Discontinuity, Nonlinearity, and Complexity
Computational Solutions of some Nonlinear Transportation Equations of Fractional Order via Two Efficient Methods
Discontinuity, Nonlinearity, and Complexity 12(1) (2023) 111125  DOI:10.5890/DNC.2023.03.009
Muhammad Abaid Ur Rehman$^1$, Jamshad Ahmad$^1$, Qazi Mahmood Ul Hassan$^2$
Download Full Text PDF
Abstract
The fundamental objective of this paper is to tackle the timefractional order transportation equations through two analytical methods, the method of qhomotopy analysis (qHAM) and the method of reduced differential transform (RDTM) through numerical computation and simulations. The fractional derivative is considered in Caputo's sense. Three examples have been employed to illustrate the preciseness and effectiveness of the proposed methods for theoretical and numerical analysis purpose. The techniques provide seriesform solution that converges sharply to the exact solution as the noninteger order approaches the integer order. Also, the graphical depictions of solutions are provided to compare the results of these methods.
References

[1]  Bhrawyn, A.H. and Baleanu, D. (2013), A spectral LegendreGaussLobatto collocation method for a spacefractional advection diffusion equation with variable coefficients, Reports on Mathematical Physics, 72, 219233.


[2]  Bagley, R.L. and Torvik, P.J. (1983), Fractional calculusa different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21, 741748.


[3]  Caputo, M. (1967), Linear models of dissipation whose Q is almost frequency independent II, Geophysical Journal International, 13, 529539.


[4]  Iddrisu, M.M. and Tetteh, K.I. (2017), The gamma function and its analytical applications. Journal of Advances in Mathematics and Computer Science, 23, 116.


[5]  Kumar, S., Kumar, A., Abbas, S., Al Qurashi, M., and Baleanu, D. (2020), A modified analytical approach with existence and uniqueness for fractional Cauchy reactiondiffusion equations, Advances in Difference Equations, 28, 118.


[6]  Bansu, H. and Kumar, S. (2019), Numerical solution of space and time fractional telegraph equation: a meshless approach, International Journal of Nonlinear Sciences and Numerical Simulation, 20, 325337.


[7]  Abuasad, S., Hashim, I., Karim, A., and Ariffin, S. (2019), Modified fractional reduced differential transform method for the solution of multiterm timefractional diffusion equations, Advances in Mathematical Physics, 2019, https://doi.org/10.1155/2019/5703916.


[8]  Singh, J., Secer, A., Swroop, R., and Kumar, D. (2019), A reliable analytical approach for a fractional model of advectiondispersion equation, Nonlinear Engineering, 8, 107116.


[9]  Golbabai, A., Nikanand, O., and Molavi, M.A. (2019), Numerical approximation of time fractional advectiondispersion model arising from solute transport in rivers, Journal of Pure and Applied Mathematics, 10, 117131.


[10]  Shah, R., Khan, H., Mustafa, S., Kumam, P., and Arif, M. (2019), Analytical solutions of fractionalorder diffusion equations by natural transform decomposition method, Entropy, 21, 557.


[11]  Khan, H., Shah, R., Kumam, P., Baleanu, D., and Arif, M. (2019), An efficient analytical technique, for the solution of fractionalorder telegraph equations,
Mathematics, 7, 426.


[12]  Khan, H., Shah, R., Kumam, P., and Arif, M. (2019), Analytical solutions of fractionalorder heat and wave equations by the natural transform decomposition method, Entropy, 21, 597.


[13]  Shah, K., Khalil, H., and Khan, R.A. (2018), Analytical solutions of fractional order diffusion equations by natural transform method, Iranian Journal of Science and Technology, Transactions A: Science, 42, 14791490.


[14]  Sungu, I.C. and Demir, H. (2015), A new approach and solution technique to solve time fractional nonlinear reactiondiffusion equations, Mathematical Problems in Engineering, https://doi.org/10.1155/2015/457013.


[15]  Kumar, S. (2014), A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling, 38, 31543163.


[16]  Iyiola, O.S. and Zaman, F.D. (2014), A fractional diffusion equation model for cancer tumor, AIP Advances, 4, 107121.


[17]  \.{I}bi\c{s}, B. and Bayram, M. (2014), Approximate solution of timefractional advectiondispersion equation via fractional variational iteration method, The Scientific World Journal, 2014(6), 769713, https://doi.org/10.1155/2014/769713.


[18]  Momani, S. and Odibat, Z. (2008), Numerical solutions of the spaceātime fractional advectiondispersion equation, Numerical Methods for Partial Differential Equations, 24, 14161429.


[19]  Eltayeb, H., Abdalla, Y.T., Bachar, I., and Khabir, M.H. (2019), Fractional telegraph equation and its solution by natural transform decomposition method, Symmetry, 11, 334.


[20]  Yavuz, M. and Ya\c{s}k\i ran, B. (2017), Approximateanalytical solutions of cable equation using conformable fractional operator, New Trends in Mathematical Sciences, 5, 209219.


[21]  Oyjinda, P. and Pochai, N. (2019), Numerical simulation of an air pollution model on industrial areas by considering the influence of multiple point sources, International Journal of differential Equations, https://doi.org/10.1155/2019/2319831.


[22]  Ahmad, J. and MohyudDin, S.T. (2015), Solving wave and diffusion equations on cantor sets, Proceedings of the Pakistan Academy of Sciences, 52, 7177.


[23]  Bahar, E. and G\"{u}rarslan, G. (2017), Numerical solution of advectiondiffusion equation using operator splitting method, International Journal of Engineering and Applied Sciences, 9, 7688.


[24]  Ziane, D., Belgacem, R., and Bokhari, A. (2019), A new modified Adomian decomposition method for nonlinear partial differential equations, Open Journal of Mathematical Analysis, 3, 8190.


[25]  Rani, A., Hssan, Q., Ayub, K., Ahmad, J., and Zulfiqar, A. (2020), Soliton solutions of nonlinear evolution equations by basic $(G'/G)$expansion method, Mathematical Modelling of Engineering Problems, 7(2), 242250.


[26]  Prakasha, D.G., Veeresha, P., and Baskonus, H.M. (2019), Residual power series method for fractional SwiftHohenberg equation, Fractal and Fractional, 3, 9.


[27]  Shqair, M., ElAjou, A., and Nairat, M. (2019), Analytical solution for multienergy groups of neutron diffusion equations by a residual power series method, Mathematics, 7, 633.


[28]  Ahmad, J. and MohyudDin, S.T. (2014), An efficient algorithm for some highly nonlinear fractional pdes in mathematical physics, Plos One, 9(12), e109127.


[29]  Ahmed, N., Shah, N.A., and Vieru, D. (2019), Twodimensional advectiondiffusion process with memory and concentrated source, Symmetry, 11, 879.


[30]  Khalouta, A. and Kadem, A. (2019), A new computational for approximate analytical solutions of nonlinear timefractional wavelike equations with variable coefficients, AIMS Mathematics, 5, 114.
