The analytical analysis of fractional order Fokker-Planck equations

Khan, Hassan; Farooq, Umar; Tchier, Fairouz; Khan, Qasim; Singh, Gurpreet; Sitthithakerngkiet, Kanokwan

doi:10.3934/math.2022665

Cited by 4 publications

(2 citation statements)

References 33 publications

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“…This paper focuses on the nonlinear space-time FFPE, which is given by 2017) used an inventive reconciliation of q-homotopy analysis and Laplace transform to procure the numerical solution. Odibat and Momani (2007) received the numerical solution of FFPE by variational iteration method and ADM. Hassan et al (2022) achieved the approximate solution of FFPE using decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. Saravanan and Magesh (2014) derived numerical solution by both fractional variational iteration method and fractional reduced differential transform method.…”

Section: Introductionmentioning

confidence: 99%

“…Prakash and Kaur (2017) used an inventive reconciliation of q-homotopy analysis and Laplace transform to procure the numerical solution. Odibat and Momani (2007) received the numerical solution of FFPE by variational iteration method and ADM. Hassan et al. (2022) achieved the approximate solution of FFPE using decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator.…”

Section: Introductionmentioning

confidence: 99%

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Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Zhang,

Wang

2024

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PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Zhang,

Wang

2024

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Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

Khan,

Suen,

Khan

2024

Partial Differential Equations in Applied Mathematics

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A Novel Efficient Approach for Solving Nonlinear Caputo Fractional Differential Equations

Liaqat,

Khan,

Anjum

et al. 2024

Advances in Mathematical Physics

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Several scientific areas utilize fractional nonlinear partial differential equations (PDEs) to model various phenomena, yet most of these equations lack exact solutions (Ex‐Ss). Consequently, techniques for obtaining approximate solutions (App‐S), which sometimes yield Ex‐Ss, are essential for solving these equations. This study employs a novel technique by combining the residual function and modified fractional power series (FPS) with the Aboodh transform (A‐T) to solve various nonlinear problems within the framework of the Caputo derivative. Studies on absolute error (Abs‐E), relative error (Rel‐E), residual error (Res‐E), and recurrence error (Rec‐E) validate the accuracy and effectiveness of our approach. We apply the limit principle at infinity to determine the coefficients of the series solution terms. In contrast, other methods, such as variational iteration, homotopy perturbation, and Elzaki Adomian decomposition, rely on integration, while the residual power series method (RPSM) employs differentiation, both of which face challenges in fractional scenarios. Moreover, the efficiency of our approach in solving nonlinear problems without depending on Adomian and He polynomials makes it more effective than various approximate series solution techniques. Our method yields results that are very similar to those obtained from the differential transform, the homotopy perturbation, the analytical computational, and Adomian decomposition methods (ADMs). This demonstrates that our technique is a suitable alternative tool for solving nonlinear models.

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The analytical analysis of fractional order Fokker-Planck equations

Cited by 4 publications

References 33 publications

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

A Novel Efficient Approach for Solving Nonlinear Caputo Fractional Differential Equations

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