Separation variable method combined with integral bifurcation method for solving time-fractional reaction-diffusion models. (English) Zbl 1476.35316

Rui, Weiguo; Zhang, Hui

Comput. Appl. Math. 39, No. 4, Paper No. 299, 26 p. (2020).

Summary: 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.

MSC:

35R11	Fractional partial differential equations
26A33	Fractional derivatives and integrals
34A08	Fractional ordinary differential equations
35K57	Reaction-diffusion equations

Keywords:

time-fractional reaction-diffusion model; separation variable method; homogenous balanced principle; integral bifurcation method; exact solution

References:

[1]	Ahmed, E.; Elgazzar, AS, On fractional order differential equations model for nonlocal epidemics, Phys A, 379, 2, 607-614 (2007) · doi:10.1016/j.physa.2007.01.010
[2]	Ahmet, B.; Özkan, G.; Esin, A.; Yusuf, P., Functional variable method for the nonlinear fractional differential equations, AIP Conf, 1648, 1, 623-630 (2015)
[3]	Anh, VV; Leonenko, NN, Scaling laws for fractional diffusion-wave equations with singular initial data, Stat Probab Lett, 48, 3, 239-252 (2000) · Zbl 0970.35174 · doi:10.1016/S0167-7152(00)00003-1
[4]	Aronson, DG; Stewart, WE; Ray, WH; Cobley, CC, Density-dependent interaction systems, Dynamics and modelling of reactive systems (1980), New York: Academic Press, New York
[5]	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 · Zbl 1499.65593
[6]	Bakkyaraj T, Sahadevan R (2014b) On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn 77(4):1309-1322
[7]	Bakkyaraj, T.; Sahadevan, R., Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville derivative, Nonlinear Dyn, 80, 1, 447-455 (2015) · Zbl 1345.34003 · doi:10.1007/s11071-014-1881-4
[8]	Bakkyaraj, T.; Sahadevan, R., Approximate analytical solution of two coupled time fractional nonlinear Schrodinger equations, Int J Appl Comput Math, 2, 1, 113-135 (2016) · Zbl 1420.65083 · doi:10.1007/s40819-015-0049-3
[9]	Baleanu, D.; Machado, JAT; Luo, AC, Fractional dynamics and control (2011), Berlin: Springer Science and Business Media, Berlin
[10]	Blasiak, S., Time-fractional heat transfer equations in modeling of the non-contacting face seals, Int J Heat Mass Transf, 100, 79-88 (2016) · doi:10.1016/j.ijheatmasstransfer.2016.04.040
[11]	Chen, J.; Liu, F.; Anh, V., Analytical solution for the time-fractional telegraph equation by the method of separating variables, J Math Anal Appl, 338, 2, 1364-1377 (2008) · Zbl 1138.35373 · doi:10.1016/j.jmaa.2007.06.023
[12]	Daftardar-Gejji, V.; Jafari, H., Adomian decomposition: a tool for solving a system of fractional differential equations, J Math Anal Appl, 301, 2, 508-518 (2005) · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039
[13]	Dorcak L (2002) Numerical models for the simulation of the fractional-order control systems. 2002. arXiv:math/0204108
[14]	Efe, MÖ, Fractional order systems in industrial automation—a survey, IEEE Trans Ind Inform, 7, 4, 582-591 (2011) · doi:10.1109/TII.2011.2166775
[15]	El-Sayed, AMA; Rida, SZ; Arafa, AAM, Exact solutions of fractional-order biological population model, Commun Theor Phys, 52, 12, 992-996 (2009) · Zbl 1184.92038 · doi:10.1088/0253-6102/52/6/04
[16]	Elsayed, MEZ; Yasser, AA; Reham, MAS, 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 (2016)
[17]	Eslami, M.; Vajargah, BF; Mirzazadeh, M.; Biswas, A., Applications of first integral method to fractional partial differential equations, Indian J Phys, 88, 2, 177-184 (2014) · doi:10.1007/s12648-013-0401-6
[18]	Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys Rev Lett, 91, 3, 034-101 (2003) · doi:10.1103/PhysRevLett.91.034101
[19]	Harris, PA; Garra, R., Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method, Nonlinear Stud, 20, 4, 471-481 (2013) · Zbl 1305.35151
[20]	Hayat, T.; Nadeem, S.; Asghar, S., Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl Math Comput, 151, 1, 153-161 (2004) · Zbl 1161.76436
