An Avant-Garde Handling of Temporal-Spatial Fractional Physical Models

Imad Jaradat 1 , Marwan Alquran 2 , Qutaibeh Katatbeh 3 , Feras Yousef 4 , Shaher Momani 5 , 6  and Dumitru Baleanu 7 , 8
  • 1 Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan
  • 2 Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan
  • 3 Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan
  • 4 Department of Mathematics, Faculty of Science, The University of Jordan, 11942, Amman, Jordan
  • 5 Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE
  • 6 Department of Mathematics, Faculty of Science, The University of Jordan, 11942, Amman, Jordan
  • 7 Department of Mathematics, Cankaya University, Ankara, Turkey
  • 8 Institute of Space Sciences, Magurele, Bucharest, Romania
Imad Jaradat
  • Corresponding author
  • Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Marwan Alquran
  • Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Qutaibeh Katatbeh
  • Department of Mathematics & Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Feras Yousef, Shaher Momani
  • Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE
  • Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Dumitru Baleanu
  • Department of Mathematics, Cankaya University, Ankara, Turkey
  • Institute of Space Sciences, Magurele, Bucharest, Romania
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In the present study, we dilate the differential transform scheme to develop a reliable scheme for studying analytically the mutual impact of temporal and spatial fractional derivatives in Caputo’s sense. We also provide a mathematical framework for the transformed equations of some fundamental functional forms in fractal 2-dimensional space. To demonstrate the effectiveness of our proposed scheme, we first provide an elegant scheme to estimate the (mixed-higher) Caputo-fractional derivatives, and then we give an analytical treatment for several (non)linear physical case studies in fractal 2-dimensional space. The study concluded that the proposed scheme is very efficacious and convenient in extracting solutions for wide physical applications endowed with two different memory parameters as well as in approximating fractional derivatives.

  • [1]

    R. R. Nigmatullin, To the theoretical explanation of the “universal response”, Phys. Stat. Solidi B. 123 (1984), 739–745.

  • [2]

    C. Coussot, S. Kalyanam, R. Yapp and M. Insana, Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 56 (2009), 715–725.

    • Crossref
    • PubMed
    • Export Citation
  • [3]

    S. Butera and M. D. Paola, A physically based connection between fractional calculus and fractal geometry, Ann. Phys. 350 (2014), 146–158.

  • [4]

    F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comp. Acous. 9 (2001), 1417–1436.

    • Crossref
    • Export Citation
  • [5]

    M. Du, Z. Wang and H. Hu, Measuring memory with the order of fractional derivative, Sci. Rep. 3 (2013), ID:3431.

  • [6]

    S. Momani, An algorithm for solving the fractional convection-diffusion equation with nonlinear source term, Commun. Nonlin. Sci. Numer. Simul. 12 (2007), 1283–1290.

    • Crossref
    • Export Citation
  • [7]

    S. Momani and A. Yildirim, Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He’s homotopy perturbation method, Int. J. Comput. Math. 87 (2010), 1057–1065.

    • Crossref
    • Export Citation
  • [8]

    Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev. 63 (2010), 010801–52.

  • [9]

    B. N. Lundstrom, M. H. Higgs, W. J. Spain and A. L. Fairhall, Fractional differentiation by neocortical pyramidal neurons, Nat. Neurosci. 11 (2008), 1335–1342.

  • [10]

    A. S. Balankin, J. Bory-Reyes and M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric, Phys. A Stat. Mech. Appl. 444 (2016), 345–359.

  • [11]

    I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.

  • [12]

    T. Atanackovic, S. Pilipovic, B. Stankovic and D. Zorica, Fractional calculus with applications in mechanics: Vibrations and diffusion processes, Wiley-ISTE, London, Hoboken, 2014.

  • [13]

    V. E. Tarasov, Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media, Springer, New York, 2011.

  • [14]

    R. Herrmann, Fractional calculus: An introduction for physicists, World Scientific, Singapore, 2011.

  • [15]

    R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol. 30 (1986), 133–155.

  • [16]

    F. C. Meral, T. J. Royston and R. Magin, Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlin. Sci. Numer. Simul. 15 (2010), 939–945.

  • [17]

    M. D. Paola, A. Pirrotta and A. Valenza, Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results, J. Mech. Mater. 43 (2011), 799–806.

  • [18]

    R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), 299–307.

  • [19]

    K. N. Le, W. McLean and K. Mustapha, Numerical solution of the time-fractional Fokker–Planck equation with general forcing, SIAM J. Numer. Anal. 54 (2016), 1763–1784.

  • [20]

    B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), 825–843.

  • [21]

    A. H. Bhrawy, J. F. Alzaidy, M. A. Abdelkawy and A. Biswas, Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations, Nonlin. Dyn. 84 (2016), 1553–1567.

  • [22]

    Y. Yang, Y. Chen, Y. Huang and H. Wei, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, Comput. Math. Appl. 73 (2017), 1218–1232.

  • [23]

    M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation method, SIAM J. Sci. Comput. 36 (2014), A40–A62.

  • [24]

    M. G. Sakar, F. Uludag and F. Erdogan, Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Appl. Math. Model. 40 (2016), 6639–6649.

  • [25]

    S. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004), 499–513.

  • [26]

    K. Vishal, S. Kumar and S. Das, Application of homotopy analysis method for fractional Swift Hohenberg equation – revisited, Appl. Math. Model. 36 (2012), 3630–3637.

