zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. (English) Zbl 1189.65254
Summary: Variational iteration method has been used to handle linear and nonlinear differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this work, a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the variational iteration method with those obtained by Adomian decomposition method reveals that the first method is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
MSC:
65M99Numerical methods for IVP of PDE
26A33Fractional derivatives and integrals (real functions)
76A02Foundations of fluid mechanics
References:
[1]J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288–291
[2]He, J. H.: Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. Technol. 15, No. 2, 86-90 (1999)
[3]He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. Mech. engrg. 167, 57-68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[4]Podlubny, I.: Fractional differential equations, (1999)
[5]Al-Khaled, K.; Momani, S.: An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl. math. Comput. 165, 473-483 (2005) · Zbl 1071.65135 · doi:10.1016/j.amc.2004.06.026
[6]Mainardi, F.; Luchko, Y.; Pagnini, G.: The fundamental solution of the space–time fractional diffusion equation, Frac. calc. Appl. anal. 4, 153-192 (2001) · Zbl 1054.35156
[7]Hanyga, A.: Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. lond. A 458, 933-957 (2002) · Zbl 1153.35347 · doi:10.1098/rspa.2001.0904
[8]Huang, F.; Liu, F.: The time fractional diffusion and fractional advection-dispersion equation, Anziam 46, 1-14 (2005) · Zbl 1072.35218 · doi:10.1017/S1446181100008282
[9]Huang, F.; Liu, F.: The fundamental solution of the space–time fractional advection-dispersion equation, J. appl. Math. comput. 18, 21-36 (2005) · Zbl 1086.35003 · doi:10.1007/BF02936577
[10]Momani, S.: Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. math. Comput. 170, No. 2, 1126-1134 (2005) · Zbl 1103.65335 · doi:10.1016/j.amc.2005.01.009
[11]Momani, S.: An explicit and numerical solutions of the fractional KdV equation, Math. comput. Simul. 70, 110-118 (2005) · Zbl 1119.65394 · doi:10.1016/j.matcom.2005.05.001
[12]Debnath, L.; Bhatta, D.: Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics, Frac. calc. Appl. anal. 7, 21-36 (2004) · Zbl 1076.35096
[13]Adomian, G.: A review of the decomposition method in applied mathematics, J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[14]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[15]Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. Comput. 111, 53-69 (2000) · Zbl 1023.65108 · doi:10.1016/S0096-3003(99)00047-8
[16]Wazwaz, A. M.; El-Sayed, S.: A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. math. Comput. 122, 393-405 (2001) · Zbl 1027.35008 · doi:10.1016/S0096-3003(00)00060-6
[17]Öziş, T.; Yildirim, A.: Comparison between adomians method and he’s homotopy perturbation method, Comput. math. Appl. 56, No. 5, 1216-1224 (2008) · Zbl 1155.65344 · doi:10.1016/j.camwa.2008.02.023
[18]Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation, Chaos solitons fractals 27, No. 5, 1119-1123 (2006) · Zbl 1086.65113 · doi:10.1016/j.chaos.2005.04.113
[19]He, J. H.: Variational iteration method for delay differential equations, Commun. nonlinear sci. Numer. simul. 2, No. 4, 235-236 (1997) · Zbl 0924.34063
[20]He, J. H.: Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo jet-engines 14, No. 1, 23-28 (1997)
[21]He, J. H.: Approximate solution of non linear differential equations with convolution product nonlinearities, Comput. methods appl. Mech. engrg. 167, 69-73 (1998) · Zbl 0932.65143 · doi:10.1016/S0045-7825(98)00109-1
[22]He, J. H.: Variational iteration method- a kind of non-linear analytical technique: some examples, Int. J. Nonlinear mech. 34, 699-708 (1999)
[23]He, J. H.: Variational iteration method for autonomous ordinary differential systems, Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[24]He, J. H.: Variational theory for linear magneto-electro-elasticity, Int. J. Nonlinear sci. Numer. simul. 2, No. 4, 309-316 (2001) · Zbl 1083.74526 · doi:10.1515/IJNSNS.2001.2.4.309
[25]He, J. H.: Variational principle for nano thin film lubrication, Int. J. Nonlinear sci. Numer. simul. 4, No. 3, 313-314 (2003)
