An Approximate Method for Solving Space-Time Fractional Advection-Dispersion Equation
Subject Areas : International Journal of Industrial Mathematicsاسماعیل بابلیان 1 , محمد ادبی تبارفیروزجا 2 , ذهرام عاقلی 3
1 - Department of Computer Science, Kharazmi University, Tehran, Iran.
2 - Department of Mathematics, Qaemshahr Branch, Islamic Azad
University, Qaemshahr, Iran.
3 - Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.
Keywords: Basic functions, Approximate method, Caputo derivative, Fuzzy-transform, Space-time fractional advection-dispersion,
Abstract :
In this research work, we have shown that it is possible to use fuzzy transform method (FTM) for approximate solution of space-time fractional advection-dispersion equation. In numerical methods, in order to approximate a function on a particular interval, only a restricted number of points are employed. However, what makes the F-transform preferable to other methods is that it makes use of all points in the interval.
[1] D. Baleanu, A. C. Luo, Discontinuity and Complexity in Nonlinear Physical Systems, J. T. Machado (Ed.). Springer (2014).
[2] A. D. Benson, S. W. Wheatcraft, M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research 36 (2020) 1403-1412.
[3] W. Chen, Y. Shen, Approximate solution for a class of second-order ordinary differential equations by the fuzzy transform, Journal of Intelligent & Fuzzy Systems 27 (2014) 73-82.
[4] A. S. Deshpande, V. Daftardar-Gejji, Y. V. Sukale, On Hopf bifurcation in fractional dynamical systems, Chaos, Solitons & Fractals 98 (2017) 189-198.
[5] A. M. El-Sayed, S. H. Behiry, W. E. Raslan, Adomians decomposition method for solving an intermediate fractional advectiondispersion equation, Computers & Mathematics with Applications 59 (2010) 1759-1765.
[6] X. M. Gu, T. Z. Huang, C. C. Ji, B. Carpentieri, A. A. Alikhanov, Fast iterative method with a second-order implicit difference scheme for time-space fractional convection-diffusion equation, Journal of Scientific Computing 72 (2017) 957-985.
[7] O. Guner, A. Bekir, The Exp-function method for solving nonlinear space-time fractional differential equations in mathematical physics, Journal of the Association 6of Arab Universities for Basic and Applied Sciences 7 (2014) 57-85 .
[8] G. Hariharan, R. Rajaraman, A new coupled wavelet-based method applied to the nonlinear reaction-diffusion equation arising in mathematical chemistry, Journal of Mathematical Chemistry 51 (2013) 2386-2400.
[9] A. Khastan, I. Perfilieva, Z. Alijani, A new fuzzy approximation method to Cauchy problems by fuzzy transform, Fuzzy Sets and Systems 288 (2016) 75-95.
[10] A. Khastan, Z. Alijani, I. Perfilieva, Fuzzy transform to approximate solution of twopoint boundary value problems, Mathematical Methods in the Applied Sciences 6 (2016) 45-56.
[11] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, Elsevier B. V, Netherlands (2006).
[12] X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis 47 (2009) 2108-2131.
[13] C. Li, Z. Zhao, Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers & Mathematics with Applications 62 (2011) 855-875.
[14] R. L. Magin, O. Abdullah, D. Baleanu, X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, Journal of Magnetic Resonance 190 (2014) 255-270.
[15] M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics 172 (2001) 65-77.
[16] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports 339 (2000) 1-77.
[17] C. Ming, F. Liu, L. Zheng, I. Turner, V. Anh, Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid, Computers & Mathematics with Applications 72 (2000) 2084-2097.
[18] S. Momani, Z. Odibat, Numerical solutions of the space-time fractional advectiondispersion equation, Numerical Methods for Partial Differential Equations 24 (2008) 1416-1429.
[19] A. Neamaty, M. Nategh, B. Agheli, TimeSpace Fractional Burgers’ Equation on Time Scales, Journal of Computational and Nonlinear Dynamics 12 (2017) 310-322.
[20] A. Neamaty, M. Nategh, B. Agheli, Local non-integer order dynamic problems on time scales revisited, International Journal of Dynamics and Control 14 (2000) 1-13.
[21] I. Perfilieva, Fuzzy transforms: Theory and applications, Fuzzy sets and systems 157 (2006) 993-1023.
[22] R. K. Pandey, O. P. Singh, V. K. Baranwal, An analytic algorithm for the spacetime fractional advection-dispersion equation, Computer Physics Communications 182 (2011) 1134-1144.
[23] I. Perfilieva, Fuzzy transforms in image compression and fusion, Acta Mathematica Universitatis Ostraviensis 15 (2007) 27-37.
[24] M. A. Z. Raja, R. Samar, E. S. Alaidarous, E. Shivanian, Bio-inspired computing platform for reliable solution of Bratu-type equations arising in the modeling of electrically conducting solids, Applied Mathematical Modelling 40 (2016) 5964-5977.
[25] J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, Journal of Computational and Applied Mathematics 193 (2006) 243-268.
[26] E. Scalas, The application of continuoustime random walks in finance and economics, Physica A: Statistical Mechanics and its Applications 362 (2000) 225-239.
[27] I. M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8 (2012) 9043-9052.
[28] A. Suzuki, Y. Niibori, S. Fomin, V. Chugunov, T. Hashida, Prediction of reinjection effects in fault-related subsidiary structures by using fractional derivative-based mathematical models for sustainable design of geothermal reservoirs, Geothermics 57 (2015) 196-204.
[29] S. Tomasiello, An alternative use of fuzzy transform with application to a class of delay differential equations, International Journal of Computer Mathematics 5 (2016) 1-8.
[30] M. Jani, E. Babolian, S. Javadi, D. Bhatta, Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation, Numerical Algorithms 75 (2017) 1041-1063.
[31] S. Javadi, E. Babolian, M. Jani, A numerical scheme for space-time fractional advectiondispersion equation, International Journal of Nonlinear Analysis and Applications 7 (2015) 331-343.
[32] V. V. Uchaikin, R. T. Sibatov, Fractional theory for transport in disordered semiconductors, Communications in Nonlinear Science and Numerical Simulation 13 (2008) 715-727.
[33] A. Yildirim, H. Koak, Homotopy perturbation method for solving the space-time fractional advection-dispersion equation, Advances in Water Resources 32 (2009) 1711-1716.
[34] X. Zhang, M. Lv, J. W. Crawford, I. M. Young, The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: defining the fractional dispersive flux with the Caputo derivatives, Advances in water resources 30 (2007) 1205-1217.
[35] X. Zhang, L. Liu, Y., Wu, B. Wiwatanapataphee, Nontrivial soluions for a fractional advection dispersion equation in anomalous diffusion, Applied Mathematics Letters 66 (2017) 1-8.