Abstract
In this study, we propose an efficient iterative technique for investigating the analytical approximate solutions of the Caputo derivative-based fractional diffusion-wave equations. The proposed technique involves the Sumudu transform (ST) and the Daftardar-Gejji and Jafari method (DGJM). Further, we give various illustrations of linear and nonlinear diffusion-wave equations and demonstrate the solutions figuratively. The proposed technique is free from complex calculations and works without any discretization.
References
1.
Kumar
, M.
, 2022
, “An Efficient Numerical Scheme for Solving a Fractional-Order System of Delay Differential Equations
,” Int. J. Appl. Comput. Math.
, 8
(5
), pp. 1--13. 10.1007/s40819-022-01466-32.
Liang
, Y.
, and Chen
, W.
, 2018
, “Continuous Time Random Walk Model With Asymptotical Probability Density of Waiting Times Via Inverse Mittag-Leffler Function
,” Commun. Nonlinear Sci. Numer. Simul.
, 57
, pp. 439
–448
.10.1016/j.cnsns.2017.10.0143.
Liang
, Y.
, Yu
, Y.
, and Magin
, R. L.
, 2022
, “Computation of the Inverse Mittag–Leffler Function and Its Application to Modeling Ultraslow Dynamics
,” Fractional Calculus Appl. Anal.
, 25
(2
), pp. 439
–452
.10.1007/s13540-022-00020-84.
Tandel
, P.
, Patel
, H.
, and Patel
, T.
, 2022
, “Tsunami Wave Propagation Model: A Fractional Approach
,” J. Ocean Eng. Sci.
, 7
(6
), pp. 509
–520
.10.1016/j.joes.2021.10.0045.
Acosta
, S.
, Chan
, J.
, Johnson
, R.
, and Palacios
, B.
, 2024
, “Pseudodifferential Models for Ultrasound Waves With Fractional Attenuation
,” SIAM J. Appl. Math.
, 84
(4
), pp. 1609
–1630
.10.1137/24M16340116.
Jafari
, H.
, Yousefi
, S.
, Firoozjaee
, M.
, Momani
, S.
, and Khalique
, C. M.
, 2011
, “Application of Legendre Wavelets for Solving Fractional Differential Equations
,” Comput. Math. Appl.
, 62
(3
), pp. 1038
–1045
.10.1016/j.camwa.2011.04.0247.
Liu
, L.
, Zhang
, S.
, Chen
, S.
, Liu
, F.
, Feng
, L.
, Turner
, I.
, Zheng
, L.
, and Zhu
, J.
, 2023
, “An Application of the Distributed-Order Time-and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures
,” Fractal Fractional
, 7
(3
), p. 239
.10.3390/fractalfract70302398.
Kuttler
, C.
, Maslovskaya
, A. G.
, and Moroz
, L. I.
, 2021
, “Numerical Simulation of Time-Fractional Diffusion-Wave Processes Applied to Communication in Bacterial Populations
,” Days on Diffraction (DD)
, May 31–June 4, St. Petersburg, Russia, pp. 1
–6
.10.1109/DD52349.2021.95986489.
González-Olvera
, M.
, Torres
, L.
, Hernández-Fontes
, J.
, and Mendoza
, E.
, 2021
, “Time Fractional Diffusion Equation for Shipping Water Events Simulation
,” Chaos, Solitons Fractals
, 143
, p. 110538
.10.1016/j.chaos.2020.11053810.
Daftardar-Gejji
, V.
, and Bhalekar
, S.
, 2008
, “Solving Fractional Diffusion-Wave Equations Using a New Iterative Method
,” Fractional Calculus Appl. Anal.
, 11
(2
), pp. 193
–202
.11.
Sandev
, T.
, Metzler
, R.
, and Tomovski
, Ž.
, 2011
, “Fractional Diffusion Equation With a Generalized Riemann–Liouville Time Fractional Derivative
,” J. Phys. A: Math. Theor.
, 44
(25
), p. 255203
.10.1088/1751-8113/44/25/25520312.
Ou
, C.
, Cen
, D.
, Vong
, S.
, and Wang
, Z.
, 2022
, “Mathematical Analysis and Numerical Methods for Caputo-Hadamard Fractional Diffusion-Wave Equations
,” Appl. Numer. Math.
, 177
, pp. 34
–57
.10.1016/j.apnum.2022.02.01713.
Jia
, J.
, and Wang
, H.
, 2023
, “Analysis of Asymptotic Behavior of the Caputo–Fabrizio Time-Fractional Diffusion Equation
,” Appl. Math. Lett.
, 136
, p. 108447
.10.1016/j.aml.2022.10844714.
Abuomar
, M. M.
, Syam
, M. I.
, and Azmi
, A.
, 2022
, “A Study on Fractional Diffusion—Wave Equation With a Reaction
,” Symmetry
, 14
(8
), p. 1537
.10.3390/sym1408153715.
