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The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method. (English) Zbl 1146.35304
Summary: A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations with initial conditions by the variational iteration method. The exact and numerical solutions obtained by the variational iteration method are compared with that obtained by Adomian decomposition method. The results show that the variational iteration method is much easier, more convenient, and more stable and efficient than Adomian decomposition method. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.
MSC:
35A35Theoretical approximation to solutions of PDE
35S10Initial value problems for pseudodifferential operators
26A33Fractional derivatives and integrals (real functions)
References:
[1]Ablowitz, M. J.; Clarkson, P. A.: Solitons: nonlinear evolution equations and inverse scattering, (1991) · Zbl 0762.35001
[2]Miura, M. R.: Bäcklund transformation, (1978)
[3]Gu, C. H.: Darboux transformation in solitons theory and geometry applications, (1999)
[4]Gardner, C. S.: Method for solving the Korteweg – de Vries equation, Phys. rev. Lett. 19, 1095 (1967)
[5]Hirota, R.: Exact solution of the Korteweg – de Vries equation for multiple collisions of solitons, Phys. rev. Lett. 27, 1192 (1971) · Zbl 1168.35423 · doi:10.1103/PhysRevLett.27.1192
[6]Malfliet, W.: Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60, 659 (1992) · Zbl 1219.35246 · doi:10.1119/1.17120
[7]Inc, M.; Fan, E. G.: Extended tanh-function method for finding travelling wave solutions of some nonlinear partial differential equations, Z. naturforsch. A 60, 7 (2005)
[8]Yan, Z.; Zhang, H. Q.: New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. lett. A 252, 291 (1999) · Zbl 0938.35130 · doi:10.1016/S0375-9601(98)00956-6
[9]Inc, M.; Evans, D. J.: A study for obtaining more solitary pattern solutions of fifth-order KdV-like equations, Int. J. Comput. math. 81, 473 (2004) · Zbl 1061.35112 · doi:10.1080/00207160410001661285
[10]Wang, M. L.: Exact solutions for a compound KdV – Burgers equation, Phys. lett. A 215, 279 (1996) · Zbl 0972.35526 · doi:10.1016/0375-9601(96)00103-X
[11]Yan, Z.; Zhang, H. Q.: New explicit solitary wave solutions and periodic wave solutions for Whitham – Broer – Kaup equation in shallow water, Phys. lett. A 285, 355 (2001) · Zbl 0969.76518 · doi:10.1016/S0375-9601(01)00376-0
[12]Fu, Z.: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. lett. A 290, 72 (2001) · Zbl 0977.35094 · doi:10.1016/S0375-9601(01)00644-2
[13]Yan, Z.: The extended Jacobian elliptic function expansion method and its application in the generalized Hirota – satsuma coupled KdV system, Chaos solitons fractals 15, 575 (2003) · Zbl 1037.35074 · doi:10.1016/S0960-0779(02)00145-5
[14]Ray, S. S.; Bera, R. K.: An approximate solutions of a nonlinear fractional differential equation by Adomian decomposition method, Appl. math. Comput. 167, No. 2, 561-571 (2005) · Zbl 1082.65562 · doi:10.1016/j.amc.2004.07.020
[15]Bagley, R. L.; Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity, J. rheol. 27, 201-210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724
[16]Bagley, R. L.; Torvik, P. J.: Fractional calculus — A different approach to the analysis of viscoelastically damped structures, Aiaa j. 21, 741-748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142
[17]Bagley, R. L.; Torvik, P. J.: Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j. 23, 918-925 (1985) · Zbl 0562.73071 · doi:10.2514/3.9007
[18]Ichise, M.; Nagayanagi, Y.; Kojima, T.: An analog simulation of non integer order transfer functions for analysis of electrode processes, J. electron. Chem. interf. Electrochem. 33, 253-265 (1971)
[19]Sun, H. H.; Onaral, B.; Tsao, Y.: Application of positive reality princible to metal electrode linear polarization phenomena, IEEE trans. Bimed. eng. 31, 664-674 (1984)
[20]Sun, H. H.; Abdelvahab, A. A.; Onaral, B.: Linear approximation of transfer function with a pole of fractional order, IEEE trans. Automat. control 29, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[21]Mandelbrot, B.: Some noises with 1/f spectrum, a Bridge between direct current and white noise, IEEE trans. Inform. theory 13, 289-298 (1967) · Zbl 0148.40507 · doi:10.1109/TIT.1967.1053992
