RESEARCH PAPER
Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy
1
Department of Mathematics, Faculty of Sciences, Sudan University of Science and Technology, HGX7+M5F, Al Khartoum, Sudan
2
Department of Mathematics, Faculty of Sciences, University of Ha’il, Ha’il 2440, Saudi Arabia
3
Department of Mathematics, Faculty of Sciences and Arts-Alkamil, University of Jeddah, Hamzah Ibn Al Qasim St, Al Sharafeyah, Jeddah 23218, Saudi Arabia
4
Department of Mathematics, Faculty of Sciences, Taif University, Taif 21944, Saudi Arabia
Submission date: 2022-09-08
Acceptance date: 2023-02-05
Online publication date: 2023-07-11
Publication date: 2023-09-01
Acta Mechanica et Automatica 2023;17(3):372-380
KEYWORDS
fractional-order heat-like and wave-like equationsinitial-boundary value problemsAdomian decomposition method
ABSTRACT
This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-like equations, a graphical depiction of the solutions to three instructive cases was constructed. Solution graphs were arrived at for integer and fractional-order problems. The derived and exact solutions to integer-order problems were found to be in excellent agreement. The subject of the present research endeavour is the convergence of fractional-order solutions. This strategy is considered to be the most successful way of addressing fractional-order initial-boundary value issues in science and engineering. This strategy is presented here.
REFERENCES (34)
1.
Eltaib M Abd Elmohmoud, Mohamed Z Mohamed. Numerical treatment of some fractional nonlinear equations by Elzaki transform. Journal of Taibah University for Science.2022;16(1): 774-787.
2.
Metzler, R Nonnenmacher TF. Space-and time-fractional diffusion and wave equations fractional Fokker–Planck equations and physical motivation. Chem. Phys.2002;284:67–90.
3.
Metzler, R Klafter J. The random walk’s guide to anomalous diffusion A fractional dynamics approach. Phys Rep.2000; 339: 1–77.
4.
M Subramanian, M Manigandan, C Tunç, T N Gopal, J Alzabut. On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order. Journal of Taibah University for Science.2022;16(1):1-23.
5.
Imran Talib, Ali Raza, Abdon Atangana, Muhammad Bilal Riaz. Numerical study of multi- order fractional differential equations with constant and variable coefficients. Journal of Taibah University for Science.2022;16(1):608-620.
6.
Jothimani, K Kaliraj, K Hammouch, Z Ravichandran C. New results on controllability in the framework of fractional integro differential equations with nondense domain. Eur Phys J Plus. 2019;134(9):441.
7.
Khan H, Shah R, Kumam P, Arif M. Analytical Solutions of Fractional- Order Heat and Wave Equations by the Natural Transform Decomposition Method. Entropy. 2019; 21: 597.
8.
Lei Y, Wang H, Chen X,Yang X, You Z, Dong S, Gao J. Shear property high-temperature rheological performance and low-temperature flexibility of asphalt mastics modified with bio-oil. Constr Build Mater. 2018;174:30–37.
9.
Zhang ZY. Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation.Proc R Soc.2020;476:2019-0564.
10.
Mohamed Z Mohamed, Amjad E Hamza, Abdelilah Kamal H Sedeeg. An Efficient Approximate Solutions of the fractional coupled Burger’s equation by conformable double Sumudu transforms. Ain shams journal.2022;16: 101-879.
11.
Shah R, Khan H, Arif M, et al. Application of Laplace–Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy. 2019;21(4):335.
12.
Shah HF, ur Rahman K, Shahzad M G. Numerical solution of fractional order smoking model via Laplace Adomian decomposition method. Alex Eng J. 2018;57(2):1061–1069.
13.
Rubbab Q, Nazeer M, Ahmad F, et al. Numerical simulation of advection– diffusion equation with Caputo Fabrizio time fractional derivative in cylindrical domains applications of pseudo-spectral collocation method. Alexandria Eng J. 2021;60(1):1731–1738.
14.
Agbata B C, Shior MM, Olorunni shola OA, Ezugorie IG, Obeng- Denteh. Analysis of Homotopy Perturbation Method (HPM) and its Application for Solving Infectious Disease Models. IJMSS 2021,9: 27-38.
15.
Hamoud A, Dawood L, Ghadle K, Atshan S. Usage of the modified variational iteration technique for solving Fredholm integro-differential equations. Int J Mech Prod Eng Res Dev.2019; 9(2):895–902.
16.
Osman M, Gong ZT, Mohammed A. Differential transform method for solving fuzzy fractional wave equation. J Comput Anal Appl.2021; 29(3):431–453.
17.
Omer Acan1, Omer Firat, Yildiray Keskin, Galip Oturanc. Conformable variational iteration method. NTMSCI.2017;5(1):172-178.
18.
G Adomian. A review of the decomposition method in applied mathematics. J Math Anal Appl.1988;135:501–544.
19.
G. Adomian Solving Frontier Problems of Physics The Decomposition method. Kluwer Academic Publishers Boston. 1994.
20.
S Momani, Z Odibat. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals. 2007;31(5):1248–1255.
21.
Z Odibat, S Momani. Approximate solutions for boundary value problems of time-fractional wave equation. Appl Math Comput. 2006;181(1):767–774.
22.
V Marinca. An approximate solution for one-dimensional weakly nonlinear oscillations. J Nonlinear Sci Numer Simul.2002;3(2): 107–110.
23.
TH Hao. Search for variational principles in electrodynamics by Lagrange method. Int J Nonlinear Sci Numer Simul.2005;6(2): 209–210.
24.
Mustafa S. Hajira, Khan H, Shah R, Masood S. A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations. Fractal Fract. 2021;5:206.
25.
Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity An Introduction to Mathematical Models. World Scientific Singapore. 2010.
26.
Anh VV, Leonenko NN. Harmonic analysis of random fractional diffusion–wave equations. Appl Math Comput.2003;141:77–85.
27.
Ali EJ. A new technique of initial boundary value problems using Adomian decomposition method. Int Math Forum. 2012;7:799–814.
28.
Mohamed Z Mohamed, Tarig M Elzaki. Applications of new integral transform for linear and nonlinear fractional partial differential equations. Journal of King Saud University – Science. 2020;32:544–549.
29.
Mustafa S, Hajira, Khan Hm Shah R, Masood S. A Novel Analytical Approach for the Solution of Fractional-Order Diffusion-Wave Equations. Fractal Fract. 2021;5:206.
30.
Alia A, Shaha K, Lib Y, Khana RA. Numerical treatment of coupled system of fractional order partial differential equations. J Math Comput Sci. 2019;19:74–85.
31.
Hassan Khan, Rasool Shah, Poom Kumam, and Muhammad Arif. Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method. mdpi journal entropy. 2019;21:597.
32.
S Sarwar, Salem Alkhalaf, S Iqbal, MA Zahid. A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations. Computers & Mathematics with Applications. 2020;70(5):942-953.
33.
Shah R, Khan H, Mustafa S, Kumam P, Arif M. Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method. Entropy,2019, 557.
34.
M A AL-Jawary. An Efficient Treatments For Linear And Nonlinear Heat-Like And Wave-Like Equations With Variable Coefficients. IOSR Journal of Mathematics.2015;11(4):01-13.
| eISSN: | 2300-5319 |
| ISSN: | 1898-4088 |
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