Derakhshan
, M.
, and Aminataei
, A.
, 2021
, “An Iterative Method for Solving Fractional Diffusion-Wave Equation Involving the Caputo–Weyl Fractional Derivative
,” Numer. Linear Algebra Appl.
, 28
(2
), p. e2345
.10.1002/nla.234516.
Mustafa
, S.
, Hajira
, Khan
, H.
, Shah
, R.
, and Masood
, S.
, 2021
, “A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations
,” Fractal Fractional
, 5
(4
), p. 206
.10.3390/fractalfract504020617.
Liu
, Z.
, Cheng
, A.
, and Li
, X.
, 2018
, “A Novel Finite Difference Discrete Scheme for the Time Fractional Diffusion-Wave Equation
,” Appl. Numer. Math.
, 134
, pp. 17
–30
.10.1016/j.apnum.2018.07.00118.
Heydari
, M.
, Hooshmandasl
, M.
, Ghaini
, F. M.
, and Cattani
, C.
, 2015
, “Wavelets Method for the Time Fractional Diffusion-Wave Equation
,” Phys. Lett. A
, 379
(3
), pp. 71
–76
.10.1016/j.physleta.2014.11.01219.
Wang
, K.
, and Liu
, S.
, 2016
, “A New Sumudu Transform Iterative Method for Time-Fractional Cauchy Reaction–Diffusion Equation
,” SpringerPlus
, 5
(1
), p. 865
.10.1186/s40064-016-2426-820.
Daftardar-Gejji
, V.
, and Jafari
, H.
, 2006
, “An Iterative Method for Solving Nonlinear Functional Equations
,” J. Math. Anal. Appl.
, 316
(2
), pp. 753
–763
.10.1016/j.jmaa.2005.05.00921.
Prakash
, A.
, Kumar
, M.
, and Baleanu
, D.
, 2018
, “A New Iterative Technique for a Fractional Model of Nonlinear Zakharov–Kuznetsov Equations Via Sumudu Transform
,” Appl. Math. Comput.
, 334
, pp. 30
–40
.10.1016/j.amc.2018.03.09722.
Kumar
, M.
, and Daftardar-Gejji
, V.
, 2019
, “Exact Solutions of Fractional Partial Differential Equations by Sumudu Transform Iterative Method
,” Fractional Calculus and Fractional Differential Equations
, Trends in Mathematics, Springer, Singapore, pp. 157
–180
.10.1007/978-981-13-9227-6_823.
Anaç
, H.
, Merdan
, M.
, and Kesemen
, T.
, 2020
, “Solving for the Random Component Time-Fractional Partial Differential Equations With the New Sumudu Transform Iterative Method
,” SN Appl. Sci.
, 2
(6
), pp. 1--10. 10.1007/s42452-020-2625-324.
Mittag-Leffler
, G.
, 1903
, “Sur la Nouvelle Fonction E(x)
,” C. R. Acad. Sci. Paris
, 137
, pp. 554
–558
.25.
Purohit
, S.
, 2013
, “Solutions of Fractional Partial Differential Equations of Quantum Mechanics
,” Adv. Appl. Math. Mech.
, 5
(5
), pp. 639
–651
.10.4208/aamm.12-m129826.
Belgacem
, F. B. M.
, and Karaballi
, A. A.
, 2006
, “Sumudu Transform Fundamental Properties Investigations and Applications
,” Int. J. Stochastic Anal.
, 2006
(1
), pp. 1–23.10.1155/JAMSA/2006/9108327.
Amer
, Y.
, Mahdy
, A.
, and Youssef
, E.
, 2018
, “Solving Systems of Fractional Nonlinear Equations of Emden Fowler Type by Using Sumudu Transform Method
,” Global J. Pure Appl. Math.
, 14
(1
), pp. 91
–113
.https://www.tu.edu.sa/Attachments/f7a4c8f9-0b17-4caf-8546-bf338cd9ed57_.pdf28.
Kumar
, M.
, 2024
, “Exact Solutions of (1 + 2)-Dimensional Non-Linear Time-Space Fractional PDEs
,” Arab J. Math. Sci.
, 30
(1
), pp. 30
–42
.10.1108/AJMS-11-2021-028229.
Jafari
, H.
, and Seifi
, S.
, 2009
, “Homotopy Analysis Method for Solving Linear and Nonlinear Fractional Diffusion-Wave Equation
,” Commun. Nonlinear Sci. Numer. Simul.
, 14
(5
), pp. 2006
–2012
.10.1016/j.cnsns.2008.05.00830.
Jafari
, H.
, and Daftardar-Gejji
, V.
, 2006
, “Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by Adomian Decomposition
,” Appl. Math. Comput.
, 180
(2
), pp. 488
–497
.10.1016/j.amc.2005.12.031Copyright © 2025 by ASME
You do not currently have access to this content.