[21]	He, JH, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys Lett A, 376, 257-259 (2012) · Zbl 1255.26002 · doi:10.1016/j.physleta.2011.11.030
[22]	Henry, BI; Langlands, TAM; Wearne, SL, 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 (2006) · doi:10.1103/PhysRevE.74.031116
[23]	Jamil, M.; Khan, AN; Shahid, N., Fractional magnetohydrodynamics Oldroyd-B fluid over an oscillating plate, Therm Sci, 17, 4, 997-1011 (2013) · doi:10.2298/TSCI110731140J
[24]	Jiang, H.; Liu, F.; Turner, I.; Burrage, K., Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput Math Appl, 64, 10, 3377-3388 (2012) · Zbl 1268.35124 · doi:10.1016/j.camwa.2012.02.042
[25]	Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput Math Appl, 51, 9-10, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[26]	Jumarie, G., Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J Appl Math Comput, 24, 1-2, 31-48 (2007) · Zbl 1145.26302 · doi:10.1007/BF02832299
[27]	Jumarie, G., Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order, Appl Math Lett, 23, 12, 1444-1450 (2010) · Zbl 1202.30068 · doi:10.1016/j.aml.2010.08.001
[28]	Li, ZB; He, JH, Fractional complex transform for fractional differential equations, Math Comput Appl, 15, 5, 970-973 (2010) · Zbl 1215.35164
[29]	Li, H.; Jiang, W., A space-time spectral collocation method for the 2-dimensional nonlinear Riesz space fractional diffusion equations, Math Meth Appl Sci, 41, 16, 6130-6144 (2018) · Zbl 1402.65126 · doi:10.1002/mma.5124
[30]	Li, ZB; Zhu, WH; He, JH, Exact solutions of time-fractional heat conduction equation by the fractional complex transform, Therm Sci, 16, 2, 335-338 (2012) · doi:10.2298/TSCI110503069L
[31]	Liu, F.; Burrage, K., Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput Math Appl, 62, 3, 822-833 (2011) · Zbl 1228.93114 · doi:10.1016/j.camwa.2011.03.002
[32]	Lohmann, AW; Mendlovic, D.; Zalevsky, Z.; Dorsch, RG, Some important fractional transformations for signal processing, Opt Commun, 125, 1, 18-20 (1996) · doi:10.1016/0030-4018(95)00748-2
[33]	Luchko, Y., 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 (2010) · Zbl 1189.35360 · doi:10.1016/j.camwa.2009.08.015
[34]	Malaguti, L.; Marcelli, C., Sharp profiles in degenerate and double degenerate Fisher-KPP equations, J Differ Equ, 195, 2, 471-496 (2003) · Zbl 1042.34056 · doi:10.1016/j.jde.2003.06.005
[35]	Manzoor, T.; Mehmood, Z.; Zahid, MA, Anote on fractional order in thermo-elasticity of shape memory alloys’ dampers, Int J Heat Mass Transf, 114, 597-606 (2017) · doi:10.1016/j.ijheatmasstransfer.2017.06.060
[36]	Metzler, R.; Glöckle, WG; Nonnenmacher, TF, Fractional model equation for anomalous diffusion, Phys A, 211, 1, 13-24 (1994) · doi:10.1016/0378-4371(94)90064-7
[37]	Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys Rev Lett, 82, 18, 3563-3567 (1999) · doi:10.1103/PhysRevLett.82.3563
[38]	Momani, S.; Zaid, O., Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput Math Appl, 54, 7, 910-919 (2007) · Zbl 1141.65398 · doi:10.1016/j.camwa.2006.12.037
[39]	Moustafa, ES, MHD of a fractional viscoelastic fluid in a circular tube, Mech Res Commun, 33, 2, 261-268 (2006) · Zbl 1192.76067 · doi:10.1016/j.mechrescom.2005.02.017
[40]	Odibat, ZM; Shaher, M., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput Math Appl, 58, 2199-2208 (2009) · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[41]	Pablo, AD; Vázquez, JL, Travelling waves and finite propagation in a reaction-diffusion equation, J Differ Equ, 93, 1, 19-61 (1991) · Zbl 0784.35045 · doi:10.1016/0022-0396(91)90021-Z
[42]	Pei, SC; Ding, JJ, Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing, IEEE Trans Signal Process, 55, 10, 4839-4850 (2007) · Zbl 1390.94058 · doi:10.1109/TSP.2007.896271
[43]	Rui, W., 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 (2017) · Zbl 1510.35385 · doi:10.1016/j.cnsns.2016.11.018