[22]Hartley, T. T.: Chaos in a fractional order Chua system, IEEE trans. Circuits syst. 42, 485-490 (1995)
[23]Mainardi, F.: Fractional calculus: some basic problem in continuum and statistical mechanics, Fractals and fractional calculus in continuum and mechanics, 291-348 (1997)
[24]Rossikhin, Y. A.; Shitikova, M. V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. mech. Rev. 50, 15-67 (1997)
[25]Magin, R. L.: Fractional calculus in bioengineering, Crit. rev. Biomed. eng. 32, 1-104 (2004)
[26]Podlubny, I.: Fractional differential equations, (1999)
[27]A. Luchko, R. Groneflo, The initial value problem for some fractional differential equations with the Caputo derivative, preprint series, A08-98, Fachbereich Mathematik und Informatik, Freie Universität Berlin, 1998
[28]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[29]Oldham, K. B.; Spainer, J.: The fractional calculus, (1974)
[30]Caputo, M.: Linear model of dissipation whose q is almost frequency independent, J. roy. Astr. soc. 13, 529-539 (1967)
[31]Atanackovic, T. M.; Stankovic, B.: On a system of differential equations with fractional derivatives arising in rod theory, J. phys. A 37, 1241-1250 (2004) · Zbl 1059.35011 · doi:10.1088/0305-4470/37/4/012
[32]Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. Comput. 131, 517-529 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[33]Daftardar-Gejji, V.; Babakhani, A.: Analysis of a system of fractional differential equations, J. math. Anal. appl. 293, 511-522 (2004) · Zbl 1058.34002 · doi:10.1016/j.jmaa.2004.01.013
[34]Daftardar-Gejji, V.; Babakhani, A.: Adomian decomposition: A tool for solving a system of fractional differential equations, J. math. Anal. appl. 301, 508-518 (2005) · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039
[35]Daftardar-Gejji, V.; Babakhani, A.: An iterative method for solving nonlinear functional equations, J. math. Anal. appl. 316, 753-763 (2006) · Zbl 1087.65055 · doi:10.1016/j.jmaa.2005.05.009
[36]Momani, S.: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos solitons fractals 28, 930-937 (2006) · Zbl 1099.35118 · doi:10.1016/j.chaos.2005.09.002
[37]Biler, P.; Funaki, T.; Woyczynski, W. A.: Fractal burger equations, J. differential equations 148, 9-46 (1998) · Zbl 0911.35100 · doi:10.1006/jdeq.1998.3458
[38]Biler, P.; Karch, G.; Woyczynski, W. A.: Critical nonlinearity exponent and self-similar asyptotics for Lévy conservation laws, Ann. henri Poincaré 5, 613-637 (2001) · Zbl 0991.35009 · doi:10.1016/S0294-1449(01)00080-4 · doi:numdam:AIHPC_2001__18_5_613_0
[39]Mann, J. A.; Woyczynski, W. A.: Growing fractal interfaces in the presence of self-similar hopping surface diffusion, Phys. A 291, 159-183 (2001) · Zbl 0972.82078 · doi:10.1016/S0378-4371(00)00467-2
[40]Stanescu, D.; Kim, D.; Woyczynski, W. A.: Numerical study of interacting particles approximation for integro-differential equations, J. comput. Phys. 206, 706-726 (2005) · Zbl 1070.65004 · doi:10.1016/j.jcp.2004.12.023
[41]He, J. H.: A new approach to nonlinear partial differential equations, Commun. nonlinear sci. Numer. simul. 2, No. 4, 230-235 (1997) · Zbl 0923.35046 · doi:10.1016/S1007-5704(97)90029-0
[42]He, J. H.: Commun. nonlinear sci. Numer. simul., Commun. nonlinear sci. Numer. simul. 2, No. 4, 235-236 (1997)
[43]He, J. H.: Approximate anlytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. Mech. engrg. 167, No. 1 – 2, 57-68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[44]He, J. H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. Mech. engrg. 167, No. 1 – 2, 69-73 (1998) · Zbl 0932.65143 · doi:10.1016/S0045-7825(98)00109-1
[45]He, J. H.: Variational iteration method a kind of non-linear analytical technique; some examples, Internat. J. Non-linear mech. 34, 699-708 (1999)
[46]Ganji, D. D.; Jannatabadi, M.; Mohseni, E.: Application of he’s variational iteration method to nonlinear Jaulent – Miodek equations and comparing it with ADM, J. comput. Appl. math. 207, 35-45 (2007) · Zbl 1120.65107 · doi:10.1016/j.cam.2006.07.029
[47]Tian, L.; Yin, J.: Shock – peakon and shock – compacton solutions for K(p,q) equation by variational iteration method, J. comput. Appl. math. 207, 46-52 (2007) · Zbl 1119.65099 · doi:10.1016/j.cam.2006.07.026
