Adomian decomposition method for solving the diffusion-convection-reaction equations

Abstract Adomian decomposition method is used to solve the explicit and numerical solutions of three types of the diffusion–convection–reaction (DECR) equations. The calculations are carried out for three different types of the DCRE such as, the Black–Scholes equation used in financial market option pricing, Fokker–Planck (FP) equation for lazer filed and Fokker–Planck equation from plasma physics. The behaviour of the approximate solutions of the distribution functions is shown graphically and compared with that obtained by other theories such as the variational iteration method.

[1]  J. Rogers.  Chaos , 1876 .

[2]  G. Adomian.  Stochastic Burgers' equation , 1995 .

[3]  G. Wei,et al.  A unified approach for the solution of the Fokker-Planck equation , 2000, physics/0004074.

[4]  Stanislav Spichak,et al.  Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusion , 1999 .

[5]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[6]  N. G. van Kampen.  Short first-passage times , 1993 .

[7]  Michael A. Lieberman,et al.  Theory of electron cyclotron resonance heating. II. Long time and stochastic effects , 1973 .

[8]  Daniel Lesnic,et al.  Blow-up solutions obtained using the decomposition method , 2006 .

[9]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[10]  Peter G. Kazansky,et al.  Vecksler-Macmillan phase stability for neutral atoms accelerated by a laser beam , 2003 .

[11]  Abdul-Majid Wazwaz,et al.  Exact solutions for heat-like and wave-like equations with variable coefficients , 2004, Appl. Math. Comput..

[12]  Abdul-Majid Wazwaz.  Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions , 2001, Appl. Math. Comput..

[13]  Andrew G. Glen,et al.  APPL , 2001 .

[14]  M. A. Abdou.  Adomian decomposition method for solving the telegraph equation in charged particle transport , 2005 .

[15]  Ji-Huan He,et al.  Variational iteration method for autonomous ordinary differential systems , 2000, Appl. Math. Comput..

[16]  M. A. Abdou,et al.  Maximum-entropy approach with higher moments for solving Fokker–Planck equation , 2002 .

[17]  Shaher Momani,et al.  Non-perturbative analytical solutions of the space- and time-fractional Burgers equations , 2006 .

[18]  A. A. Soliman,et al.  New applications of variational iteration method , 2005 .

[19]  Abdul-Majid Wazwaz,et al.  New solitary-wave special solutions with compact support for the nonlinear dispersive K(m, n) equations , 2002 .

[20]  B. M. Fulk.  MATH , 1992 .

[21]  M. A. Abdou,et al.  On the Diffusion-Convection-Reaction Equations , 1999 .

[22]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .

[23]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[24]  Abdul-Majid Wazwaz,et al.  General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations mK(n, n) in higher dimensional spaces , 2002, Math. Comput. Simul..

[25]  G. Adomian,et al.  Non-perturbative analytical solution of the general Lotka-Volterra three-species system , 1996 .

[26]  George Adomian,et al.  The Decomposition Method , 1989 .