Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

Shuang-Shuang Zhou; Saima Rashid; Asia Rauf; Khadija Tul Kubra; Abdullah M. Alsharif; Shuang-Shuang Zhou; Saima Rashid; Asia Rauf; Khadija Tul Kubra; Abdullah M. Alsharif

doi:10.3934/math.2021703

AIMS Mathematics

2021, Volume 6, Issue 11: 12114-12132. doi: 10.3934/math.2021703

Previous Article Next Article

Research article

Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

1.
School of Science, Hunan City University, Yiyang 413000, China
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Government College Women University, Faisalabad, Pakistan
4.
Department of Mathematics, Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 10 April 2021 Accepted: 11 August 2021 Published: 20 August 2021
MSC : 26A51, 26A33, 26D07, 26D10, 26D15

For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between $ 0 $ and $ 1 $, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain $ T $. For the solution of the challenges, we proposed certain relevant cases.
- direct problem,
- inverse problem,
- fractional derivative,
- multinomial Mittag-Leffler function
Citation: Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Khadija Tul Kubra, Abdullah M. Alsharif. Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time[J]. AIMS Mathematics, 2021, 6(11): 12114-12132. doi: 10.3934/math.2021703

Related Papers:

Abstract

For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between $ 0 $ and $ 1 $, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain $ T $. For the solution of the challenges, we proposed certain relevant cases.

