Solution of moving boundary space-time fractional Burger’s equation. (English) Zbl 1437.35604
Summary: The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35R11 | Fractional partial differential equations |
| 35R37 | Moving boundary problems for PDEs |
References:
| [1] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier (2006), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands · Zbl 1092.45003 |
| [2] | West, B. J.; Bolognab, M.; Grigolini, P., Physics of Fractal Operators (2003), New York, NY, USA: Springer, New York, NY, USA |
| [3] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Yverdon, Switzerland: Gordon and Breach, Yverdon, Switzerland · Zbl 0818.26003 |
| [4] | He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology & Society, 15, 2, 86-90 (1999) |
| [5] | Cui, M., Compact finite difference method for the fractional diffusion equation, Journal of Computational Physics, 228, 20, 7792-7804 (2009) · Zbl 1179.65107 · doi:10.1016/j.jcp.2009.07.021 |
| [6] | El-Sayed, A. M. A.; Gaber, M., The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Physics Letters A, 359, 3, 175-182 (2006) · Zbl 1236.35003 · doi:10.1016/j.physleta.2006.06.024 |
| [7] | El-Sayed, A. M. A.; Behiry, S. H.; Raslan, W. E., Adomian’s decomposition method for solving an intermediate fractional advection-dispersion equation, Computers & Mathematics with Applications, 59, 5, 1759-1765 (2010) · Zbl 1189.35358 · doi:10.1016/j.camwa.2009.08.065 |
| [8] | Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters, 21, 2, 194-199 (2008) · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022 |
| [9] | He, J. H., A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation, 2, 4, 230-235 (1997) · Zbl 0923.35046 |
| [10] | Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus: Models and Numerical Methods. Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3 (2012), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA · Zbl 1248.26011 |
| [11] | Podlubny, I., Fractional Differential Equations, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010 |
| [12] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Amsterdam, The Netherlands: Gordon and Breach Science Publishers, Amsterdam, The Netherlands · Zbl 0818.26003 |
| [13] | Jumarie, G., On the fractional solution of the equation f(x + y) = f(x)f(y) and its application to fractional Laplace’s transform, Applied Mathematics and Computation, 219, 4, 1625-1643 (2012) · Zbl 1291.26003 · doi:10.1016/j.amc.2012.08.004 |
| [14] | Jumarie, G., An approach via fractional analysis to non-linearity induced by coarse-graining in space, Nonlinear Analysis. Real World Applications, 11, 1, 535-546 (2010) · Zbl 1195.37054 · doi:10.1016/j.nonrwa.2009.01.003 |
| [15] | Jumarie, G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Applied Mathematics Letters, 22, 3, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003 |
| [16] | Wang, G.-W.; Liu, X.-Q.; Zhang, Y.-Y., Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Communications in Nonlinear Science and Numerical Simulation, 18, 9, 2321-2326 (2013) · Zbl 1304.35624 · doi:10.1016/j.cnsns.2012.11.032 |
| [17] | Wang, H.; Xia, T.-C., The fractional supertrace identity and its application to the super Jaulent-Miodek hierarchy, Communications in Nonlinear Science and Numerical Simulation, 18, 10, 2859-2867 (2013) · Zbl 1306.37083 · doi:10.1016/j.cnsns.2013.02.005 |
| [18] | Hammouch, Z.; Mekkaoui, T., Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, Internationaal Journal of Applied Mathematical Research, 1, 2, 206-212 (2012) |
| [19] | Almeida, R.; Torres, D. F. M., Fractional variational calculus for nondifferentiable functions, Computers and Mathematics with Applications, 61, 10, 3097-3104 (2011) · Zbl 1222.49026 · doi:10.1016/j.camwa.2011.03.098 |
| [20] | Wu, G.-C., A fractional variational iteration method for solving fractional nonlinear differential equations, Computers and Mathematics with Applications, 61, 8, 2186-2190 (2011) · Zbl 1219.65085 · doi:10.1016/j.camwa.2010.09.010 |
| [21] | Zhang, S.; Zong, Q. A.; Liu, D.; Gao, Q., A generalized exp-function method for fractional Riccati differential equations,, Communications in Fractional Calculus, 1, 1, 48-51 (2010) |
