Symmetry Analysis of Time Fractional Convection-reaction-diffusion Equation with a Delay

Yıl 2019, Cilt: 2 Sayı: 3, 113 - 124, 01.10.2019

Aminu Nass , Kassimu Mpungu

Öz

Lie symmetry theory of partial differential equations with both fractional and delay phenomena is considered. A complete group classification of time fractional convection-reaction-diffusion equation with a delay is presented. The Minimal symmetry algebra is found to be one dimensional. The classification is used to find symmetry reductions and exact solutions.

Anahtar Kelimeler

Lie symmetries; Fractional Delay; B$\ddot{a}$ckland Operator; Mittag-Leffler function

Kaynakça

• 1] Kolmanovskii, V.; Myshkis, A., Applied Theory of Functional Differential Equations,The Netherlands: Kluwer Academic Publishers, (1992).
• [2] Ping S., Hongyong Z., Xuebing Z., Dynamic analysis of a fractional order delayedpredatorprey system with harvesting, Theory in Biosciences - Springer, 135 (2016), 5972.
• [3] Abdallah, C., Dorato, P., Benitez-Read, J., and Byrne, R.,Delayed positive feed-back can stabilize oscillatory system, ACC. San Francisco, 3106-3107 (1993).
• [4] Joseph, W.-H. S., Jianhong, W., Xingfu, Z.,A reaction-diusion model for a singlespecies with age structure. I Travelling wavefronts on unbounded domains, Proceedings of theRoyal Society of London. Series A Mathematical, Physical and Engineering Sciences, 457,1841 (2001).
• [5] Meleshko, SV., Moyo, S., On the complete group classication of the reactiondiusionequation with a delay, Journal of Mathematical Analysis and Applications, 2016 (2016),963815.
• [6] Long, F-S., Meleshko, SV., On the complete group classification of the one-dimensionalnonlinear Klein-Gordon equation with a delay, Mathematical Methods in the Applied Sciences,bf 39(12) (2016), 3255-3270.
• [7] Tanthanuch J. Symmetry analysis of the nonhomogeneous inviscid burgers equation withdelay, Communications in Nonlinear Science and Numerical Simulation 17 (2012), 49784987.
• [8] Pue-on, P., Meleshko, S.V. Group classification of second-order delay ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation, 15 (2010),1444-1453.
• [9] Tanthanuch, J., Meleshko, S.V. On denition of an admitted Lie group for functionaldifferential equations, Communications in Nonlinear Science and Numerical Simulation, 9(2004), 117-125.
• [10] Long, F-S., Meleshko, SV., Symmetry analysis of the nonlinear two dimensional Klein-Gordon equation with a time-varying delay, Mathematical Methods in the Applied Sciences,40 (2017), 4658-4673.
• [11] Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko S.V., Symmetries ofIntegro-Dierential Equation, Springer: New York, (2010).
• [12] Kiryakova, V., Generalized Fractional Calculus and Applications, Pitman Research Notesin Mathematics Series. Longman Scientic and Technical, Longma Group, Harlow 1994.
• [13] Kilbas,A. A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of FractionalDifferential Equations, in: Jan V. Mill (Ed.), North-Holland Mathematics Studies, Elsevier,Amsterdam, The Netherlands, 204 (2006).
• [14] Vasily E.T,On chain rule for fractional derivatives, Communications in Nonlinear Scienceand Numerical Simulation, 30(1-3) (20016) , 1-4.
• [15] Laskri,Y., Nasser-eddine Tatar, The critical exponent for an ordinary fractional differential problem, Computers and Mathematics with Applications 59 (2010), 1266-1270.
• [16] Singla, K., Gupta, R.K., Spacetime fractional nonlinear partial differential equations:symmetry analysis and conservation laws, Nonlinear Dyn 89(1) (2017), 321331.
• [17] Chen, C., and Yao-Lin, J., Lie Group Analysis and Invariant Solutions for NonlinearTime-Fractional Diusion-Convection Equations, Commun. Theor. Phys. 68 (2017), 295.
• [18] Wang, G.W., Xu, T.Z., Feng, T. , Lie Symmetry Analysis and Explicit Solutions of theTime Fractional Fifth-Order KdV Equation, PLoS ONE9(2), (2014)
• [19] El Kinani, E. H. and Ouhadan, A., Lie symmetry analysis of some time fractional partialdifferential equations, Int. J. Mod. Phys. Conf. Ser. 38(2015), 15600751-15600758.
