Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves

Document Type: Research Paper

Author

1 School of Science, Xi'an University of Architecture and Technology, Xi’an, China

2 University National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China

Abstract

The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of the equation.

Keywords

[1] Mancas, S.C., Adams, R., Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations, Chaos, Solitons & Fractals, 126, 2019, 385-39.

[2] Gupta, A.K., Ray, S.S., On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water, Chaos, Solitons & Fractals, 116, 2018, 376-38.

[3] Cevikel, A.C., New exact solutions of the space-time fractional KdV-Burgers and non-linear fractional foam drainage equation, Thermal Science, 22, 2018, S15-S24.

[4] Kim, J.M., Chun, C.B.  New Exact Solutions to the KdV-Burgers-Kuramoto Equation with the Exp-Function Method, Abstract and Applied Analysis, 2012, 892420.

[5] He, J.H., Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical simulation, 14(6), 2013, 363-366.

[6] He, J.H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012, 916793.

[7] He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations, Chaos Solitons & Fractals, 30(3), 2006,700-708.

[8] Biswas, A., Zhou, Q., Moshokoa, S.P., et al. Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik, 145, 2017, 14-17.

[9] El-Kalaawy, O.H., Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation, Computers and Mathematics with Applications, 72, 2016, 1013-1041.

[10] El-Kalaawy, O.H., New Variational principle-exact solutions and conservation laws for modified ion-acoustic shock waves and double layers with electron degenerate in plasma, Physics of Plasmas, 24(3), 2017, 032308.

[11] He, J.H. A modified Li-He’s variational principle for plasma, International Journal of Numerical Methods for Heat and Fluid Flow, 2019, DOI: 10.1108/HFF-06-2019-0523.

[12] He, J.H., Ji, F.Y., Taylor series solution for Lane-Emden equation, Journal of Mathematical Chemistry, 57(8), 2019, 1932–1934.

[13] He, J.H., The simplest approach to nonlinear oscillators, Results in Physics, 15, 2019, 102546.

[14] He, J.H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012, 857612. 

[15] He, J.H., Homotopy perturbation method with two expanding parameters, Indian Journal of Physics, 88, 2014, 193-196.

[16] Adamu, M.Y., Ogenyi, P., New approach to parameterized homotopy perturbation method, Thermal Science, 22(4), 2018, 1865-1870.

[17] Ban, T., Cui, R.Q., He’s homotopy perturbation method for solving time-fractional Swift-Hohenerg equation, Thermal Science, 22(4), 2018, 1601-1605.

[18] Liu, Z.J., Adamu, M.Y., Suleiman, E., et al. Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations, Thermal Science, 21, 2017, 1843-1846.

[19] Wu, Y., He, J.H., Homotopy perturbation method for nonlinear oscillators with coordinate dependent mass, Results in Physics, 10, 2018, 270–271.

[20] Anjum, N., He, J.H., Laplace transform: Making the variational iteration method easier, Applied Mathematics Letters, 92, 2019, 134-138.

[21] He, J.H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 2006, 1141-1199.

[22] He, J.H., Kong, H.Y., Chen, R.X., Variational iteration method for Bratu-like equation arising in electrospinning, Carbohydrate Polymers, 105, 2014, 229-230.

[23] He, J.H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53(11), 2014, 3698-3718. 

[24] He, J.H., Fractal calculus and its geometrical explanation, Results in Physics, 10, 2018, 272-276.

[25] Li, X.X., Tian, D., He, C.H., He, J.H., A fractal modification of the surface coverage model for an electrochemical arsenic sensor, Electrochimica Acta, 296, 2019, 491-493.

[26] Wang, Q.L., Shi, X.Y., He, J.H., Li, Z.B., Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals, 26(6), 2018, 1850086.

[27] Wang, Y., Deng, Q.G., Fractal derivative model for tsunami travelling, Fractals, 27(1), 2019, 1950017.

[28] Liu, H.Y., Yao, S.W., Yang, H.W., Liu, J., A fractal rate model for adsorption kinetics at solid/solution interface, Thermal Science, 23(4), 2019, 2477-2480.

[29] Wang, Y., Yao, S.W., Yang, H.W., A fractal derivative model for snow’s thermal insulation property, Thermal Science, 23(4), 2019, 2351-2354.

[30] Wang, Y., An, J.Y., Wang, X.Q., A variational formulation for anisotropic wave traveling in a porous medium, Fractals, 27(4), 2019, 1950047.

[31] He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons & Fractals, 19(4), 2004, 847-851.

[32] Ain, Q.T., He, J.H. On two-scale dimension and its applications, Thermal Science, 23(3B), 2019, 1707-1712.

[33] He, J.H., Ji, F.Y., Two-scale mathematics and fractional calculus for thermodynamics, Thermal Science, 23(4), 2019, 2131-2133.

[34] He, J.H., An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams, Applied Mathematics Letters, 52, 2016, 1-3.

[35] He, J.H., Hamilton's principle for dynamical elasticity, Applied Mathematics Letters, 72, 2017, 65-69.

[36] He, J.H., Generalized equilibrium equations for shell derived from a generalized variational principle, Applied Mathematics Letters, 64, 2017, 94-100.

[37] He, J.H., Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, 2019, DOI: 10.1108/HFF-07-2019-0577.

[38] He, J.H., Sun, C., A variational principle for a thin film equation, Journal of Mathematical Chemistry, 57(9), 2019, 2075–2081.

[39] Wang, K. L., He, C.H., A remark on Wang's fractal variational principle, Fractals, 2019, DOI: 10.1142/S0218348X19501342.