# Local Fractional Operator for a One-Dimensional Coupled Burger Equation of Non-Integer Time Order Parameter

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2018.50.1.3## Keywords:

Caputo derivative, coupled Burger equation, exact solution, fractional differential equations, modified DTM## Abstract

In this study, approximate solutions of a system of time-fractional coupled Burger equations were obtained by means of a local fractional operator (LFO) in the sense of the Caputo derivative. The LFO technique was built on the basis of the standard differential transform method (DTM). Illustrative examples used in demonstrating the effectiveness and robustness of the proposed method show that the solution method is very efficient and reliable as "? unlike the variational iteration method "? it does not depend on any process of identifying Lagrange multipliers, even while still maintaining accuracy.

## References

Oderinua, R.A., The Reduced Differential Transform Method for the Exact Solutions of Advection, Burgers and Coupled Burgers Equations, Theory and Applications of Mathematics & Computer Science, 2(1), pp. 10-14, 2012.

Burger, J.M., A Mathematical Model Illustrating the Theory of Turbulence, Advanced in Applied Mechanics, 1, pp. 171-179, 1948.

Cole, J.D., On A Quasilinear Parabolic Equations Occurring in Aerodynamics, Quarterly of Applied Mathematics, 9, pp. 225-236, 1951.

Srivastava, V.K., Singh, S. & Awasthi, M.K., Numerical Solution of Coupled Burgers' Equation by An Implicit Finite Difference Scheme, AIP Advances, 3, 082131, 2013.

Mittal, R.C. & Arora, G., Numerical Solution of the Coupled Viscous Burgers' Equation, Communications in Nonlinear Science and Numerical Simulation, 16(3), pp. 1304-1313, 2011.

Deghan, M., Asgar, H. & Mohammad, S., The Solution of Coupled Burgers' Equations using Adomian-Pade Technique, Applied Mathematics and Computation, 189, pp. 1034-1047, 2007.

Soliman, A.A., The Modified Extended Tanh-function Method for Solving Burgers-type Equations, Physica A, 361(2), pp. 394-404, 2006.

Abdou, M.A. & Soliman, A.A., Variational Iteration Method for Solving Burger's and Coupled Burger's Equations, Journal of Computational and Applied Mathematics, 181(2), pp. 245-251, 2005.

Esipov, S.E., Coupled Burgers' Equations: A Model of Polydispersive Sedimentation, Physical Review E., 52, 3711, 1995.

Mokhtari, R., Toodar, A.S. & Chegini, N.G., Application of the Generalized Differential Quadrature Method in Solving Burgers' Equations, Communications in Theoretical Physics, 56(6), 1009, 2011.

Rashid, A. & Ismail, A.I.B., A Fourier Pseudospectral Method for Solving Coupled Viscous Burgers' Equations, Computational Methods in Applied Mathematics, 9(4), pp. 412-420, 2009.

Kaya, D., An Explicit Solution of Coupled Viscous Burgers' Equations by the Decomposition Method, JJMMS, 27(11), pp. 675-680, 2001.

Yang, X.J., Machado, J.A.T., Cattani, C. & Gao, F., On A Fractal LC-Electric Circuit Modeled by Local Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 47, pp. 200-206, 2017.

Machado, J.T., Kiryakova, V. & Mainardi, F., Recent History of Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 16(3), pp. 1140-1153, 2011.

Kilbas, A.A. Srivastava, H.M. & Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.

Khalil, R., Al Horani, M., Yousef, A. & Sababheh, M., A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 264, pp. 65-70, 2014.

De Oliveira, E.C. & Machado, J.A.T., A Review of Definitions for Fractional Derivatives and Integral, Mathematical Problems in Engineering, 2014, Article ID: 238459, 2014. DOI:10.1155/2014/238459

Yang, X.J., Fractional Derivatives of Constant and Variable Orders Applied to Anomalous Relaxation Models in Heat Transfer Problems, Thermal Science, 21(3), pp. 1161-1171, 2017.

