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Separation variable method combined with integral bifurcation method for solving time-fractional reaction–diffusion models

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Abstract

In this paper, based on the previous works, we improve a computational method on solving time-fractional partial differential equation. Using the improved method, a series of time-fractional reaction–diffusion models with Fisher–KPP type are studied from mathematical point of view. Different kinds of exact solutions of four time-fractional reaction–diffusion models are obtained. The forms of these solutions include parametric form, explicit form, and implicit form. Most of them (solutions) have degenerate property according as time increase. The dynamical properties of some representative exact solutions are illustrated by graphs.

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References

  • Ahmed E, Elgazzar AS (2007) On fractional order differential equations model for nonlocal epidemics. Phys A 379(2):607–614

    MathSciNet  Google Scholar 

  • Ahmet B, Özkan G, Esin A, Yusuf P (2015) Functional variable method for the nonlinear fractional differential equations. AIP Conf 1648(1):623–630

    Google Scholar 

  • Anh VV, Leonenko NN (2000) Scaling laws for fractional diffusion-wave equations with singular initial data. Stat Probab Lett 48(3):239–252

    MATH  Google Scholar 

  • Aronson DG (1980) Density-dependent interaction systems. In: Stewart WE, Ray WH, Cobley CC (eds) Dynamics and modelling of reactive systems. Academic Press, New York

    Google Scholar 

  • Bakkyaraj T, Sahadevan R (2014a) An approximate solution to some classes of fractional nonlinear partial differential-difference equation using Adomian decomposition method. J Fract Calc Appl 5(1):37–52

  • Bakkyaraj T, Sahadevan R (2014b) On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn 77(4):1309–1322

  • Bakkyaraj T, Sahadevan R (2015) Invariant analysis of nonlinear fractional ordinary differential equations with Riemann–Liouville derivative. Nonlinear Dyn 80(1):447–455

    MathSciNet  MATH  Google Scholar 

  • Bakkyaraj T, Sahadevan R (2016) Approximate analytical solution of two coupled time fractional nonlinear Schrodinger equations. Int J Appl Comput Math 2(1):113–135

    MathSciNet  MATH  Google Scholar 

  • Baleanu D, Machado JAT, Luo AC (eds) (2011) Fractional dynamics and control. Springer Science and Business Media, Berlin

    MATH  Google Scholar 

  • Blasiak S (2016) Time-fractional heat transfer equations in modeling of the non-contacting face seals. Int J Heat Mass Transf 100:79–88

    Google Scholar 

  • Chen J, Liu F, Anh V (2008) Analytical solution for the time-fractional telegraph equation by the method of separating variables. J Math Anal Appl 338(2):1364–1377

    MathSciNet  MATH  Google Scholar 

  • Daftardar-Gejji V, Jafari H (2005) Adomian decomposition: a tool for solving a system of fractional differential equations. J Math Anal Appl 301(2):508–518

    MathSciNet  MATH  Google Scholar 

  • Dorcak L (2002) Numerical models for the simulation of the fractional-order control systems. 2002. arXiv:math/0204108

  • Efe MÖ (2011) Fractional order systems in industrial automation—a survey. IEEE Trans Ind Inform 7(4):582–591

    Google Scholar 

  • El-Sayed AMA, Rida SZ, Arafa AAM (2009) Exact solutions of fractional-order biological population model. Commun Theor Phys 52(12):992–996

    MathSciNet  MATH  Google Scholar 

  • Elsayed MEZ, Yasser AA, Reham MAS (2016) The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. J Assoc Arab Univ Basic Appl Sci 19(1):59–69

    Google Scholar 

  • Eslami M, Vajargah BF, Mirzazadeh M, Biswas A (2014) Applications of first integral method to fractional partial differential equations. Indian J Phys 88(2):177–184

    Google Scholar 

  • Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett 91(3):034–101

    Google Scholar 

  • Harris PA, Garra R (2013) Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method. Nonlinear Stud 20(4):471–481

    MathSciNet  MATH  Google Scholar 

  • Hayat T, Nadeem S, Asghar S (2004) Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model. Appl Math Comput 151(1):153–161

    MathSciNet  MATH  Google Scholar 

  • He JH (2012) Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys Lett A 376:257–259

    MathSciNet  MATH  Google Scholar 

  • Henry BI, Langlands TAM, Wearne SL (2006) Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. Phys Rev E Stat Nonlinear Soft Matter Phys 74(3):031116

    MathSciNet  Google Scholar 

  • Jamil M, Khan AN, Shahid N (2013) Fractional magnetohydrodynamics Oldroyd-B fluid over an oscillating plate. Therm Sci 17(4):997–1011