References

[1]	M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, 1969.
[2]	A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited: Amsterdam, The Netherlands, 204, 2006.
[3]	J. Liouville, Memoir on some questions of geometry and mechanics, and on a new kind of calculation to solve these questions, J. de l'École Pol. tech, 13 (1832), 1–69.
[4]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[5]	W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl, 332 (2007), 709–726. doi: 10.1016/j.jmaa.2006.10.040
[6]	Y. Li, F. Liub, I. W. Turner, T. Li, Time-fractional diffusion equation for signal smoothing, Appl. Math. Comput., 326 (2018), 108–116.
[7]	H. Nasrolahpour, A note on fractional electrodynamics, Comm. Nonlin. Sci. Numer. Sim., 18 (2013), 2589–2593. doi: 10.1016/j.cnsns.2013.01.005
[8]	A. Esen, T. A. Sulaiman, H. Bulut, H. M. Baskonus, Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation, Optik, 167 (2018), 150–156. doi: 10.1016/j.ijleo.2018.04.015
[9]	K. Ogata, Modern Control Engineering, Prentice Hall, 2010.
[10]	M. Senol, O. Tasbozan, A. Kurt, Numerical solutions of fractional Burgers' type equations with conformable derivative, Chinese J. Phy., 58 (2019), 75–84. doi: 10.1016/j.cjph.2019.01.001
[11]	S. Chen, Y. Liu, L. Wei, B. Guan, Exact solutions to fractional Drinfel'd-Sokolov-Wilson equations, Chinese J. Phy., 56 (2018), 708–720. doi: 10.1016/j.cjph.2018.01.010
[12]	Q. Feng, A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation, Chinese J. Phy., 56 (2018), 2817–2828. doi: 10.1016/j.cjph.2018.08.006
[13]	A. M. Wazwaz, Partial Differential Equations: Methdos and Applications, Balkema, Leiden, 2002.
[14]	G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1976.
[15]	J. Hietarinta, A search for bilinear equations passing Hirota's three-soliton condition. I. KdV type bilinear equations, J. Math. Phy., 28 (1987), 1732–1742. doi: 10.1063/1.527815
[16]	R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004.
[17]	R. Hirota, Exact solutions of the Sine-Gordan equation for multiple collisions of solitons, J. Phy. Soc. Japan, 33 (1972), 1459–1463. doi: 10.1143/JPSJ.33.1459
[18]	G. Adomian, A new approach to nonlinear partial differential equations, J. Math. Anal. Appl., 102 (1984), 420–434. doi: 10.1016/0022-247X(84)90182-3
[19]	S. Rashid, A. Khalid, S. Sultana, Z. Hammouch, R. Shah, A. M. Alsharif, A novel analytical view of time-fractional Korteweg-De Vries equations via a new integral transform, Symmetry, 13 (2021), 1254. doi: 10.3390/sym13071254
[20]	V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl., 316 (2006), 753–763. doi: 10.1016/j.jmaa.2005.05.009
[21]	H. Jafari, Iterative Methods for Solving System of Fractional Differential Equations, Ph.D. Thesis, Pune Uni, Pune City, India, 2006.
[22]	S. Rashid, K. T. Kubra, S. U. Lehre, Fractional spatial diffusion of a biological population model via a new integral transform in the settings of power and Mittag-Leffler nonsingular kernel, Phy. Scr., 96 (2021), 114003. doi: 10.1088/1402-4896/ac12e5
[23]	F. B. M. Belgacem, R. Silambarasan, Theory of natural transform, Math. Engg. Sci. Aeros., 3 (2012), 99–124.
[24]	V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Comp. Math. Appl., 59 (2010), 1801–1809. doi: 10.1016/j.camwa.2009.08.018
[25]	T. M. Elzaki, On the connections between Laplace and Elzaki transforms, Adv. Appl. Math., 6 (2011), 1–11.
[26]	S. Rashid, K. T. kubra, A. Rauf, Y.-M. Chu, Y. S. Hamed, New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method, Phy. Scr., 96 (2021), 105204. doi: 10.1088/1402-4896/ac0bce
[27]	Z. H. Khan, W. A. Khan, N-transform properties and applications, NUST J. Eng. Sci., 1 (2008), 127–133.
[28]	F. B. M. Belgacem, R. Silambarasan, Theory of the natural transform, Math. Engg. Sci. Aeros., 3 (2012), 99–124.
[29]	F. B. M. Belgacem, R. Silambarasan, Advances in the natural transform, In AIP Conference Proceedings, 1493 (2012), 106–110.
[30]	M. R. Spiegel, Schaum's Outline of Theory and Problems of Laplace Transform, McGraw-Hill, New York, NY, USA, 1965.
[31]	F. B. Belgacem, A. Karaballi, Sumudu transform fundamental properties investigations and applications, International Journal of Stochastic Analysis, 2006 (2006), Article ID 91083.
[32]	G. K. Watugala, Sumudu transform-a new integral transform to solve differential equations and control engineering problems, Mathematical engineering in industry, 6 (1998), 319–329.
[33]	M. S. Rawashdeh, H. Al-Jammal, New approximate solutions to fractional nonlinear systems of partial differential equations using the FNDM, Adv. Diff. Eqs, 235 (2016), 1-19.
[34]	S. Rashid, A. Khalid, O. Bazighifan, G. I. Oros, New modifications of integral inequalities via $\wp$-convexity pertaining to fractional calculus and their applications, Mathematics, 9 (2021), 1753. doi: 10.3390/math9151753
[35]	S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y.-M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, (2021), in Press.
[36]	S. Rashid, S. Sultana, Z. Hammouch, F. Jarad, Y. S. Hamed, Novel aspects of discrete dynamical type inequalities within fractional operators having generalized h-discrete Mittag-Leffler, Chaos Soliton. Fract., 151 (2021), 111204. doi: 10.1016/j.chaos.2021.111204
[37]	S.-S. Zhou, S. Rashid, E. Set, A. Garba, Ahmad, Y. S. Hamed, On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications, AIMS Math., 6 (2021), 9154–9176. doi: 10.3934/math.2021532
[38]	Y.-M. Chu, S. Rashid, F. Jarad, M. A. noor, H. Kalsoom, More new results on integral inequalities for generalized K-fractional conformable integral operators, Dics. Cont. Dyn. Ser. S, 14 (2021), 2119.