Yang, X.J, Zhang, Z.H, Machado, T.J.A. & Baleanu, D., On Local Fractional Operators View of Computational Complexity Diffusion and Relaxation Defined on Cantor Sets, Thermal Science, 20(Suppl.3), pp. 755-767, 2016.

Momani, S., Non-perturbative Analytical Solutions of the Space- and Time-fractional Burgers Equations, Chaos, Solitons and Fractals, 28(4), pp. 930-937, 2006.

Yang, X.J., Gao, F. & Srivastava, H.M., Exact Travelling Wave Solutions for the Local Fractional Two-dimensional Burgers-type Equations, Computers and Mathematics with Applications, 73(2), pp. 203-210, 2017.

Rao, C.S. & Satyanarayana, E., Solutions of Burgers Equation, International Journal of Nonlinear Science, 9(3), pp. 290-295, 2010.

Fletcher, C.A.J., Burgers Equation: A Model for All Reasons, in Numerical Solutions of Partial Differential Equations, (Ed. J. Noye), North-Holland, Amsterdam, pp. 139-225, 1982.

Kim, Y.J. & Tzavaras, A.E., Diffusive ?-waves and Metastability in the Burgers Equation, SIAM Journal on Mathematical Analysis, 33(3), pp. 607-633, 2001.

Tamsir, M., Srivastava, V.K. & Jiwari, R., An Algorithm Based on Exponential Modified Cubic B-spline Differential Quadrature Method for Nonlinear Burgers' Equation, Applied Mathematics and Computation. 290, pp. 111-124, 2016.

Edeki, S.O., Akinlabi, G.O. & Adeosun, S.A., Analytic and Numerical Solutions of Time-fractional Linear Schrdinger Equation, Communications in Mathematics and Applications, 7(1), pp. 1-10, 2016a.

Caputo, M., & Mainardi, F., Linear Models of Dissipation in Anelastic Solids, Rivista Del Nuovo Cimento, 1(2), pp. 161-198, 1979.

Mainardi, F., On the Initial Value Problem for the Fractional Diffusion-Wave Equation, in: S. Rionero, T. Ruggeeri (Eds.), Waves and Stability in Continuous Media, World Scientific, Singapore, pp. 246-256, 1994.

Zhou, J.K., Differential Transformation and its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China, 1986.

Edeki, S.O., Akinlabi, G.O. & Adeosun, S.A., On A Modified Transformation Method for Exact and Approximate Solutions of Linear Schrdinger Equations, AIP Conference Proceedings, 1705, 020048; DOI:10.1063/1.4940296, 2016b.

Jang, B., Solving Linear and Nonlinear Initial Value Problems by the Projected Differential Transform Method, Computer Physics Communications, 181(5), pp. 848-854, 2010.

Edeki, S.O. Akinlabi, G.O. & Akeju, A.O., A Handy Approximation Technique for Closed-form and Approximate Solutions of Time-Fractional Heat and Heat-Like Equations with Variable Coefficients, Proceedings of the World Congress on Engineering, II, WCE 2016, June 29 - July 1, London, United Kingdom, 2016.

Akinlabi, G.O. & Edeki, S.O., On Approximate and Closed-form Solution Method for Initial-value Wave-like Models, International Journal of Pure and Applied Mathematics, 107(2), pp. 449-456, 2016.

Tamsir, M., & Srivastava, V.K., Revisiting the Approximate Analytical Solution of Fractional-order Gas Dynamics Equation, Alexandria Engineering Journal, 55(2), pp. 867-874, 2016.

Srivastava, V.K., Tarmsir, M., Awasthi, M.K. & Singh, S., One-Dimensional Coupled Burgers' Equation and Its Numerical Solution by An Implicit Logarithmic Finite-difference Method, AIP Advances, 4, 037119, 2014. DOI:10.1063/1.4869637