• [20] Alshamrani, M. M., Zedan, H. A., Abu-Nawas, M. , Lie group method and fractionaldifferential equations, JNSA, 10, Issue 8(2017),4175-4180.
• [21] Singla, K., Gupta, R.K., Generalized Lie symmetry approach for fractional order systemsof differential equations, III Journal of Mathematical Physics 58 (2017), 061501.
• [22] Lukashchuk, S.Yu., Makunin, A.V., Group classification of nonlinear time-fractionaldiffusion equation with a source term, Appl.Math.Comput. 257(2015), 335-343.
• [23] Hu, J., Ye, Y.J., Shen, S.F., Zhang, J., Lie symmetry analysis of the time fractionalKdV-type equation, Appl.Math.Comput. 233(2014), 439-444.
• [24] Gazizov, R. K., Kasatkin A. A., Lukashchuk, S. Y., Continuous transformationgroups of fractional differential equations, Vestn. USATU 9(2007), 12535
• [25] Gazizov R. K., Kasatkin A. A., Lukashchuk S. Y., Symmetry properties of fractionaldiffusion equations, Phys. Scr. T136 (2009), 0140-161.
• [26] Gaur, M., Singh, K., Symmetry analysis of time-fractional potential Burger's equation,Math. Commun. 22 (2017), 111.
• [27] Ovsiannikov, L. V., Group Analysis of Differential Equations, Nauka, Moscow (1978).
• [28] Bluman, G., Cheviakov, A., Anco S., Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag New York, 168 (2010).
• [29] Aminu, M. N., Symmetry Analysis and Invariants Solutions of Laplace Equation on Surfacesof Revolution, Advances in Mathematics: Scientic Journal, 3(1) (2014), 23-31 .
• [30] Ibragimov, N. H., Elementary Lie group analysis and ordinary differential equations, JohnWiley and Sons, Chichester (1999).
• [31] Azad, H., Mustafa, M. T., Ziad, M., Group classification, optimal system and optimalsymmetry reductions of a class of Klien Gorden equations, system and optimal symmetryreductions of a class of Klien Gorden equations, Communications in nonlinear science andnumerical simulation, (2009).
• [32] Ahmad, Y. A., On the Conservation Laws for a Certain Class of Nonlinear Wave Equation via a New Conservation Theorem, Communications in Nonlinear Science and Numerical Simulation, 17(4), (2012), 15661575.
• [33] Nass, M. N., Fredericks, E., W-symmetries of jump-diffusion It stochastic differentialequations, Nonlinear Dyn, 90(4) (2017) , 2869-2877.
• [34] Wafo, S. C., Mahomed, F. M, Integration of stochastic ordinary differential equationsfrom a symmetry standpoint, Journal of Physics A: Mathematical and General; 34 (2001),177-194.
• [35] Unal, G., Symmetries of It^o and Stratonovich Dynamical Systems and their ConservedQuantities, Nonlinear Dynamics, 32 (2003), 417-426.
• [36] Fredericks, E., Mahomed, F. M., Formal Approach for Handling Lie Point Symmetriesof Scalar First-Order Ito Stochastic Ordinary Differential Equations, Journal of NonlinearMathematical Physics 15 (2008), 44-59.
• [37] Nass,, A, M., Fredericks, E., Lie Symmetry of It^o Stochastic Differential EquationDriven by Poisson Process, American Review of Mathematics and Statistics, 4(1) (2016),17-30.
• [38] Nass, A, M., Fredericks, E., Symmetry of Jump-Diffusion Stochastic Differential Equations, Global and Stochastic Analysis, 3(1) (2016), 11-23.
• [39] Zhang, Q., Ran, M., Xu, D., Analysis of the compact difference scheme for the semilinearfractional partial differential equation with time delay, Applicable Analysis, 96(11) (2017),1867-1884.
• [40] Hale, J. K., Verduyn, L., Sjoerd, M., Introduction to Functional Differential Equations,Springer, New York, (1993).
• [41] Morgadoa, L., Fordb, N.J., Limac, P.M., Analysis and numerical methods for fractionaldifferential equations with delay, Journal of Computational and Applied Mathematics, 252(2013), 159-168.
• [42] Ermk, J., Hornek, J., Kisela, T., Stability regions for fractional differential systemswith a time delay, Communications in Nonlinear Science and Numerical Simulation, 31(1-3)(2016), 108-123.
• [43] Pimenov, V. G., Hendy, A. S., Numerical Studies for Fractional Functional DifferentialEquations with Delay Based on BDF-Type Shifted Chebyshev Approximations, Abstract andApplied Analysis, 2015.
