Application of the Adomian decomposition method for the Fokker–Planck equation

Tatari, Mehdi; Dehghan, Mehdi; Razzaghi, Mohsen

doi:10.1016/j.mcm.2006.07.010

Cited by 104 publications

(75 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…u(x, 0) = x n , n ∈ R. If α = β = 1, γ = 2 and n = 1, whence the solution is u(x, t) = x − t, which is the exact solution of the problem, see [26]) (see Figure 2a). Also the solution of example2 when α = 0.5, β = 1 and α = 0.75, β = .25 are shown in Figures 2b, 2c. …”

Section: Test Examplesmentioning

confidence: 99%

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Hikal¹,

Ibrahim²

2015

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

In this paper, the Adomian's decomposition method (ADM) is considered to solve a fractional advection-dispersion model. This model can be represented if the first order derivative in time is replaced by the Caputo fractional derivative of order α (0 < α ≤ 1). In addition, the space derivative orders are replaced by the alternative orders 0 < β ≤ 1 and 1 < γ ≤ 2. The obtained solutions are formulated in a convergent infinite series in terms of Mittage-Leffler functions. Finally, two illustrative examples are introduced to ensure the effectiveness of the used method.

show abstract

Section: Test Examplesmentioning

confidence: 99%

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Hikal¹,

Ibrahim²

2015

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

show abstract

“…In Fig. 1a) we show the solution approximation for call option problem (2) using formula (27) with k = 10, applying the ADM on homogeneous dirichlet boundary diffusion problem (24) obtained from problem (7), and so, in Fig. 1b), we presented the solution approximation profiles obtained from problem (2) applying the ADM only using the initial condition (see section 4.1 and [12]).…”

Section: Simulationsmentioning

confidence: 99%

“…The tools used to study these types of problems are methods and ideas specialized in stochastic calculus and partial differential equations: Wilmott et al [3], Courtadon [4] and Company et al [25] used finite differences methods to approximate the solution of the option valuation equations; Geske & Johnson, MacMillan, Barone-Adesi & Whaley, Barone-Adesi & Elliot, Barone-Adesi and Whaley, and Barone-Adesi developed methods of analytic approximation [5,6,7,8,9]; Gülkaç , used a series expansion method called homotopy perturbation method to find an approximate solution for the Black-Scholes equation [10], Alawneh & Al-Khaled [28] applied the Variational Iteration Method (VIM) to solve the Fockker-Planck equation and Black-Scholes equations. Cheng et al applied the homotopy analysis method [11], Bohner & Zheng [12], El-Wakil et al [26] and Tatari et al [27] used the Adomian decomposition method but they did not use boundary conditions to find the approximate solution of the Black-Scholes or Fockker-Planck equations.…”

Section: Introductionmentioning

confidence: 99%

Solution of the Black-Scholes equation via the Adomian decomposition method

Avila¹,

Estrella²,

Blanco

2013

International Journal of Applied Mathematical Research

View full text Add to dashboard Cite

The Adomian Decomposition Method (ADM) is applied to obtain a fast and reliable solution to the BlackScholes equation with boundary condition for a European option. We cast the problem of pricing a European option with boundary conditions in terms of a diffusion partial differential equation with homogeneous boundary condition in order to apply the ADM. The analytical solution of the equations is calculated in the form of an explicit series approximation.

show abstract

“…Tatari et al [10] used Adomian decomposition method while Hesama et al [11] used differential transform method to find the analytic solutions of FPE.…”

Section: Introductionmentioning

confidence: 99%

A Novel Approach for Numerical Computation of Fokker-Planck Equation

Kumar¹,

Singh²,

Nath³

2016

JMSS

View full text Add to dashboard Cite

This paper present an implementation of "modified cubic B-spline differential quadrature method (MCB-DQM)" proposed by Arora & Singh (Applied Mathematics and Computation Vol. 224(1) (2013) 161-177) for numerical computation of Fokker-Planck equations. The modified cubic B-splines are used as set of basis functions in the differential quadrature to compute the weighting coefficients for the spatial derivatives, which reduces Fokker-Planck equation into system of first-order ordinary differential equations (ODEs), in time. The well known SSP-RK43 scheme is then applied to solve the resulting system of ODEs. The efficiency of proposed method has been confirmed by three examples having their exact solutions. This shows that MCB-DQM results are capable of achieving high accuracy. Advantage of the scheme is that it can be applied very smoothly to solve the linear or nonlinear physical problems, and a very less storage space is required which causes less accumulation of numerical errors.

show abstract

Application of the Adomian decomposition method for the Fokker–Planck equation

Cited by 104 publications

References 47 publications

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Solution of the Black-Scholes equation via the Adomian decomposition method

A Novel Approach for Numerical Computation of Fokker-Planck Equation

Contact Info

Product

Resources

About