[48]He, J. H.; Wu, X. H.: Construction of solitary solution and compacton-like solution by variational iteration method, Chaos solitons fractals 29, 108-113 (2006) · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[49]Inc, M.: Numerical simulation of KdV and mkdv equations with initial conditions by the variational iteration method, Chaos solitons fractals 34, 1071-1081 (2007) · Zbl 1142.35572 · doi:10.1016/j.chaos.2006.04.069
[50]Inc, M.: Numerical doubly-periodic solution of the (2+1)-dimensional Boussinesq equation with initial conditions by the variational iteration method, Phys. lett. A 366, 20-24 (2007) · Zbl 1203.65210 · doi:10.1016/j.physleta.2007.01.066
[51]Inc, M.: Exact and numerical solitons with compact support for nonlinear dispersive K(m,p) equations by the variational iteration method, Phys. A 375, 447-456 (2007)
[52]Dehghan, M.; Tatari, M.: The use of he’s variational iterational method for solving a Fokker – Planck equation, Phys. scr. 74, 310-316 (2006) · Zbl 1108.82033 · doi:10.1088/0031-8949/74/3/003
[53]Draganescu, G.; Cofan, N.; Rujan, D.: Nonlinear vibrations of a nano sized sensor with fractional damping, J. optoelectron. Adv. mater. 7, 877-884 (2005)
[54]Draganescu, G.: Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives, J. math. Phys. 47 (2006) · Zbl 1112.74009 · doi:10.1063/1.2234273
[55]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear sci. Numer. simul. 7, 27-34 (2006)
[56]He, J. H.: Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern phys. B 20, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[57]Dehghan, M.; Shakeri, F.: Application of he’s variational iteration method for solving the Cauchy reaction – diffusion problem, J. comput. Appl. math. 214, 435-446 (2008) · Zbl 1135.65381 · doi:10.1016/j.cam.2007.03.006
[58]Dehghan, M.; Shakeri, F.: Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New astron. 13, 53-59 (2008)
[59]Shakeri, F.; Dehghan, M.: Numerical solution of a biological population model using he’s variational iteration method, Comput. math. Appl. 54, 1197-1209 (2007) · Zbl 1137.92033 · doi:10.1016/j.camwa.2006.12.076
[60]Tatari, M.; Dehghan, M.: On the convergence of he’s variational iteration method, J. comput. Appl. math. 207, 121-128 (2007) · Zbl 1120.65112 · doi:10.1016/j.cam.2006.07.017
[61]Abbasbandy, S.: A new application of he’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. comput. Appl. math. 207, 59-63 (2007) · Zbl 1120.65083 · doi:10.1016/j.cam.2006.07.012
[62]Abassy, T. A.; El-Tawil, M. A.; El Zoheiry, H.: Solving nonlinear partial differential equations using the modified variational iteration Padé technique, J. comput. Appl. math. 207, 73-91 (2007) · Zbl 1119.65095 · doi:10.1016/j.cam.2006.07.024
[63]Inokvti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in nonlinear mathematical physics, Variational method in the mechanics of solids (1978)
[64]He, J. H.: Generalized variational principles in fluids, (2003)
[65]J.H. He, Non-perturbative methods for strongly nonlinear problems, Disertation, de-Verlag in GmbH, Berlin, 2006
[66]Kaya, D.; Yokuş, A.: A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Math. comput. Simulation 60, 507-512 (2002) · Zbl 1007.65078 · doi:10.1016/S0378-4754(01)00438-4
[67]Kaya, D.: An application of the decomposition method for the KdVB equation, Appl. math. Comput. 152, 279-288 (2004) · Zbl 1053.65087 · doi:10.1016/S0096-3003(03)00566-6
[68]Momani, S.; Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. math. Comput. 162, 1351-1365 (2005) · Zbl 1063.65055 · doi:10.1016/j.amc.2004.03.014
[69]Momani, S.: Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl. math. Comput. 165, 459-472 (2005) · Zbl 1070.65105 · doi:10.1016/j.amc.2004.06.025
[70]Momani, S.: An explicit and numerical solutions of the fractional KdV equations, Math. comput. Simulation 70, 110-118 (2005) · Zbl 1119.65394 · doi:10.1016/j.matcom.2005.05.001
[71]Momani, S.: Analytic and approximate solutions of the space- and time- fractional telegraph equations, Appl. math. Comput. 170, 1126-1134 (2005) · Zbl 1103.65335 · doi:10.1016/j.amc.2005.01.009
[72]Momani, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006)