https://orcid.org/0000-0001-7716-3455
Abstract
The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.
Issue Section:
Research Papers
References
1.
Keshavarz
,
E.
,
Ordokhani
,
Y.
, and
Razzaghi
,
M.
, 2014
, “
Bernoulli Wavelet Operational Matrix of Fractional Order Integration and Its Applications in Solving the Fractional Order Differential Equations
,” Appl. Math. Model.
,
38
, pp. 6038
–6051
.10.1016/j.apm.2014.04.0642.
Syam
,
M. I.
, and
R
,
M. A.
, 2019
, “
Fractional Differential Equations With Atangana-Baleanu Fractional Derivative: Analysis and Applications
,” Chaos, Solitons Fractals X
,
2
, pp. 100
–113
.10.1016/j.csfx.2019.1000133.
Jafari
,
H.
,
Yousefi
,
S. A.
,
Firoozjaee
,
M. A.
,
Momani
,
S.
, and
Khalique
,
C. M.
, 2011
, “
Application of Legendre Wavelets for Solving Fractional Differential Equations
,” Comput. Math. Appl.
,
62
, pp. 1038
–1045
.10.1016/j.camwa.2011.04.0244.
Samraiz
,
M. E.
, and
Set
,
E.
,
Hasnain
,
M.
, and
Rahman
,
G.
, 2019
, “
On an Extension of Hadamard Fractional Derivative
,” J. Inequalities Appl.
,
2019
(1
), pp. 1
–15
.10.1186/s13660-019-2218-05.
Al-Refai
,
M.
, and
Pal
,
K.
, 2019
, “
New Aspects of Caputo-Fabrizio Fractional Derivative
,” Prog. Fract. Differ. Appl.
,
5
(2
), pp. 157
–166
.10.18576/pfda/0502066.
Guo
,
W.
,
Wang
,
Y.
,
Zhao
,
F.
, and
Dai
,
F.
, 2019
, “
Riesz Fractional Derivative Eliteguided Sine Cosine Algorithm
,” Appl. Soft Comput.
,
81
, p. 105481
.10.1016/j.asoc.2019.04.0447.
Khader
,
M. M.
, and
Sweilam
,
N. H.
, 2013
, “
On the Approximate Solutions for System of Fractional Integrodifferential Equations Using Chebyshev Pseudo-Spectral Method
,” Appl. Math. Modell.
,
37
, pp. 9819
–9828
.10.1016/j.apm.2013.06.0108.
Wang
,
J.
,
Xu
,
T. Z.
,
Wei
,
Y. Q.
, and
Xie
,
J. Q.
, 2018
, “
Numerical Simulation for Coupled Systems of Nonlinear Fractional Order Integro-Differential Equations Via Wavelets Method
,” Appl. Math. Comput.
,
324
, pp. 36
–50
.10.1016/j.amc.2017.12.0109.
Shah
,
K.
,
Amin
,
R.
,
Ali
,
G.
,
Mlaiki
,
N.
, and
Abdeljawad
,
T.
, 2022
, “
Algorithm for the Solution of Nonlinear Variable Order Pantograph Fractional Integro Differential Equations Using Haar Method
,” Fractals
,
30
(8
), p. 2240225
.10.1142/S0218348X2240225310.
Sweilam
,
N. H.
,
Khader
,
M. M.
, and
Al-Bar
,
R. F.
, 2007
, “
Numerical Studies for a Multi-Order Fractional Differential Equation
,” Phys. Lett. A.
,
371
, pp. 26
–33
.10.1016/j.physleta.2007.06.01611.
He
,
J.
, “
Approximate Analytical Solution for Seepage Flow With Fractional Derivatives in Porous Media
,” Comput. Methods Appl. Mech. Eng.
,
167
(1
), pp. 57
–68
.10.1016/S0045-7825(98)00108-X12.
Ganji
,
D. D.
, and
Rajabi
,
A.
, 2006
, “
Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations
,” Int. Commun. Heat Mass Transfer
,
33
(3
), pp. 391
–400
.10.1016/j.icheatmasstransfer.2005.11.00113.
He
,
J.
, “
Homotopy Perturbation Technique
,” Comput. Methods Appl. Mech. Eng.
,
178
(3
), pp. 257
–262
.10.1016/S0045-7825(99)00018-314.
Hashim
,
I.
, 2006
, “
Adomian Decomposition Method for Solving BVPs for Fourth-Order Integro-Differential Equations
,” J. Comput. Appl. Math.
,
193
, pp. 658
–664
.10.1016/j.cam.2005.05.03415.
Hashim
,
I.
,
Abdulaziz
,
O.
, and
Momani
,
S.
, 2009
, “
Homotopy Analysis Method for Fractional IVPs
,” Commun. Nonlinear Sci. Numer. Simul.
,
14
, pp. 674
–684
.10.1016/j.cnsns.2007.09.01416.
Khader
,
M. M.
,
Sweilam
,
N. H.
, and
Mahdy
,
A. M. S.
, 2013
, “
Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM
,” Appl. Math. Inf. Sci.
