# Approximate Solutions Of Time-fractional Fourth-order Differential Equations With Variable Coefficients Xiaohua Wang

Teaching Affairs Office, Zhejiang Guangsha Vocational and Technical University of Construction, Dongyang 322100, China

Accepted: August 5, 2022
Publication Date: November 24, 2022

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

## ABSTRACT

In this work, the homotopy analysis method (HAM) is proposed to obtain semi-analytical solutions of timefractional fourth-order partial differential equations (PDEs) with variable coefficients, by the Caputo fractional derivative in the time direction. Convergence of this method has been considered and some illustrative examples show the effect of changing homotopy auxiliary parameter ¯h on the convergence of the approximate solution. Comparison of obtained results with other techniques such as Adomian decomposition method and modified variational iteration method, in literature demonstrate that our utilized method is powerful and reliable technique. Moreover, the absolute errors of considered problems in the integer differential order cases, show that the reported results are very closed to the exact solutions.

Keywords: Homotopy analysis method; time-fractional fourth-order equation; variable coefficient; Caputo fractional derivative.

## REFERENCES

1.  M. S. Hashemi and D. Baleanu. Lie symmetry analysis of fractional differential equations. Chapman and Hall/CRC, 2020.
2.  I. Podlubny, (1999) “An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications" Math. Sci. Eng 198: 340.
3.  A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and applications of fractional differential equations. 204.elsevier, 2006.
4.  M. Inc, A. Yusuf, A. Isa Aliyu, and M. Hashemi, (2018) “Soliton solutions, stability analysis and conservation laws for the brusselator reaction diffusion model with time- and constant-dependent coefficients" European Physical Journal Plus 133(5): DOI: 10.1140/epjp/i2018-11989-8.
5.  M. S. Hashemi, A. Haji-Badali, and P. Vafadar, (2014) “Group invariant solutions and conservation laws of the fornberg- whitham equation" Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 69(8-9): 489–496. DOI: 10.5560/ZNA.2014-0037.
6.  A. Akbulut, M. Sajjad Hashemi, and H. Rezazadeh, (2021) “New conservation laws and exact solutions of coupled Burgers’ equation"Waves in Random and Complex Media: DOI: 10.1080/17455030.2021.1979691.
7.  M. Hashemi, (2018) “Some new exact solutions of (2+1)- dimensional nonlinear Heisenberg ferromagnetic spin chain with the conformable time fractional derivative" Optical and Quantum Electronics 50(2): DOI: 10 .1007/s11082-018-1343-1.
8.  R. Gazizov and A. Kasatkin, (2013) “Construction of exact solutions for fractional order differential equations by the invariant subspace method" Computers and Mathematics with Applications 66(5): 576–584. DOI: 10.1016/j.camwa.2013.05.006.
9.  R. Sahadevan and T. Bakkyaraj, (2015) “Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations" Fractional Calculus and Applied Analysis 18(1): 146–162. DOI: 10.1515/fca-2015-0010.
10.  A. Zafar, M. Raheel, K. Hosseini, M. Mirzazadeh, S. Salahshour, C. Park, and D. Y. Shin, (2021) “Diverse approaches to search for solitary wave solutions of the fractional modified Camassa–Holm equation" Results in
Physics 31: DOI: 10.1016/j.rinp.2021.104882.
11.  H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, (2019) “New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gard-ner equation" Modern Physics Letters B 33(17): DOI: 10.1142/S0217984919501963.
12.  M. S. M. Shehata, H. Rezazadeh, E. H. M. Zahran, E. Tala-Tebue, and A. Bekir, (2019) “New Optical Soliton Solutions of the Perturbed Fokas-Lenells Equation" Communications in Theoretical Physics 71(11): 1275–1280. DOI: 10.1088/0253-6102/71/11/1275.
13.  H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2016) “Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method" Boletim da Sociedade Paranaense de Matematica 34(2): 213–229. DOI: 10.5269/bspm.v34i2.25501.
14.  H. Rezazadeh, (2018) “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity" Optik 167: 218–227. DOI: 10.1016/j.ijleo.2018.04.026.
15.  M. Eslami and H. Rezazadeh, (2016) “The first integral method for Wu–Zhang system with conformable timefractional derivative" Calcolo 53(3): 475–485. DOI: 10.1007/s10092-015-0158-8.
16.  S. Abbasbandy, E. Shivanian, K. H. AL-Jizani, and S. N. Atluri, (2021) “Pseudospectral meshless radial point interpolation for generalized biharmonic equation subject to simply supported and clamped boundary conditions" Engineering Analysis with Boundary Elements 125: 23–32. DOI: 10.1016/j.enganabound.2021.01.004.
17.  S. Abbasbandy, E. Shivanian, and K. H. AL-Jizani, (2021) “On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo-spectral meshfree radial point interpolation" Numerical Methods for Partial Differential Equations 37(1): 462–478. DOI: 10.1002/num.22536.
18.  M. Hashemi and A. Akgül, (2021) “On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods" Engineering with Computers 37(2): 1147–1158. DOI: 10.1007/s00366-019-00876-0.