[39]	S.-S. Zhou, S. Rashid, A. Rauf, F. Jarad, Y. S. Hamed, K. M. Abualnaja, Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Math., 6 (2021), 8001-8029.
[40]	S. Rashid, S. Sultana, F. Jarad, H. Jafari, Y. S. Hamed, More efficient estimates via h-discrete fractional calculus theory and applications, Chaos Soliton. Fract., 147 (2021), 110981.
[41]	Y.-M. Chu, S. Rashid, T. Abdeljawad, A. Khalid, H. Kalsoom, On new generalized unified bounds via generalized exponentially harmonically s-convex functions on fractal sets, Adv. Diff. Equ., 2021 (2021), 1–33. doi: 10.1186/s13662-020-03162-2
[42]	H. Ge-Jile, S. Rashid, F. B. Farooq, S. Sultana, Some inequalities for a new class of convex functions with applications via local fractional integral, J. Funct. Space., 2021 (2021), 1–17.
[43]	S. Rashid, S. Parveen, H. Ahmad, Y.-M. Chu, New quantum integral inequalities for some new classes of generalized $\Psi$-convex functions and their scope in physical systems, Open. Phy., 19 (2021), 35–50.
[44]	S. Rashid, F. Jarad, K. M. Abualnaja, On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative, AIMS Math., 6 (2021), 10920–10946,
[45]	R. Hilfer, Applications of Fractional Calculus in Physics, World Science, Publishing: River Edge, NJ, USA, 2000.
[46]	I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[47]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin Heidelberg, 2014.
[48]	N. Tran, V. V. Au, Y. Zhou, N. H. Tuan, On a final value problem for fractional reaction-diffusion equation with Riemann-Liouville fractional derivative, Math. Meth. Appl. Sci., 43 (2020), 3086–3098.
[49]	N. H. Tuan, D. Baleanu, T. N. Thach, D. O. Regan, N. H. Can, Final value problem for nonlinear time fractional reaction-diffusion equation with discrete data, J. Comput. Appl. Math., 376 (2020), 112883. doi: 10.1016/j.cam.2020.112883
[50]	S. Kenichi, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426–447. doi: 10.1016/j.jmaa.2011.04.058
[51]	F. Al-Musalhi, N. Al-Salti, E. Karimov, Initial boundary value problems for a fractional differential equation with hyper-bessel operator, Fract. Cal. Appl. Anal., 21 (2018), 200–219. doi: 10.1515/fca-2018-0013
[52]	M. Kirane, S. A. Malik, M. A. Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci., 36 (2013), 1056–1069. doi: 10.1002/mma.2661
[53]	C. S. Liu, W. Chen, Z. Fu, A multiple-scale MQ-RBF for solving the inverse Cauchy problems in arbitrary plane domain, Eng. Anal. Bound. Elem., 68 (2016), 11–16. doi: 10.1016/j.enganabound.2016.02.011
[54]	G. Hu, F. Qu, B. Zhang, Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric, Math. Meth. Appl. Sci., 33 (2010), 147–156.
[55]	A. V. Glushak, T. A. Manaenkova, Direct and inverse problems for an abstract differential equation containing Hadamard fractional derivatives, Diff. Equ., 47 (2011), 1307–1317 doi: 10.1134/S0012266111090084
[56]	W. Fan, F, Liu, X. Jiang, I. Turner, some noval numerical techniques for an inverse problem of the multi-term time fractional partial differential equation, J. Comput. Appl. Math., 25 (2017), 1618–1638.
[57]	M. Ali, S. Aziz, S. A. Malik, Inverse source problems for a space-time fractional differential equation, Inverse Prob. Sci. Eng., 28 (2019), 1–22.
[58]	S. Tarar, R. Tinaztepe, S. Ulusoy, Determination of an unknown source term in a apace-time fractional diffusion equation, Journal of Fractional Calculus and Applications, 6 (2015), 83–90.
[59]	M. Slodicka, Determination of a solely time-dependent source in a semilinear parabolic problem by means of boundary measurements, J. Comput. Appl. Math., 289 (2015), 433–440. doi: 10.1016/j.cam.2014.10.004
[60]	S. A. Malik, S. Aziz, Identification of an unknown source term for a time fractional fourth-order parabolic equation, Elect. J. Diff. Equ., 2016 (2016), 1-20. doi: 10.1186/s13662-015-0739-5
[61]	E. Karimov, S. Pirnafasov, Higher order multi-term time fractional partial differential equations involving Caputo-Fabrizo derivative, Elect. J. Diff. Equ., 243 (2017), 1–11.
[62]	P. Feng, E. T. Karimov, Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations, J. Inverse Ill-Posed P., 23 (2015), 339–353.
[63]	M. Ali, S. Aziz, S. A. Malik, Inverse problem for a space-time fractional diffusion equation: Application of fractional Sturm-Liouville operator, Math. Meth. Appl. Sci., 41 (2018), 2733–2744. doi: 10.1002/mma.4776
[64]	W. Rundell, X. Xu, L. Zuo, The determination of an unknown boundary condition in a fractional diffusion equation, Appl. Anal., 92 (2013), 1511–1526. doi: 10.1080/00036811.2012.686605
[65]	H. Sun, G. Li, X. Jia, Simultaneous inversion for the diffusion and source coefficients in the multi-term TFDE, Inverse Prob. Sci. Eng., 336(2018), 114–126.
[66]	S. G. Samko, A. A. Kilbas, D. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Science Publishers, 1993.
[67]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
[68]	Y. Luchko, R. Gorenflo, An Operational Method for Solving Fractional Differential Equations with the Caputo Derivatives, Acta Math. Vietnam, 24(1999), 207–233.
[69]	S. A. Malik, A. Ilyas, A. Samreen, Simultaneous determination of a source term and diffusion concentration for a multi-term space-time fractional diffusion equation, Math. Model. Anal., 26 (2021), 411–431. doi: 10.3846/mma.2021.11911
[70]	M. Ali, S. Aziz, S. A. Malik, Inverse problem for a multi-term fractional differential equation: Operational Calculus Approach, Frac. Calc. Appl. Anal., 23 (2020), 799–821. doi: 10.1515/fca-2020-0040
[71]	E. I. Moiseev, On the basis property of systems of sines and cosines, Doklady AN SSSR, 275 (1984), 794–798.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)