,
7
(5
), pp. 2013
–2020
.10.12785/amis/07054117.
Rawashdeh
,
E.
, 2006
, “
Numerical Solution of Fractional Integro-Differential Equations by Collocation Method
,” Appl. Math. Comput.
,
176
, pp. 1
–6
.10.1016/j.amc.2005.09.05918.
Wang
,
G.
, and
Wazwaz
,
A. M.
, 2022
, “
On the Modified Gardner Type Equation and Its Time Fractional Form
,” Chaos, Solitons Fractals
,
155
, p. 111694
.10.1016/j.chaos.2021.11169419.
Wang
,
G.
, and
Wazwaz
,
A. M.
, 2022
, “
A New
Dimensional KdV Equation and mKdV Equation With Their Corresponding Fractional Forms
,” Fractals
,
30
(4
), p. 2250081
.10.1142/S0218348X2250081520.
Wang
,
G.
, 2021
, “
A New (3 + 1)-Dimensional Schrodinger Equation: Derivation, Soliton Solutions and Conservation Laws
,” Nonlinear Dyn.
,
104
(2
), pp. 1595
–1602
.10.1007/s11071-021-06359-621.
Wang
,
G.
, 2021
, “
Symmetry Analysis, Analytical Solutions and Conservation Laws of a Generalized KdV-Burgers-Kuramoto Equation and Its Fractional Version
,” Fractals
,
29
(4
), p. 2150101
.10.1142/S0218348X2150101222.
Wang
,
G.
, 2021
, “
A Novel
Dimensional Sine-Gorden and a Sinh-Gorden Equation: Derivation, Symmetries and Conservation Laws
,” Appl. Math. Lett.
,
113
, p. 106768
.10.1016/j.aml.2020.10676823.
Wang
,
G.
,
Yang
,
K.
,
Gu
,
H.
,
Guan
,
F.
, and
Kara
,
A. H.
, 2020
, “
A
Dimensional Sine-Gordon and Sinh-Gordon Equations With Symmetries and Kink Wave Solutions
,” Nucl. Phys. B
,
953
(4
), p. 114956
.10.1016/j.nuclphysb.2020.11495624.
Hu
,
W.
,
Wang
,
Z.
,
Zhao
,
Y.
, and
Deng
,
Z.
, 2020
, “
Symmetry Breaking of Infinite-dimensional Dynamic System
,” Appl. Math. Lett.
,
103
, p. 106207
.10.1016/j.aml.2019.10620725.
Hu
,
W.
,
Ye
,
J.
, and
Deng
,
Z.
, 2020
, “
Internal Resonance of a Flexible Beam in a Spatial Tethered System
,” J. Sound Vib.
,
475
, p. 115286
.10.1016/j.jsv.2020.11528626.
Hu
,
W.
,
Xu
,
M.
,
Song
,
J.
,
Gao
,
Q.
, and
Deng
,
Z.
, 2021
, “
Coupling Dynamic Behaviors of Flexible Stretching Hub-Beam System
,” Mech. Syst. Signal Process.
,
151
, p. 107389
.10.1016/j.ymssp.2020.10738927.
Zhou
,
Q.
,
Huang
,
Z.
,
Sun
,
Y.
,
Triki
,
H.
,
Liu
,
W.
, and
Biswas
,
A.
, 2023
, “
Collision Dynamics of Three-Solitons in an Optical Communication System With Third-Order Dispersion and Nonlinearity
,” Nonlinear Dyn.
,
111
, pp. 5757
–5765
.10.1007/s11071-022-08138-328.
Zhou
,
Q.
,
Triki
,
H.
,
Xu
,
J.
,
Zeng
,
Z.
,
Liu
,
W.
, and
Biswas
,
A.
, 2022
, “
Perturbation of Chirped Localized Waves in a Dual-Power Law Nonlinear Medium
,” Chaos, Solitons Fractals
,
160
, p. 112198
.10.1016/j.chaos.2022.11219829.
Zhou
,
Q.
,
Zhong
,
Y.
,
Triki
,
H.
,
Sun
,
Y.
,
Xu
,
S.
,
Liu
,
W.
, and
Biswas
,
A.
, 2022
, “
Chirped Bright and Kink Solitons in Nonlinear Optical Fibers With Weak Nonlocality and Cubic-Quintic-Septic Nonlinearity
,” Chin. Phys. Lett.
,
39
, p. 044202
.10.1088/0256-307X/39/4/04420230.
Darweesh
,
A.
,
Al-Khaled
,
K.
, and
Abue Al Yaqeen
,
O.
, 2023
, “
Haar Wavelets Method for Solving Class of Coupled Systems of Linear Fractional Fredholm Integro-Differential Equations
,” Heliyon
, accepted, pp. 1
–27
.10.2139/ssrn.437752031.
Darweesh
,
A.
, and
Al-Khaled
,
K.
, 2020
, “
Sinc-Galerkin Method for Solving Higher Order Fractional Boundary Value Problems
,” Nonlinear Dyn. Syst. Theory
,
20
(3
), pp. 267
–281
.https://www.e-ndst.kiev.ua/v20n3/4(73).pdf32.