19.  M. Hashemi, (2021) “Numerical study of the onedimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method" Engineering with Computers 37(4): 3397–3407. DOI: 10.1007/s00366-020-01001-2.
20.  M. S. Hashemi, E. Darvishi, and D. Baleanu, (2016) “A geometric approach for solving the density-dependent diffusion Nagumo equation" Advances in Difference Equations 2016(1): DOI: 10.1186/s13662-016-0818-2.
21.  M. Hajiketabi and S. Abbasbandy, (2018) “The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation" Engineering Analysis with Boundary Elements 87: 36–46. DOI: 10.1016/j.enganabound.2017.11.008.
22.  L. N. Trefethen. Spectral methods in MATLAB. SIAM, 2000.
23.  S. Kheybari, M. T. Darvishi, and M. S. Hashemi, (2019) “Numerical simulation for the space-fractional diffusion equations" Applied Mathematics and Computation 348: 57–69. DOI: 10.1016/j.amc.2018.11.041.
24.  W. Bao, Y. Feng, and C. Su, (2022) “UNIFORM ERROR BOUNDS OF TIME-SPLITTING SPECTRAL METHODS FOR THE LONG-TIME DYNAMICS OF THE NONLINEAR KLEIN–GORDON EQUATION WITH WEAK NONLINEARITY" Mathematics of Computation 91(334): 811–842. DOI: 10.1090/mcom/3694.
25.  S. Noeiaghdam, D. Sidorov, A.-M. Wazwaz, N. Sidorov, and V. Sizikov, (2021) “The numerical validation of the adomian decomposition method for solving volterra integral equation with discontinuous kernels using the cestac method" Mathematics 9(3): 1–15. DOI: 10.3390/math9030260.
26.  S. Rashid, K. T. Kubra, and J. L. G. Guirao, (2021) “Construction of an approximate analytical solution for multi-dimensional fractional zakharov–kuznetsov equation via aboodh adomian decomposition method" Symmetry 13(8): DOI: 10.3390/sym13081542.
27.  T.-T. Lu andW.-Q. Zheng, (2021) “Adomian decomposition method for first order PDEs with unprescribed data" Alexandria Engineering Journal 60(2): 2563–2572. DOI: 10.1016/j.aej.2020.12.021.
28.  O. González-Gaxiola and A. Biswas, (2019) “Optical solitons with Radhakrishnan–Kundu–Lakshmanan equation by Laplace–Adomian decomposition method" Optik 179: 434–442. DOI: 10.1016/j.ijleo.2018.10.173.
29.  W. Qiu, D. Xu, and J. Guo, (2021) “Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation" Applied Mathematics and Computation 392: DOI: 10.1016/j.amc.2020.125693.
30.  W. Qiu, D. Xu, and J. Guo, (2021) “The Crank-Nicolsontype Sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel" Applied Numerical Mathematics 159: 239–258. DOI: 10.1016/j.apnum.2020.09.011.
31.  S.-J. Liao, (1997) “A kind of approximate solution technique which does not depend upon small parameters - II. An application in fluid mechanics" International Journal of Non-Linear Mechanics 32(5): 815–822. DOI: 10.1016/s0020-7462(96)00101-1.
32.  S. Liao, (2004) “On the homotopy analysis method for nonlinear problems" Applied Mathematics and Computation 147(2): 499–513. DOI: 10.1016/S0096-3003(02)00790-7.
33.  S. Saratha, G. Sai Sundara Krishnan, and M. Bagyalakshmi, (2021) “Analysis of a fractional epidemic model by fractional generalised homotopy analysis method using modified Riemann - Liouville derivative" Applied Mathematical Modelling 92: 525–545. DOI: 10.1016/j.apm.2020.11.019.
34.  L. Noeiaghdam, S. Noeiaghdam, and D. Sidorov. “Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation”. In: 1847. 1. Cited by: 4; All Open Access, Bronze Open Access. 2021. DOI: 10.1088/1742-6596/1847/1/012010.
35.  H. Jafari, J. G. Prasad, P. Goswami, and R. S. Dubey, (2021) “SOLUTION of the LOCAL FRACTIONAL GENERALIZED KDV EQUATION USING HOMOTOPY ANALYSIS METHOD" Fractals 29(5): DOI: 10.1142/S0218348X21400144.
36.  P. Jain, M. Kumbhakar, and K. Ghoshal, (2022) “Application of homotopy analysis method to the determination of vertical sediment concentration distribution with shearinduced diffusivity" Engineering with Computers 38: 2609–2628. DOI: 10.1007/s00366-021-01491-8.
37.  A. Khaliq and E. Twizell, (1987) “A Family of Second Order Methods for Variable Coefficient Fourth Order Parabolic Partial Differential Equations" International Journal of Computer Mathematics 23(1): 63–76. DOI: 10.1080/00207168708803608.
38.  M. Dehghan and J. Manafian, (2009) “The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method" Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences 64(7-8): 420–430. DOI: 10.1515/zna-2009-7-803.
39.  C. Andrade and S. McKee, (1977) “High accuracy A.D.I. methods for fourth order parabolic equations with variable coefficients" Journal of Computational and Applied Mathematics 3(1): 11–14. DOI: 10.1016/0771-050X(77)90019-5.
40.  J. Biazar and H. Ghazvini, (2007) “He’s variational iteration method for fourth-order parabolic equations" Computers and Mathematics with Applications 54(7-8): 1047–1054. DOI: 10.1016/j.camwa.2006.12.049.
41.  A.-M. Wazwaz, (2001) “Analytic treatment for variable coefficient fourth-order parabolic partial differential equations" Applied Mathematics and Computation 123(2): 219–227. DOI: 10.1016/S0096-3003(00)00070-9.   0.9
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