Some inverse problems for time-fractional diffusion equation with nonlocal Samarskii-Ionkin type condition
Corresponding Author
Muhammad Ali
Department of Sciences & Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan
Correspondence
Muhammad Ali, Department of Sciences & Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan.
Email: [email protected]
Communicated by: T. E. Simos
Search for more papers by this authorSara Aziz
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Search for more papers by this authorCorresponding Author
Muhammad Ali
Department of Sciences & Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan
Correspondence
Muhammad Ali, Department of Sciences & Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan.
Email: [email protected]
Communicated by: T. E. Simos
Search for more papers by this authorSara Aziz
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
Search for more papers by this authorAbstract
Two inverse problems for time-fractional diffusion equation having a family of nonlocal boundary conditions are discussed. In first inverse problem, initial distribution is determined provided that the data at final temperature is given. Second inverse problem addresses the recovery of temporal component of source term whenever total energy of the system is known. A bi-orthogonal system of functions is used to write the series solution by Fourier's method. The classical nature of the solution of both inverse problems is established by using the estimates of Mittag-Leffler function and by imposing some regularity conditions on given datum.
CONFLICT OF INTEREST
We hereby declare that we do not have any conflict of interest to declare.
REFERENCES
- 1Laskin N. Time fractional quantum mechanics. Chaos Solitons Fractals. 2017; 102: 16-28.
- 2Sohail A, Bég OA, Li Z, Celik S. Physics of fractional imaging in biomedicine. Prog Biophys Mol Biol. 2018; 140: 13-20. https://doi.org/10.1016/jpbiomolbio.2018.03.002
- 3Monje CA, Chen YQ, Vinagre BM, Xue D, Feliu V. Fractional-Order Systems and Control, Fundamentals and Applications. London:Springer; 2010.
10.1007/978-1-84996-335-0 Google Scholar
- 4Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. London:World Scientific; 2010.
10.1142/p614 Google Scholar
- 5Sheng H, Chen YQ, Qiu TS. Fractional Processes and Fractional-Order Signal Processing Techniques and Applications. London:Springer; 2011.
- 6Magin RL. Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl. 2010; 59: 586-1593.
- 7Sun H, Zhang Y, Baleanu D, Chen W. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul. 2018; 64: 213-231.
- 8Ferreira JA, Pena G, Romanazzi G. Anomalous diffusion in porous media. Appl Math Model. 2016; 40: 1850-1862.
- 9de Mesquita JJ, Lopes A, Jansen J, de Melo P. Using the forced oscillation technique to evaluate respiratory resistance in individuals with silicosis. J Bras Pneumol. 2006; 32: 213-220.
- 10Scher H, Montroll E. Anomalous transit-time dispersion in amor- phous solid. Phys Rev B. 1975; 12: 2455-2477.
- 11Salari L, Rondoni L, Giberti C, Klages R. A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics. Chaos. 2015; 25: 73113.
- 12Hughes DB. Random Walks and Random Environments, Vol. 1: Random Walks. New York:Oxford University Press; 1995.
- 13Metzler R, Klafter J. The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys Rep. 2000; 339: 1-77.
- 14Evangelista LR, Lenzi EK. Fractional Diffusion Equations and Anomalous Diffusion. Cambridge:Cambridge University Press; 2018.
10.1017/9781316534649 Google Scholar
- 15Neto FDM, Neto AJS. An Introduction to Inverse Problems with Applications. New York:Springer; 2013.
10.1007/978-3-642-32557-1 Google Scholar
- 16Hasanov A, Pektaş B. Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method. Comput Math Appl. 2013; 65: 42-57.
- 17Ruan Z, Wang Z. Identification of a time-dependent source term for a time fractional diffusion problem. Appl Anal. 2017; 96: 1638-1655.
- 18Ismailov MI, Kanca F, Lesnic D. Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions. Appl Math Comp. 2011; 218: 4138-4146.
- 19Ali M, Malik SA. An inverse problem for a family of time fractional diffusion equations. Inverse Probl Sci Eng. 2017; 25: 1299-1322.
- 20Slodicka M, Siskova K. An inverse source problem in a semilinear time-fractional diffusion equation. Comput Math Appl. 2016; 72: 1655-1669.
- 21Ali M, Aziz S, Malik SA. Inverse source problem for a space-time fractional diffusion equation. Fract Calc Appl Anal. 2018; 21: 844-863.
- 22Ali M, Aziz S, Malik SA. Inverse problem for a space-time fractional diffusion equation: Application of fractional Sturm-Liouville operator. Math Methods Appl Sci. 2018; 41: 2733-2747.
- 23Ma YK, Prakash P, Deiveegan A. Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation. Chaos, Solitons Fractals. 2018; 108: 39-48.
- 24Samko SG, Kilbas AA, Marichev DI. Fractional Integrals and Derivatives: Theory and Applications. Amsterdam:Gordon and Breach Science Publishers; 1993.
- 25Gara R, Gorenflo R, Polito F, Tomovski Z. Hilfer-prabhakar derivatives and some applications. Appl Math Comput. 2014; 242: 576-589.
- 26Haubold HJ, Mathai AM, Saxena RK. Mittag-leffler functions and their applications. J Appl Math. 2011; 2011: 298628.
- 27Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag-Leffler Functions, Related Topics and Applications. Berlin Heidelberg:Springer-Verlag; 2014.
10.1007/978-3-662-43930-2 Google Scholar
- 28Podlubny I. Fractional Differential Equations. San Diego:Academic Press; 1999.
- 29Sadybekova M, Dildabekb G, Tengayevac A. Constructing a Basis from Systems of Eigenfunctions of one not Strengthened Regular Boundary Value Problem. Filomat. 2017; 31: 981-987.
- 30Frames ChristensenO. Bases An Introductory Course. Boston, New,York:Birkhäuser; 2008.
- 31Duan JS, Wang Z, Fu SZ. The zeros of the solutions of the fractional oscillator equation. Fract Calc Appl Sci. 2014; 17: 10-22.
- 32Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier; 2006.
10.1016/S0304-0208(06)80001-0 Google Scholar
- 33AL-Musalhi F, AL-Salti N, Kerbal S. Inverse problems of a fractional differential equation with Bessel operator. Math Model Nat Phenom. 2017; 12: 105-113.