Aman
,
S.
,
Khan
,
I.
,
Ismail
,
Z.
,
Salleh
,
M. Z.
, and
Al-Mdallal
,
Q. M.
, 2017
, “
Heat Transfer Enhancement in Free Convection Flow of CNTs Maxwell Nanofluids With Four Different Types of Molecular Liquids
,” Sci. Rep.
,
7
(1
), pp. 1
–13
.10.1038/s41598-017-01358-333.
Bergman
,
T. L.
,
Lavine
,
A. S.
,
Incropera
,
F. P.
, and
DeWitt
,
D. P.
, 2011
, Introduction to Heat Transfer
,
Wiley
, Hoboken, NJ.34.
Khan
,
N. S.
,
Gul
,
T.
,
Islam
,
S.
,
Khan
,
I.
,
Alqahtani
,
A. M.
, and
Alshomrani
,
A. S.
, 2017
, “
Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder With Heat Transfer
,” Appl. Sci.
,
7
(3
), p. 271
.10.3390/app703027135.
Aaiza
,
G.
,
Khan
,
I.
, and
Shafie
,
S.
, 2015
, “
Energy Transfer in Mixed Convection Mhd Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled With Saturated Porous Medium
,” Nanoscale Res. Lett.
,
10
(1
), pp. 1
–14
.10.1186/s11671-015-1144-436.
Yardley
,
J.
, 2012
, Introduction to Molecular Energy Transfer
,
Elsevier
, Amsterdam, The Netherlands.37.
Hussanan
,
A.
,
Ismail
,
Z.
,
Khan
,
I.
,
Hussein
,
A. G.
, and
Shafie
,
S.
, 2014
, “
Unsteady Boundary Layer MHD Free Convection Flow in a Porous Medium With Constant Mass Diffusion and Newtonian Heating
,” Eur. Phys. J. Plus
,
129
(3
), pp. 1
–16
.10.1140/epjp/i2014-14046-x38.
Middleman
,
S.
, 1998
, “
An Introduction to Mass and Heat Transfer: Principles of Analysis and Design
,” Eur. J. Eng. Edu.
,
23
(3
), p. 37
.https://www.tandfonline.com/doi/epdf/10.1080/0304379980892351639.
Djennadi
,
S.
,
Shawagfeh
,
N.
, and
Arqub
,
O. A.
, 2021
, “
A Numerical Algorithm in Reproducing Kernel-Based Approach for Solving the Inverse Source Problem of the Time-Space Fractional Diffusion Equation
,” Partial Diff. Equ. Appl. Math.
,
4
, p. 100164
.10.1016/j.padiff.2021.10016440.
Do
,
Q. H.
,
Ngo
,
H. T.
, and
Razzaghi
,
M.
, 2021
, “
A Generalized Fractional-Order Chebyshev Wavelet Method for Two-Dimensional Distributed-Order Fractional Differential Equations
,” Commun. Nonlinear Sci. Numer. Simul.
,
95
, p. 105597
.10.1016/j.cnsns.2020.10559741.
Yuttanan
,
B.
,
Razzaghi
,
M.
, and
Vo
,
T. N.
, 2021
, “
Legendre Wavelet Method for Fractional Delay Differential Equations
,” Appl. Numer. Math.
,
168
, pp. 127
–142
.10.1016/j.apnum.2021.05.02442.
Podlubny
,
I.
,
Chechkin
,
A.
,
Skovranek
,
T.
,
Chen
,
Y.
, and
Jara
,
B. M. V.
, 2009
, “
Matrix Approach to Discrete Fractional Calculus II: Partial Fractional Differential Equations
,” J. Comput. Phys.
,
228
(8
), pp. 3137
–3153
.10.1016/j.jcp.2009.01.01443.
Qutaiba
,
W.
, and
Ibraheem
,
M. H.
, 2023
, “
Determination of Time-Dependent Coefficient in Time Fractional Heat Equation
,” Partial Diff. Equ. Appl. Math.
,
7
, p. 100492
.10.1016/j.padiff.2023.10049244.
Diethelm
,
K.
, 2010
, The Analysis of Fractional Differential Equations
,
Springer-Verlag
,
Berlin, Germany
.45.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
, 2006
, Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies, 204),
Elsevier Science B.V
,
Amsterdam, The Netherlands
.46.
Podlubny
,
I.
, 2007
, Fractional Differential Equations, Physics and Engineering
,
Springer, Academic Press
,
San Diego
, CA, p. 1999
.47.
Miller
,
K. S.
, and
Ross
,
B.
, 1993
, An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.48.
Moreno
,
S. G.
, and
García-Caballero
,
E. M.
, 2009
, “
Non-Standard Orthogonality for the Little q-Laguerre Polynomials
,” Appl. Math. Lett.
,
22
, pp. 1745
–1749
.10.1016/j.aml.2009.05.017Copyright © 2023 by ASME
You do not currently have access to this content.
