Geodesic
Inverse problem for fractional order pseudo-parabolic equation with involution
Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 119-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we consider an inverse problem on recovering the right-hand side of a fractional pseudo-parabolic equation with an involution operator. The major obstacle for considering the inverse problems is related with the well-posedness of the problem. Inverse problems are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed since the solution is highly sensitive to variations in the final data. The advantage of this paper is two-fold. On the one hand, we investigate the solvability of the direct problem and prove the solvability to this problem. On the other hand, we study the inverse problem based on this direct problem and prove the solvability results in this problem, too. First, we investigate the Cauchy problem for the time-fractional pseudo-parabolic equation with the involution operator, and secondly, we consider the inverse problem on recovering the right-hand side from an overdetermined final condition and prove that it is solvable. To achieve our goals, we use methods corresponding to the different areas of mathematics such as the theory of partial differential equations, mathematical physics, and functional analysis. In particular, we use the $\mathcal{L}$-Fourier analysis method to establish the existence and uniqueness of solutions to this problem on the Sobolev space. The classical and generalized solutions of the inverse problem are studied.
Keywords: fractional differential equation, inverse problem, involution
Mots-clés : pseudo-parabolic equation. 
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D. Serikbaev. Inverse problem for fractional order pseudo-parabolic equation with involution. Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 119-135. http://geodesic.mathdoc.fr/item/UFA_2020_12_4_a10/

[1] B. Ahmad, A. Alsaedi, M. Kirane, R. Tapdigoglu, “An inverse problem for space and time fractional evolution equations with an involution perturbation”, Quaest. Math., 40:2 (2017), 151–160 | DOI | MR | Zbl

[2] N. Al-Salti, M. Kirane, B.T. Torebek, “On a class of inverse problems for a heat equation with involution perturbation”, Hacet. J. Math. Stat., 48:3 (2019), 669–681 | MR

[3] T.B. Benjamin, J.L. Bona, J.J. Mahony, “Model equations for long waves in nonlinear dispersive systems”, Philosoph. Trans. Royal Soc. A: Mathem. Phys. Engineer. Sci., 272:1220 (1972), 47–78 | MR | Zbl

[4] G.I. Barenblatt, J. Garcia-Azorero, A. De Pablo, J.L. Vazquez, “Mathematical model of the non-equilibrium water-oil displacement in porous strata”, Appl. Anal., 65:1–2 (1997), 19–45 | DOI | MR | Zbl

[5] P.J. Chen, M.E. Gurtin, “On a theory of heat conduction involving two temperatures”, Z. Angew. Math. Phys., 19:4 (1968), 614–627 | DOI | Zbl

[6] Marcel Dekker, New York, 2003 | MR | Zbl

[7] C. Fetecau, C-a. Fetecau, M. Kamran, D. Vieru, “Exact solutions for the flow of a generalized Oldroyd-B fluid induced by aconstantly accelerating plate between two side walls perpendicular to the plate”, J. Non-Newtonian Fluid Mech., 156:3 (2009), 189–201 | DOI | MR | Zbl

[8] V.E. Fedorov, R.R. Nazhimov, “Inverse problems for a class of degenerate evolution equations with Riemann-Liouville derivative”, Fract. Calc. Appl. Anal., 22:2 (2019), 271–286 | DOI | MR | Zbl

[9] A. Hazanee, D. Lesnic, M.I. Ismailov, N.B. Kerimov, “Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions”, Appl. Math. Comput., 346 (2019), 800–815 | MR | Zbl

[10] M. Kirane, N. Al-Salti, “Inverse problems for a nonlocal wave equation with an involution perturbation”, J. Nonl. Sci. Appl., 9:3 (2016), 1243–1251 | DOI | MR | Zbl

[11] N. Kinash, J. Janno, “Inverse problems for a perturbed time fractional diffusion equation with final overdetermination”, Math. Meth. Appl. Sci., 41:5 (2018), 1925–1943 | DOI | MR | Zbl

[12] I.A. Kaliev, M.M. Sabitova, “Problems of determining the temperature and density of heat sources from the initial and final temperatures”, J. Appl. Ind. Math., 4:3 (2010), 332–339 | DOI | MR

[13] M. Kirane, B. Samet, B.T. Torebek, “Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data”, Electron. J. Diff. Equat., 2017:257 (2017), 1–13 | MR

[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, North-Holland, 2006 | MR | Zbl

[15] Y. Kian, M. Yamamoto, “Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations”, Inverse Probl., 35:11 (2019), 115006 | DOI | MR | Zbl

[16] Y. Luchko, R. Gorenflo, “An operational method for solving fractional differential equations with the Caputo derivatives”, Acta Math. Vietnam., 24:2 (1999), 207–233 | MR | Zbl

[17] A.S. Lyubanova, A. Tani, “An inverse problem for pseudo-parabolic equation of filtration. The existence, uniqueness and regularity”, Appl. Anal., 90:10 (2011), 1557–1571 | DOI | MR | Zbl

[18] A.S. Lyubanova, A. Tani, “On inverse problems for pseudoparabolic and parabolic equations of filtration”, Inverse Probl. Sci. Eng., 19:7 (2011), 1023–1042 | DOI | MR | Zbl

[19] A.S. Lyubanova, A.V. Velisevich, “Inverse problems for the stationary and pseudoparabolic equations of diffusion”, Appl. Anal., 98:11 (2019), 1997–2010 | DOI | MR | Zbl

[20] J.J. Nieto, “Maximum principles for fractional differential equations derived from Mittag-Leffler functions”, Appl. Math. Lett., 23:10 (2010), 1248–1251 | DOI | MR | Zbl

[21] I. Orazov, M.A. Sadybekov, “On a class of problems of determining the temperature and density of heat sources given initial and final temperature”, Siber. Math. J., 53:1 (2012), 146–151 | DOI | MR | Zbl

[22] W. Rundell, “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data”, Appl. Anal., 10:3 (1980), 231–242 | DOI | MR | Zbl

[23] M. Ruzhansky, N. Tokmagambetov, B. T. Torebek, “Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations”, J. Inverse Ill-posed Probl., 27:6 (2019), 891–911 | DOI | MR | Zbl

[24] W. Rundell, Z. Zhang, “Recovering an unknown source in a fractional diffusion problem”, J. Comput. Phys., 368 (2018), 299–314 | DOI | MR | Zbl

[25] T. Simon., “Comparing Frechet and positive stable laws”, Electron. J. Probab., 19:16 (2014), 1–25 | MR

[26] M. M. Slodic̆ka, K. S̆is̆kova, K. V. Bockstal, “Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation”, Appl. Math. Lett., 91 (2019), 15–21 | DOI | MR | Zbl

[27] D. Tong, Y. Liu, “Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe”, Inter. J. Engin. Sci., 43:3–4 (2005), 281–289 | DOI | MR | Zbl

[28] B.T. Torebek, R. Tapdigoglu, “Some inverse problems for the nonlocal heat equation with Caputo fractional derivative”, Math. Methods Appl. Sci., 40:18 (2017), 6468–6479 | DOI | MR | Zbl