    Google Scholar 

  • Jiang H, Liu F, Turner I, Burrage K (2012) Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput Math Appl 64(10):3377–3388

    MathSciNet  MATH  Google Scholar 

  • Jumarie G (2006) Modified Riemann–Liouville derivative and fractional Taylor series of non-differentiable functions further results. Comput Math Appl 51(9–10):1367–1376

    MathSciNet  MATH  Google Scholar 

  • Jumarie G (2007) Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution. J Appl Math Comput 24(1–2):31–48

    MathSciNet  MATH  Google Scholar 

  • Jumarie G (2010) Cauchy’s integral formula via the modified Riemann–Liouville derivative for analytic functions of fractional order. Appl Math Lett 23(12):1444–1450

    MathSciNet  MATH  Google Scholar 

  • Li ZB, He JH (2010) Fractional complex transform for fractional differential equations. Math Comput Appl 15(5):970–973

    MathSciNet  MATH  Google Scholar 

  • Li H, Jiang W (2018) A space-time spectral collocation method for the 2-dimensional nonlinear Riesz space fractional diffusion equations. Math Meth Appl Sci 41(16):6130–6144

    MathSciNet  MATH  Google Scholar 

  • Li ZB, Zhu WH, He JH (2012) Exact solutions of time-fractional heat conduction equation by the fractional complex transform. Therm Sci 16(2):335–338

    Google Scholar 

  • Liu F, Burrage K (2011) Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. Comput Math Appl 62(3):822–833

    MathSciNet  MATH  Google Scholar 

  • Lohmann AW, Mendlovic D, Zalevsky Z, Dorsch RG (1996) Some important fractional transformations for signal processing. Opt Commun 125(1):18–20

    Google Scholar 

  • Luchko Y (2010) Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput Math Appl 59(5):1766–1772

    MathSciNet  MATH  Google Scholar 

  • Malaguti L, Marcelli C (2003) Sharp profiles in degenerate and double degenerate Fisher-KPP equations. J Differ Equ 195(2):471–496

    MATH  Google Scholar 

  • Manzoor T, Mehmood Z, Zahid MA et al (2017) Anote on fractional order in thermo-elasticity of shape memory alloys’ dampers. Int J Heat Mass Transf 114:597–606

    Google Scholar 

  • Metzler R, Glöckle WG, Nonnenmacher TF (1994) Fractional model equation for anomalous diffusion. Phys A 211(1):13–24

    Google Scholar 

  • Metzler R, Barkai E, Klafter J (1999) Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach. Phys Rev Lett 82(18):3563–3567

    Google Scholar 

  • Momani S, Zaid O (2007) Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput Math Appl 54(7):910–919

    MathSciNet  MATH  Google Scholar 

  • Moustafa ES (2006) MHD of a fractional viscoelastic fluid in a circular tube. Mech Res Commun 33(2):261–268

    MathSciNet  MATH  Google Scholar 

  • Odibat ZM, Shaher M (2009) The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput Math Appl 58:2199–2208

    MathSciNet  MATH  Google Scholar 

  • Pablo AD, Vázquez JL (1991) Travelling waves and finite propagation in a reaction-diffusion equation. J Differ Equ 93(1):19–61

    MathSciNet  MATH  Google Scholar 

  • Pei SC, Ding JJ (2007) Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing. IEEE Trans Signal Process 55(10):4839–4850

    MathSciNet  MATH  Google Scholar 

  • Rui W (2017) Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs. Commun Nonlinear Sci Numer Simul 30:253–266

    MathSciNet  MATH  Google Scholar 

  • Rui W (2018a) Applications of integral bifurcation method together with homogeneous balanced principle on investigating exact solutions of time fractional nonlinear PDEs. Nonlinear Dyn 91(1):679–712

  • Rui W (2018b) Idea of invariant subspace combined with elementary integral method for investigating exact solutions of time-fractional NPDEs. Appl Math Comput 339:158–171

  • Rui W, He B, Long Y, Chen C (2008) The integral bifurcation method and its application for solving a family of third-order dispersive PDEs. Nonlinear Anal 69(4):1256–1267

    MathSciNet  MATH  Google Scholar 

  • Sahadevan R, Bakkyaraj T (2012) Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations. J Math Anal Appl 393(2):341–347

    MathSciNet  MATH  Google Scholar 

  • Sahadevan R, Bakkyaraj T (2015) Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fract Calc Appl Anal 18(1):146–162

    MathSciNet  MATH  Google Scholar 

  • Sahadevan R, Prakash P (2016) Exact solution of certain time fractional nonlinear partial differential equations. Nonlinear Dyn 85(1):659–673

    MathSciNet  MATH  Google Scholar 

  • Sakthivel R, Mahmudov NI, Nieto JJ (2012) Controllability for a class of fractional-order neutral evolution control systems. Appl Math Comput 218(20):1033–10340