• [44] Zhang ,J., Zhang, J., Symmetry analysis for time-fractional convection-diffusion equation,e-print arXiv:1512.01319.
• [45] Polyanin, A.D., Zhurov. A.I., Exact separable solutions of delay reaction-diffusion equations and other nonlinear partial functional-differential equations, Communications in Nonlinear Science and Numerical Simulation, Vol. 19 (2014), No. 3, pp. 409-416.
• [46] Polyanin, A.D., Zhurov, A.I, Functional constraints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations, Communicationsin Nonlinear Science and Numerical Simulation, Vol. 19, No. 3 (2014), pp. 417-430.
• [47] Polyanin, A.D., Zhurov., A.I. New generalized and functional separable solutions to non-linear delay reaction-diffusion equations, International Journal of Non-Linear Mechanics, Vol.59 ( 2014), p. 16-22.
• [48] Polyanin, A.D., Zhurov. A.I. Generalized and functional separable solutions to nonlineardelay Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, Vol. 19 (2014), No. 8, pp. 2676-2689.
• [49] Polyanin, A.D., Zhurov. A.I. Non-linear instability and exact solutions to some delayreaction-diffusion systems, International Journal of Non-Linear Mechanics, Vol. 62 (2014),pp. 33-40.
• [50] Polyanin, A.D., Zhurov. A.I. Nonlinear delay reaction-diffusion equations with varyingtransfer coefficients: Exact methods and new solutions, Applied Mathematics Letters, Vol. 37(2014), pp. 43-48.
• [51] Polyanin, A.D., Zhurov. A.I. The functional constraints method: Application to non-linear delay reaction-diffusion equations with varying transfer coefficients, International Journalof Non-Linear Mechanics, Vol. 67 (2014), pp. 267-277.
• [52] Polyanin, A.D., Zhurov. A.I. The generating equations method: Constructing exact solutions to delay reaction-diffusion systems and other non-linear coupled delay PDEs, International Journal of Non-Linear Mechanics, Vol. 71 (2015), pp. 104-115. [53] Polyanin, A.D., Sorokin, V.G. Nonlinear delay reaction-diffusion equations: Traveling-wave solutions in elementary functions, Applied Mathematics Letters, Vol. 46 (2015), pp.38-43.
• [54] Ouyang, Z. G. Existence and Uniqueness of the Solutions for a Class of Nonlinear FractionalOrder Partial Differential Equations with delay, Comput. Math. Appl. 61 (2011), 860-870.
• [55] Zhu, B., Liu, L., Wu, Y. Existence and Uniqueness of Global Mild Solutions for a Class ofNonlinear Fractional Reaction-diffusion Equations with delay, Comput. Math. Appl. (2016).
• [56] Singla, K., Gupta, R.K., Generalized Lie symmetry approach for fractional order systemsof differential equations III, Journal of Mathematical Physics 58 (2017), 061501.
• [57] Singla, K., Gupta, R.K., On invariant analysis of space-time fractional nonlinear systemsof partial differential equations. II, Journal of Mathematical Physics 58 (2017), 051503.
• [58] Singla, K., Gupta, R.K., On invariant analysis of some time fractional nonlinear systemsof partial differential equations. I, Journal of Mathematical Physics 57 (2016), 101504 .
• [59] Singla, K., Gupta, R.K., Comment on Lie symmetries and group classification of a classof time fractional evolution systems,Journal of Mathematical Physics 57 (2017), 054101.
• [60] Diego, A. G., Galeano, C.H., Mantilla, J.M., Computational examples of reaction-convection-diffusion equations solution under the influence of fluid flow: First example, Applied Mathematical Modelling, 36 (2012), 50295045
• [61] Deng, D. The study of a fourth-order multistep ADI method applied to nonlinear delay reaction diffusion equations, Applied Numerical Mathematics 96(2015), 118133
• [62] Zhang, Q., Zhang, C. A new linearized compact multisplitting scheme for the nonlinearconvection-reaction-diffusion equations with delay, Commun Nonlinear Sci Numer Simulat 18(2013), 32783288.
• [63] Makungu, J., Haario, H., Mahera, W. C. A generalized 1-dimensional particle transport method for convection diffusion reaction model, Afr. Mat. (2012) 23:2139.