Liu, X., An inverse problem for a fractional diffusion equation, J. Comput. Appl. Math., 2012, vol. 236, pp. 4474–4484.
Kirane, M., Malik, S.A., and Al-Gwaiz, M.A., An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci., 2013, vol. 36, pp. 1056–1069.
Romanov, V.G., An inverse problem for a layered film on a substrate, Eurasian J. Math. Comput. Appl., 2016, vol. 4, no. 3, pp. 29–38.
Karuppiah, K., Kim, J.K., and Balachandran, K., Parameter identification of an integro-differential equation, Nonlin. Func. Anal. Appl., 2015, vol. 20, no. 2, pp. 169–185.
Ivanchov, M. and Vlasov, V., Inverse problem for a two dimensional strongly degenerate heat equation, Electronic J. Differ. Equat., 2018, vol. 77, pp. 1–17.
Huntul, M.J., Lesnic, D., and Hussein, M.S., Reconstruction of time-dependent coefficients from heat moments, Appl. Math. Comput., 2017, vol. 301, pp. 233–253.
Hazanee, A., Lesnic, D., Ismailov, M.I., and Kerimov, N.B., Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions, Appl. Math. Comput., 2019, vol. 346, pp. 800–815.
Durdiev, D.K. and Rashidov, A.Sh., Inverse problem of determining the kernel in an integro-differential equation of parabolic type, Differ. Equations, 2014, vol. 50, no. 1, pp. 110–116.
Durdiev, D.K. and Zhumaev, Zh.Zh., Problem of determining a multidimensional thermal memory in a heat conductivity equation, Methods Func. Anal. Topol., 2019, vol. 25, no. 3, pp. 219–226.
Durdiev, D.K. and Zhumaev, Zh.Zh., Problem of determining the thermal memory of a conducting medium, Differ. Equations, 2020, vol. 56, no. 6, pp. 785–796.
Durdiev, D.K. and Nuriddinov, J.Z., On investigation of the inverse problem for a parabolic integrodifferential equation with a variable coefficient of thermal conductivity, Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki, 2020, vol. 30, no. 4, pp. 572–584.
Durdiev, D.K., Shishkina, E.L., and Sitnik, S.M., The explicit formula for solution of anomalous diffusion equation in the multi-dimensional space, September 20, 2020.
Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa (Linear and Quasilinear Equations of Parabolic Type), Moscow: Nauka, 1967, p. 736.
Mathai, A.M., Saxena, R.K., and Haubold, H.J., The \(H\)-function. Theory and Application, Berlin–Heidelberg: Springer, 2010.
Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Application of Fractional Differential Equations. North-Holland Mathematical Studies, Amsterdam: Elsevier, 2006.
Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1977.
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Durdiev, U.D. Problem of Determining the Reaction Coefficient in a Fractional Diffusion Equation. Diff Equat 57, 1195–1204 (2021). https://doi.org/10.1134/S0012266121090081
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DOI: https://doi.org/10.1134/S0012266121090081