    MathSciNet  MATH  Google Scholar 

  • Sejdić E, Djurović I, Stanković L (2011) Fractional Fourier transform as a signal processing tool: an overview of recent developments. Signal Process 91(6):1351–1369

    MATH  Google Scholar 

  • Tan W, Pan W, Xu M (2003) A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int J Nonlinear Mech 38(5):645–650

    MATH  Google Scholar 

  • Tarasov VE (2016) On chain rule for fractional derivatives. Commun Nonlinear Sci Numer Simul 30(1):1–4

    MathSciNet  MATH  Google Scholar 

  • Tarasov VE, Tarasova VV (2018) Macroeconomic models with long dynamic memory: fractional calculus approach. Appl Math Comput 338:466–486

    MathSciNet  MATH  Google Scholar 

  • Tepljakov A, Petlenkov E, Belikov J (2012) Implementation and real-time simulation of a fractional-order controller using a MATLAB based prototyping platform. In: 2012 13th Biennial Baltic electronics conference (BEC). IEEE 2012, pp 145–148

  • Tripathi D, Pandey SK, Das S (2010) Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Appl Math Comput 215(10):3645–3654

    MathSciNet  MATH  Google Scholar 

  • Wu G, Lee EWM (2010) Fractional variational iteration method and its application. Phys Lett A 372(25):2506–2509

    MathSciNet  MATH  Google Scholar 

  • Wu C, Rui W (2018) Method of separation variables combined with homogenous balanced principle for searching exact solutions of nonlinear time-fractional biological population model. Commun Nonlinear Sci Numer Simul 63:88–100

    MathSciNet  MATH  Google Scholar 

  • Yang XJ, Hristov J, Srivastava HM, Ahmad B (2014) Modelling fractal waves on shallow water surfaces via local fractional Korteweg–de Vries equation. Abst Appl Anal Hindawi 2014, ID: 278672. https://doi.org/10.1155/2014/278672

  • Yang XJ (2017a) New general fractional-order rheological models with kernels of Mittag-Leffler functions. Rom Rep Phys 69(4):1–15

  • Yang XJ (2017b) Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat transfer problems. Therm Sci 21(3):1161–1171

  • Yang XJ (2018) New rheological problems involving general fractional derivatives with nonsingular power-law kernels. Proc Rom Acad 19(1):45–52

    MathSciNet  Google Scholar 

  • Yang XJ, Machado JAT (2017) A new fractional operator of variable order: application in the description of anomalous diffusion. Phys A Stat Mech Appl 481:276–283

    MathSciNet  Google Scholar 

  • Yang XJ, Srivastava HM, Machado JAT (2016a) A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. Therm Sci 20(2):753–756

  • Yang AM, Han Y, Li J et al (2016b) On steady heat flow problem involving Yang–Srivastava–Machado fractional derivative without singular kernel. Therm Sci 20(3):s717–s721

  • Yang XJ, Gao F, Srivastava HM (2017a) A new computational approach for solving nonlinear local fractional PDEs. J Comput Appl Math 339:285–296

  • Yang X, Machado JAT, Beleanu D (2017b) Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(04):1740006

  • Yang XJ, Gao F, Machado JAT, Beleanu D (2017c) A new fractional derivative involving the normalized sinc function without singular kernel. Eur Phys J Spec Top 226(16–18):3567–3575

  • Yang XJ, Machado JAT, Beleanu D (2017d) Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions. Rom Rep Phys 69(4):115

  • Yang B, Yu T, Shu H et al (2018) Perturbation observer based fractional-order sliding-mode controller for MPPT of grid-connected PV inverters: design and real-time implementation. Control Eng Pract 79:105–125

    Google Scholar 

  • Zafar AA, Vieru D, Akhtar S (2015) Magnetohydrodynamics of rotating fractional second grade fluid in porous medium. J Prime Res Math 10(1):45–58

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank reviewers very much for their useful comments and helpful suggestions on my manuscript. This work was supported by the National Natural Science Foundation of China under Grant No. 11361023 and the Chongqing Science and Technology Commission of China under Grant No. cstc2018jcyjAX0766.

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Weiguo Rui

  2. School of General Education, City College, Southwest University of Science and Technology, Mianyang Sichuan, 621000, People’s Republic of China

    Hui Zhang

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  1. Weiguo Rui
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Correspondence to Weiguo Rui.

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Communicated by José Tenreiro Machado.

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Rui, W., Zhang, H. Separation variable method combined with integral bifurcation method for solving time-fractional reaction–diffusion models. Comp. Appl. Math. 39, 299 (2020). https://doi.org/10.1007/s40314-020-01346-w

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