Numerical methods for nonlinear partial differential equations of fractional order

Applied Mathematical Modelling

Volumn 32, Issue 1, 2008, Pages 28-39

  Odibat, Zaid ^a   Momani, Shaher ^b  

^a AL BALQA APPLIED UNIVERSITY   (Jordan)

^b QATAR UNIVERSITY   (Qatar)

Author keywords

Adomian decomposition method; Caputo fractional derivative; Fractional differential equation; Lagrange multiplier; Variational iteration method

Indexed keywords

APPROXIMATION THEORY; ITERATIVE METHODS; LAGRANGE MULTIPLIERS; MATHEMATICAL MODELS; NONLINEAR EQUATIONS;

ADOMIAN DECOMPOSITION METHOD; CAPUTO FRACTIONAL DERIVATIVE; FRACTIONAL ORDER; NONLINEAR PARTIAL DIFFERENTIAL EQUATION; VARIATION ITERATION METHOD;

PARTIAL DIFFERENTIAL EQUATIONS;

APPROXIMATION THEORY; ITERATIVE METHODS; LAGRANGE MULTIPLIERS; MATHEMATICAL MODELS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

EID: 34548384362     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2006.10.025     Document Type: Article

References (39)

1. Oldham K.B., and Spanier J. The Fractional Calculus (1974), Academic Press, New York
2. Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. J. Roy. Astr. Soc. 13 (1967) 529-539
3. Schneider W., and Wyss W. Fractional diffusion equation and wave equations. J. Math. Phys. 30 (1989) 134-144
4. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons, Inc., New York
5. Beyer H., and Kempfle S. Defintion of physically consistent damping laws with fractional derivatives. Z. Angew Math. Mech. 75 (1995) 623-635
6. Mainardi F. Fractional relaxation-oscilation and fractional diffusion-wave phenomena. Chaos, Solitons Fractals 7 (1996) 1461-1477
7. Y. Luchko, R. Gorenflo, 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).
8. Podlubny I. Fractional differential equations (1999), Academic Press, San Diego, CA
9. Huang F., and Liu F. The time-fractional diffusion equation and fractional advection-dispersion equation. ANZIAM J. 46 (2005) 1-14
10. Huang F., and Liu F. The fundamental solution of the spcae-time fractional advection-dispersion equation. J. Appl. Math. Comput. 18 2 (2005) 339-350
11. Momani S. Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons Fractals 28 4 (2006) 930-937
12. Odibat Z., and Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 1 7 (2006) 15-27
13. Momani S., and Odibat Z. Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Appl. Math. Comput. 177 2 (2006) 488-494
14. Momani S., and Odibat Z. Numerical comparison of methods for solving linear differential equations of fractional order. Chaos, Solitons Fractals 31 5 (2007) 1248-1255
15. Odibat Z., and Momani S. Approximate solutions for boundary value problems of time-fractional wave equation. Appl. Math. Comput. 181 1 (2006) 767-774
16. Marinca V. An approximate solution for one-dimensional weakly nonlinear oscillations. Int. J. Nonlinear Sci. Numer. Simul. 3 2 (2002) 107-110
17. Hao T.H. Search for variational principles in electrodynamics by Lagrange method. Int. J. Nonlinear Sci. Numer. Simul. 6 2 (2005) 209-210
18. Momani S. An explicit and numerical solutions of the fractional KdV equation. Math. Comput. Simul. 70 2 (2005) 110-1118
19. Momani S., and Abuasad S. Application of He's variational iteration method to Helmholtz equation. Chaos, Solitons Fractals 27 5 (2006) 1119-1123
20. He J.H. Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 2 4 (1997) 235-236
21. He J.H. Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Int. J. Turbo Jet-Engines 14 1 (1997) 23-28
22. He J.H. Approximate solution of non linear differential equations with convolution product nonlinearities. Comput. Meth. Appl. Mech. Eng. 167 (1998) 69-73
23. He J.H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Meth. Appl. Mech. Eng. 167 (1998) 57-68
24. He J.H. Variational iteration method - a kind of non-linear analytical technique: some examples. Int. J. Nonlinear Mech. 34 (1999) 699-708
25. He J.H. Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114 (2000) 115-123
26. He J.H. Variational theory for linear magneto-electro-elasticity. Int. J. Nonlinear Sci. Numer. Simul. 2 4 (2001) 309-316
27. He J.H. Variational principle for Nano thin film lubrication. Int. J. Nonlinear Sci. Numer. Simul. 4 3 (2003) 313-314
28. He J.H. Variational principle for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons Fractals 19 4 (2004) 847-851
29. Adomian G. A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135 (1988) 501-544
30. Adomian G. Solving Frontier Problems of Physics: The Decomposition method (1994), Kluwer Academic Publishers, Boston
31. Wazwaz A. A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. Math. Comput. 111 (2000) 53-69
32. Wazwaz A., and El-Sayed S. A new modification of the Adomian decomposition method for linear and nonlinear operators. Appl. Math. Comput. 122 (2001) 393-405
33. Rèpaci A. Nonlinear dynamical systems: On the accuracy of Adomian's decomposition method. Appl. Math. Lett. 3 3 (1990) 35-39
34. Liu H.M. Variational approach to nonlinear electrochemical system. Int. J. Nonlinear Sci. Numer. Simul. 5 1 (2004) 95-96
35. Liu H.M. Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method. Chaos, Solitons Fractals 23 2 (2005) 573-576
36. Inokuti M., Sekine H., and Mura T. General use of the Lagrange multiplier in non-linear mathematical physics. In: Nemat-Nasser S. (Ed). Variational Method in the Mechanics of Solids (1978), Pergamon Press, Oxford 156-162
37. Cherruault Y. Convergence of Adomian's method. Kybernetes 18 (1989) 31-38
38. Cherruault Y., and Adomian G. Decomposition methods: A new proof of convergence. Math. Comput. Modell. 18 (1993) 103-106
39. Abbaoui K., and Cherruault Y. New ideas for proving convergence of decomposition methods. Comput. Math. Appl. 29 7 (1996) 103-108