[26]He, J. H.: Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, solitons and fractals 19, No. 4, 847-851 (2004) · Zbl 1135.35303 · doi:10.1016/S0960-0779(03)00265-0
[27]He, J. H.: Variational iteration method–some recent results and new interpretations, J. comput. Appl. math. 207, No. 1, 3-17 (2007) · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009
[28]He, J. H.; Wu, X. H.: Variational iteration method: new development and applications, Comput. math. Appl. 54, No. 7–8, 881-894 (2007) · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[29]Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics, Variational method in the mechanics of solids, 156-162 (1978)
[30]Wazwaz, A. M.: The variational iteration method for solving linear and nonlinear systems of pdes, Comput. math. Appl. 54, No. 7–8, 895-902 (2007) · Zbl 1145.35312 · doi:10.1016/j.camwa.2006.12.059
[31]Wazwaz, A. M.: The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. math. Appl. 54, No. 7–8, 926-932 (2007) · Zbl 1141.65388 · doi:10.1016/j.camwa.2006.12.038
[32]Wazwaz, A. M.: The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. math. Appl. 54, No. 7–8, 933-939 (2007) · Zbl 1141.65077 · doi:10.1016/j.camwa.2006.12.039
[33]Odibat, Z.: Reliable approaches of variational iteration method for nonlinear operators, Math. comput. Model. 48, No. 1–2, 222-231 (2008) · Zbl 1145.65314 · doi:10.1016/j.mcm.2007.09.005
[34]Yusufoglu, E.: Variational iteration method for construction of some compact and noncompact structures of Klein–Gordon equations, Int. J. Nonlinear sci. Numer. simul. 8, No. 2, 153-185 (2007)
[35]Biazar, J.; Ghazvini, H.: He’s variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear sci. Numer. simul. 8, No. 3, 311-314 (2007) · Zbl 1209.65152 · doi:10.1016/j.physleta.2007.02.049
[36]Ozer, H.: Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, Int. J. Nonlinear sci. Numer. simul. 8, No. 4, 513-518 (2007)
[37]Mokhtari, R.: Variational iteration method for solving nonlinear differential-difference equations, Int. J. Nonlinear sci. Numer. simul. 9, No. 1, 19-24 (2008)
[38]Yildirim, A.; Öziş, T.: Solutions of singular ivps of laneemden type by the variational iteration method, Nonlinear anal. TMA 70, No. 6, 2480-2484 (2009) · Zbl 1162.34005 · doi:10.1016/j.na.2008.03.012
[39]Momani, S.: Zaid odibat analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl. math. Comput. 177, No. 2, 484-492 (2006) · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025
[40]Odibat, Z.; Momani, S.: Approximate solutions for boundary value problems of time-fractional wave equation, Appl. math. Comput. 181, No. 1, 767-774 (2006) · Zbl 1148.65100 · doi:10.1016/j.amc.2006.02.004
[41]Momani, S.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, No. 1, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[42]Momani, S.; Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fractals 31, No. 5, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[43]Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order, Appl. math. Model. 32, No. 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[44]Momani, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006)
[45]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear sci. Numer. simul. 7, No. 1, 15-27 (2006)
[46]Momani, S.; Odibat, Z.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. math. Appl. 54, No. 7–8, 910-919 (2007) · Zbl 1141.65398 · doi:10.1016/j.camwa.2006.12.037
[47]Momani, S.; Odibat, Z.; Alawneh, A.: Variational iteration method for solving the space- and time-fractional KdV equation, Numer. methods partial differential equations 24, No. 1, 262-271 (2008) · Zbl 1130.65132 · doi:10.1002/num.20247
[48]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II, J. roy. Astr. soc. 13, 529-539 (1967)
[49]A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998
[50]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[51]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[52]Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation, Frac. calc. Appl. anal. 5, 367-386 (2002) · Zbl 1042.26003
[53]Cherruault, Y.: Convergence of Adomian’s method, Kybernetes 18, 31-38 (1989) · Zbl 0697.65051 · doi:10.1108/eb005812
[54]Cherruault, Y.; Adomian, G.: Decomposition methods: A new proof of convergence, Math. comput. Model. 18, 103-106 (1993) · Zbl 0805.65057 · doi:10.1016/0895-7177(93)90233-O