  • [27]

    R. K. Pandey, O. P. Singh and V. K. Baranwal, An analytic algorithm for the space-time fractional advection-dispersion equation, Comput. Phys. Commun. 182 (2011), 1134–1144.

  • [28]

    J-H. He, A short remark on fractional variational iteration method, Phys. Lett. A. 375 (2011), 3362–3364.

  • [29]

    R. Y. Molliq, M. S. M. Noorani and I. Hashim, Variational iteration method for fractional heat- and wave-like equations, Nonlin. Anal. Real World Appl. 10 (2009), 1854–1869.

  • [30]

    O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlin. Dyn. 29 (2002), 145–155.

  • [31]

    Y. Nikolova and L. Boyadjiev, Integral transforms method to solve a time-space fractional diffusion equation, Fract. Calc. Appl. Anal. 13 (2010), 57–68.

  • [32]

    S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006), 488–494.

  • [33]

    A. M. El-Sayed and M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett. A 359 (2006), 175–182.

  • [34]

    G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comput. Math. Appl. 21 (1991), 101–127.

  • [35]

    J. K. Zhou, Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan, 1986.

  • [36]

    Y. Keskin and G. Oturanç, Reduced differential transform method for partial differential equations, Int. J. Nonlin. Sci. Numer. Simul. 10 (2009), 741–749.

  • [37]

    J. Liu and G. Hou, Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Appl. Math. Comput. 217 (2011), 7001–7008.

  • [38]

    S. Kumar, A. Kumar and D. Baleanu, Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves, Nonlin. Dyn. 85 (2016), 699–715.

  • [39]

    M. Alquran, H. M. Jaradat and M. I. Syam, Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlin. Dyn. 90 (2017), 2525–2529.

  • [40]

    A. El-Ajou, O. Abu-Arqub and S. Momani, Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm, J. Comput. Phys. 293 (2015), 81–95.

  • [41]

    I. Jaradat, M. Alquran and K. Al-Khaled, An analytical study of physical models with inherited temporal and spatial memory, Eur. Phys. J. Plus 133 (2018), ID 162.

  • [42]

    M. Alquran and I. Jaradat, A novel scheme for solving Caputo time-fractional nonlinear equations: Theory and application, Nonlin. Dyn. 91 (2018), 2389–2395.

  • [43]

    A. El-Ajou, O. Abu-Arqub, Z. Al-Zhour and S. Momani, New results on fractional power series: theories and applications, Entropy 15 (2013), 5305–5323.

  • [44]

    H. Jaradat, I. Jaradat, M. Alquran, M. M. Jaradat, Z. Mustafa, K. Abohassan and R. Abdelkarim, Approximate solutions to the generalized time-fractional Ito system, Ital. J. Pure Appl. Math. 37 (2017), 699–710.

  • [45]

    I. Jaradat, M. Al-Dolat, K. Al-Zoubi and M. Alquran, Theory and applications of a more general form for fractional power series expansion, Chaos Solitons Fractals 108 (2018), 107–110.

  • [46]

    I. Jaradat, M. Alquran and R. Abdel-Muhsen, An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering, Nonlin. Dyn. 93 (2018), 1911–1922.

  • [47]

    I. Jaradat, M. Alquran and M. Al-Dolat, Analytic solution of homogeneous time-invariant fractional IVP, Adv. Differ. Equ. 2018 (2018), 143.

  • [48]

    M. Alquran, I. Jaradat, D. Baleanu and R. Abdel-Muhsen, An analytical study of (2+1)-dimensional physical models embedded entirely in fractal space, Rom. J. Phys. 64 (2019), 103.

  • [49]

    M. Alquran, I. Jaradat and S. Sivasundaram, Elegant scheme for solving Caputo-time-fractional integro-differential equations, Nonlin. Stud. 25 (2018), 385–393.

  • [50]

    A. D. Matteo and A. Pirrotta, Generalized differential transform method for nonlinear boundary value problem of fractional order, Commun. Nonlin. Sci. Numer. Simul. 29 (2015), 88–101.

  • [51]

    D. Baleanu, H. Khan, H. Jafari and R. A. Khan, On the exact solution of wave equations on cantor sets, Entropy. 17 (2015), 6229–6237.

    • Crossref
    • Export Citation
  • [52]

    M. Dehghan, J. Manafian and A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. Naturforsch. A. 65 (2010), 935–949.

  • [53]

    D. Kumar, J. Singh and S. Kumar, Analytic and approximate solutions of space-time fractional telegraph equations via laplace transform, Walailak J. Sci. Tech. 11 (2014), 711–728.

  • [54]

    S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations, Appl. Math. Comput. 217 (2011), 9905–9915.

  • [55]

    J. Singh, D. Kumar and A. Kiliçman, Homotopy perturbation method for fractional gas dynamics equation using sumudu transform, Abstr. Appl. Anal. 2013 (2013), ID 934060.

  • [56]

    S. Kumar and M. M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun. 185 (2014), 1947–1954.

    • Crossref
    • Export Citation
  • [57]

    H. Jafari, M. Alipour and H. Tajadodi, Two-dimensional differential transform method for solving nonlinear partial differential equations, Int. J. Res. Rev. Appl. Sci. 2 (2010), 47–52.

  • [58]

    M. Inc and Y. Cherruault, A new approach to solve a diffusion-convection problem, Kybernetes. 31 (2002), 536–549.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.

Search