[44]	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
[45]	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 · Zbl 1428.35670
[46]	Rui, W.; He, B.; Long, Y.; Chen, C., The integral bifurcation method and its application for solving a family of third-order dispersive PDEs, Nonlinear Anal, 69, 4, 1256-1267 (2008) · Zbl 1144.35461 · doi:10.1016/j.na.2007.06.027
[47]	Sahadevan, R.; Bakkyaraj, T., Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J Math Anal Appl, 393, 2, 341-347 (2012) · Zbl 1245.35142 · doi:10.1016/j.jmaa.2012.04.006
[48]	Sahadevan, R.; Bakkyaraj, T., Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fract Calc Appl Anal, 18, 1, 146-162 (2015) · Zbl 1499.35682 · doi:10.1515/fca-2015-0010
[49]	Sahadevan, R.; Prakash, P., Exact solution of certain time fractional nonlinear partial differential equations, Nonlinear Dyn, 85, 1, 659-673 (2016) · Zbl 1349.35406 · doi:10.1007/s11071-016-2714-4
[50]	Sakthivel, R.; Mahmudov, NI; Nieto, JJ, Controllability for a class of fractional-order neutral evolution control systems, Appl Math Comput, 218, 20, 1033-10340 (2012) · Zbl 1245.93022
[51]	Sejdić, E.; Djurović, I.; Stanković, L., Fractional Fourier transform as a signal processing tool: an overview of recent developments, Signal Process, 91, 6, 1351-1369 (2011) · Zbl 1220.94024 · doi:10.1016/j.sigpro.2010.10.008
[52]	Tan, W.; Pan, W.; Xu, M., 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 (2003) · Zbl 1346.76009 · doi:10.1016/S0020-7462(01)00121-4
[53]	Tarasov, VE, On chain rule for fractional derivatives, Commun Nonlinear Sci Numer Simul, 30, 1, 1-4 (2016) · Zbl 1489.26011 · doi:10.1016/j.cnsns.2015.06.007
[54]	Tarasov, VE; Tarasova, VV, Macroeconomic models with long dynamic memory: fractional calculus approach, Appl Math Comput, 338, 466-486 (2018) · Zbl 1427.91178
[55]	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
[56]	Tripathi, D.; Pandey, SK; Das, S., Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, Appl Math Comput, 215, 10, 3645-3654 (2010) · Zbl 1352.76131
[57]	Wu, G.; Lee, EWM, Fractional variational iteration method and its application, Phys Lett A, 372, 25, 2506-2509 (2010) · Zbl 1237.34007 · doi:10.1016/j.physleta.2010.04.034
[58]	Wu, C.; Rui, W., 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 (2018) · Zbl 1509.35364 · doi:10.1016/j.cnsns.2018.03.009
[59]	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. doi:10.1155/2014/278672 · Zbl 1468.76016
[60]	Yang XJ (2017a) New general fractional-order rheological models with kernels of Mittag-Leffler functions. Rom Rep Phys 69(4):1-15
[61]	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
[62]	Yang, XJ, New rheological problems involving general fractional derivatives with nonsingular power-law kernels, Proc Rom Acad, 19, 1, 45-52 (2018)
[63]	Yang, XJ; Machado, JAT, A new fractional operator of variable order: application in the description of anomalous diffusion, Phys A Stat Mech Appl, 481, 276-283 (2017) · Zbl 1495.35204 · doi:10.1016/j.physa.2017.04.054
[64]	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
[65]	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
[66]	Yang XJ, Gao F, Srivastava HM (2017a) A new computational approach for solving nonlinear local fractional PDEs. J Comput Appl Math 339:285-296 · Zbl 1490.35530
[67]	Yang X, Machado JAT, Beleanu D (2017b) Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(04):1740006
[68]	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
[69]	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
[70]	Yang, B.; Yu, T.; Shu, H., 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 (2018) · doi:10.1016/j.conengprac.2018.07.007
[71]	Zafar, AA; Vieru, D.; Akhtar, S., Magnetohydrodynamics of rotating fractional second grade fluid in porous medium, J Prime Res Math, 10, 1, 45-58 (2015) · Zbl 1448.76016