| [22] | Tang, B.; He, Y.; Wei, L.; Zhang, X., A generalized fractional sub-equation method for fractional differential equations with variable coefficients, Physics Letters A, 376, 38-39, 2588-2590 (2012) · Zbl 1266.34014 · doi:10.1016/j.physleta.2012.07.018 |
| [23] | Zheng, B., ( \(G^'}/G)\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Communications in Theoretical Physics, 58, 5, 623-630 (2012) · Zbl 1264.35273 · doi:10.1088/0253-6102/58/5/02 |
| [24] | Akgül, A.; Kılıçman, A.; Inc, M., Improved (G’/G)-expansion method for the space and time fractional foam drainage and KdV equations, Abstract and Applied Analysis, 2013 (2013) · Zbl 1293.35007 · doi:10.1155/2013/414353 |
| [25] | Lu, B., The first integral method for some time fractional differential equations, Journal of Mathematical Analysis and Applications, 395, 2, 684-693 (2012) · Zbl 1246.35202 · doi:10.1016/j.jmaa.2012.05.066 |
| [26] | Zhang, S.; Zhang, H., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters. A, 375, 7, 1069-1073 (2011) · Zbl 1242.35217 · doi:10.1016/j.physleta.2011.01.029 |
| [27] | Guo, S.; Mei, L.; Li, Y.; Sun, Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Physics Letters A, 376, 4, 407-411 (2012) · Zbl 1255.37022 · doi:10.1016/j.physleta.2011.10.056 |
| [28] | Abdel-Salam, E. A.; Yousif, E. A., Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.35057 · doi:10.1155/2013/846283 |
| [29] | Li, W.; Yang, H.; He, B., Exact solutions of fractional burgers and Cahn-Hilliard equations using extended fractional Riccati expansion method, Mathematical Problems in Engineering, 2014 (2014) · Zbl 1407.35214 · doi:10.1155/2014/104069 |
| [30] | Friedman, A., Partial Differential Equations of Parabolic Type (1964), Englewood Cliffs, NJ, USA: Prentice Hall, Englewood Cliffs, NJ, USA · Zbl 0144.34903 |
| [31] | Friedman, A., Variational Principles and Free-Boundary Problems (1982), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0564.49002 |
| [32] | Hastaoglu, M. A., Numerical solution of three-dimensional moving boundary problems: melting and solidification with blanketing of a third layer, Chemical Engineering Science, 42, 10, 2417-2423 (1987) · doi:10.1016/0009-2509(87)80114-8 |
| [33] | Zerroukat, M.; Chatwin, C. R., Explicit variable time-step method for implicit moving boundary problems, Communications in Numerical Methods in Engineering, 10, 3, 227-235 (1994) · Zbl 0801.65122 · doi:10.1002/cnm.1640100306 |
| [34] | Slikkerveer, P. J.; van Lohuizen, E. P.; O’Brien, S. B. G., An implicit surface tension algorithm for picard solvers of surface-tension-dominated free and moving boundary problems, International Journal for Numerical Methods in Fluids, 22, 9, 851-865 (1996) · Zbl 0865.76042 · doi:10.1002/(SICI)1097-0363(19960515)22:9LTHEXA851::AID-FLD380>3.0.CO;2-R |
| [35] | Bodard, N.; Bouffanais, R.; Deville, M. O., Solution of moving-boundary problems by the spectral element method, Applied Numerical Mathematics, 58, 7, 968-984 (2008) · Zbl 1140.76029 · doi:10.1016/j.apnum.2007.04.009 |
| [36] | Kannan, R.; Wang, Z. J., A high order spectral volume method for moving boundary problems, Proceedings of the 40th AIAA Fluid Dynamics Conference |
| [37] | Cao, W.; Huang, W.; Russell, R. D., A moving mesh method based on the geometric conservation law, SIAM Journal on Scientific Computing, 24, 1, 118-142 (2002) · Zbl 1016.65066 · doi:10.1137/S1064827501384925 |
| [38] | Li, R.; Tang, T.; Zhang, P., A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics, 177, 2, 365-393 (2002) · Zbl 0998.65105 · doi:10.1006/jcph.2002.7002 |
| [39] | Baines, M. J.; Hubbard, M. E.; Jimack, P. K., A moving mesh finite element algorithm for fluid flow problems with moving boundaries, International Journal for Numerical Methods in Fluids, 47, 10-11, 1077-1083 (2005) · Zbl 1064.76063 · doi:10.1002/fld.860 |
| [40] | Baines, M. J.; Hubbard, M. E.; Jimack, P. K.; Jones, A. C., Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions, Applied Numerical Mathematics., 56, 2, 230-252 (2006) · Zbl 1092.65081 · doi:10.1016/j.apnum.2005.04.002 |
