Approximate Analytical Solutions of Fractional Perturbed Diffusion Equation by Reduced Differential Transform Method and the Homotopy Perturbation Method

The approximate analytical solutions of differential equations with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization, or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The two different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance.

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Approximate analytical solutions of fractional Benney–Lin equation by reduced differential transform method and the homotopy perturbation method

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.03.057 ◽

2011 ◽

Vol 61 (9) ◽

pp. 2829-2842 ◽

Cited By ~ 38

Author(s):

Praveen Kumar Gupta

Keyword(s):

Perturbation Method ◽

Analytical Solutions ◽

Homotopy Perturbation Method ◽

Differential Transform Method ◽

Transform Method ◽

Homotopy Perturbation ◽

Approximate Analytical Solutions ◽

Differential Transform ◽

Reduced Differential Transform Method

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Application of He’s Homotopy Perturbation Method to Fractional Diffusion Equations

Zeitschrift für Naturforschung A ◽

10.1515/zna-2010-1-204 ◽

2010 ◽

Vol 65 (1-2) ◽

pp. 53-58 ◽

Cited By ~ 1

Author(s):

Subir Das ◽

Praveen Kumar Gupta ◽

Vinod Sankar Pandey ◽

Kabindra Nath Rai

Keyword(s):

Perturbation Method ◽

Analytical Solutions ◽

Homotopy Perturbation Method ◽

Initial Conditions ◽

Time Derivative ◽

Fractional Diffusion Equations ◽

Homotopy Perturbation ◽

Approximate Analytical Solutions ◽

And Performance ◽

Fractional Time Derivative

AbstractIn this paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of a linear external force are obtained with the help of the homotopy perturbation method (HPM). The explicit solutions of the problem for the initial condition as a function of x have been obtained. It reveals that a few iterations are needed to obtain accurate approximate analytical solutions. The numerical calculations are carried out when the initial conditions are like exponential and periodic functions and the results are depicted through graphs. The examples prove that the method is extremely effective due to its simplistic approach and performance.

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Approximate Analytical Solutions of Nonlinear Korteweg-de Vries Equations Using Multistep Modified Reduced Differential Transform Method

Mathematics and Statistics ◽

10.13189/ms.2020.081302 ◽

2020 ◽

Vol 8 (2A) ◽

pp. 9-16

Author(s):

Che Haziqah Che Hussin ◽

Ahmad Izani Md Ismail ◽

Adem Kilicman ◽

Amirah Azmi

Keyword(s):

Analytical Solutions ◽

Differential Transform Method ◽

Transform Method ◽

Approximate Analytical Solutions ◽

Differential Transform ◽

De Vries ◽

Reduced Differential Transform Method

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An Approximate Analytical Solution of the Fractional Diffusion Equation with Absorbent Term and External Force by Homotopy Perturbation Method

Zeitschrift für Naturforschung A ◽

10.1515/zna-2010-0305 ◽

2010 ◽

Vol 65 (3) ◽

pp. 182-190 ◽

Cited By ~ 16

Author(s):

Subir Das ◽

Praveen Kumar Gupta

Keyword(s):

Diffusion Equation ◽

Perturbation Method ◽

External Force ◽

Analytical Solutions ◽

Homotopy Perturbation Method ◽

Time Derivative ◽

Approximate Solutions ◽

Mathematical Tool ◽

Homotopy Perturbation ◽

Approximate Analytical Solutions

In the present paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force are obtained with the help of the powerful homotopy perturbation method (HPM). By using initial values, the approximate analytical solutions of the equation are derived. The results are deduced for different particular cases. The numerical results show that only a few iterations are needed to obtain accurate approximate solutions and these are presented graphically. The presented method is extremely simple, concise, and highly efficient as a mathematical tool in comparison with the other existing techniques.

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