| [41] | Markou, G. A.; Mouroutis, Z. S.; Charmpis, D. C.; Papadrakakis, M., The ortho-semi-torsional (OST) spring analogy method for 3D mesh moving boundary problems, Computer Methods in Applied Mechanics and Engineering, 196, 4-6, 747-765 (2007) · Zbl 1121.74482 · doi:10.1016/j.cma.2006.04.009 |
| [42] | Hubbard, M. E.; Baines, M. J.; Jimack, P. K., Consistent Dirichlet boundary conditions for numerical solution of moving boundary problems, Applied Numerical Mathematics, 59, 6, 1337-1353 (2009) · Zbl 1162.65051 · doi:10.1016/j.apnum.2008.08.002 |
| [43] | Ahmed, S. G.; Meshrif, S. A., A new numerical algorithm for 2D moving boundary problems using a boundary element method, Computers & Mathematics with Applications, 58, 7, 1302-1308 (2009) · Zbl 1189.65217 · doi:10.1016/j.camwa.2009.03.115 |
| [44] | Ko1, J. H.; Park, S. H.; Park, H. C.; Byun, D., Finite macro-element-based volume grid deformation for large moving boundary problems, International Journal for Numerical Methods in Biomedical Engineering, 26, 12, 1656-1673 (2010) · Zbl 1323.76075 · doi:10.1002/cnm.1252 |
| [45] | Liu, W.; Yao, J.; Wang, Y., Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient, International Journal of Heat and Mass Transfer, 55, 21-22, 6017-6022 (2012) · doi:10.1016/j.ijheatmasstransfer.2012.06.012 |
| [46] | Atkinson, C., Moving boundary problems for time fractional and composition dependent diffusion, Fractional Calculus & Applied Analysis, 15, 2, 207-221 (2012) · Zbl 1302.35473 · doi:10.2478/s13540-012-0015-2 |
| [47] | Yin, C.; Xu, M., An asymptotic analytical solution to the problem of two moving boundaries with fractional diffusion in one-dimensional drug release devices, Journal of Physics A: Mathematical and Theoretical, 42, 11 (2009) · Zbl 1180.82242 · doi:10.1088/1751-8113/42/11/115210 |
| [48] | Kushwaha, M. S.; Kumar, A., An approximate solution to a moving boundary problem with space-time fractional derivative in fluvio-deltaic sedimentation process, Ain Shams Engineering Journal, 4, 889-895 (2013) · doi:10.1016/j.asej.2012.12.005 |
| [49] | Li, X.; Wang, S.; Zhao, M., Two methods to solve a fractional single phase moving boundary problem, Central European Journal of Physics, 11, 10, 1387-1391 (2013) |
| [50] | Brezis, H.; Browder, F., Partial differential equations in the 20th century, Advances in Mathematics, 135, 1, 76-144 (1998) · Zbl 0915.01011 · doi:10.1006/aima.1997.1713 |
| [51] | Polyanin, A. D.; Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (2004), Boca Raton, Fla, USA: Chapman & Hall/CRC, Boca Raton, Fla, USA · Zbl 1053.35001 |
| [52] | Malfliet, W., Solitary wave solutions of nonlinear wave equations, American Journal of Physics, 60, 7, 650-654 (1992) · Zbl 1219.35246 · doi:10.1119/1.17120 |
| [53] | Estévez, P. G.; Gordoa, P. R., Non classical symmetries and the singular manifold method. Theory and six examples, Studies in Applied Mathematics, 95, 73-113 (1995) · Zbl 0840.35003 |
| [54] | Liu, S. D.; Liu, S. K.; Ye, Q. X., Explicit traveling wave solutions of nonlinear evolution equations, Mathematics in Practice and Theory, 28, 4, 289-301 (1998) · Zbl 1493.35091 |
| [55] | Wazwaz, A. M., Travelling wave solutions of generalized forms of BURgers, BURgers-KdV and BURgers-Huxley equations, Applied Mathematics and Computation, 169, 1, 639-656 (2005) · Zbl 1078.35109 · doi:10.1016/j.amc.2004.09.081 |
| [56] | Lillo, S. D., Moving boundary problems for the Burgers equation, Inverse Problems, 14, L1-L4 (1998) · Zbl 0894.35134 |
| [57] | Biondini, G.; de Lillo, S., On the Burgers equation with moving boundary, Physics Letters A, 279, 3-4, 194-206 (2001) · Zbl 0972.35095 · doi:10.1016/S0375-9601(00)00839-2 |
| [58] | Calogero, F.; Lillo, S. D., The Burgers equation on the semi-infinite and finite intervals, Nonlinearty, 2, 37-43 (1987) · Zbl 0685.35091 |
| [59] | Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers & Mathematics with Applications, 51, 9-10, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001 |
| [60] | Jumarie, G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Mathematical and Computer Modelling, 44, 3-4, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003 |
| [61] | Jumarie, G., Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Applied Mathematics Letters, 22, 11, 1659-1664 (2009) · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
