scholarly journals Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Mohammadhossein Derakhshan
Keyword(s):  
Mathematical Model ◽  
Fractional Derivative ◽  
Functional Equations ◽  
Homotopy Perturbation Method ◽  
Method Comparison ◽  
Combined Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
The Mathematical Model ◽  
Generalized Fractional Derivative

In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method  (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order μ ∈ 0,1 with λ ∈ ℤ + , θ , σ ∈ ℝ + is presented as follows: D t μ C u x , t + θ u λ x , t u x x , t − σ u x x t x , t = 0 , where for λ = 1 , θ = 1 , σ = 1 s and λ = 2 , θ = 3 , σ = 1 , equations are changed into the equal width and modified equal width equations, respectively. The analytical method which we have used for solving this equation is based on a combination of the homotopy perturbation method and Sadik transform. The convergence and error analysis are discussed in this article. Plots of the analytical results with three examples are presented to show the applicability of this numerical method. Comparison between the obtained absolute errors by the suggested method and other methods is demonstrated.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

A New Fractional Model of Nonlinear Shock Wave Equation Arising in Flow of Gases

2014 ◽  
Vol 3 (1) ◽  
pp. 43-50 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Sunil Kumar
Keyword(s):  
Shock Wave ◽  
Wave Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Homotopy Perturbation Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Nonlinear Shock ◽  
Good Agreement

Abstract In this paper, we present a numerical algorithm based on new homotopy perturbation transform method (HPTM) to solve a time-fractional nonlinear shock wave equation which describes the flow of gases. The fractional derivative is considered in the Caputo sense. The HPTM is combined form of Laplace transform, homotopy perturbation method and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient, easy to implement and computationally very attractive.

Dynamic analysis of the mathematical model of COVID-19 with demographic effects

2020 ◽  
Vol 75 (11-12) ◽  
pp. 389-396 ◽  
Author(s):  
Naeem Faraz ◽  
Yasir Khan ◽  
E. F. Doungmo Goufo ◽  
Amna Anjum ◽  
Ali Anjum
Keyword(s):  
Mathematical Model ◽  
Perturbation Method ◽  
Effective Treatment ◽  
Homotopy Perturbation Method ◽  
Simple Mathematical Model ◽  
Safety Measures ◽  
Seir Model ◽  
Homotopy Perturbation ◽  
The Mathematical Model ◽  
Demographic Effects

AbstractThe coronavirus is currently extremely contagious for humankind, which is a zoonotic tropical disease. The pandemic is the largest in history, affecting almost the whole world. What makes the condition the worst of all is no specific effective treatment available. In this article, we present an extended and modified form of SIR and SEIR model, respectively. We begin by investigating a simple mathematical model that describes the pandemic. Then we apply different safety measures to control the pandemic situation. The mathematical model with and without control is solved by using homotopy perturbation method. Obtained solutions have been presented graphically. Finally, we develop another mathematical model, including quarantine and hospitalization.

scholarly journals Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1154
Author(s):  
Temirkhan S. Aleroev ◽  
Asmaa M. Elsayed
Keyword(s):  
Fractional Derivative ◽  
Homotopy Perturbation Method ◽  
Analytic Solution ◽  
Separation Of Variables ◽  
Fourier Method ◽  
String Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
The Laplace Transform

This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a numerical solution that is based on the Laplace transform method with the homotopy perturbation method. The method of the separation of variables (the Fourier method) is constructed for the analytic solution. The derived solutions are represented by Mittag–LefLeffler type functions. Orthogonality and convergence of the solution are discussed. Finally, we present an example to illustrate the methods.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

scholarly journals An efficient approach for solution of fractional-order Helmholtz equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nehad Ali Shah ◽  
Essam R. El-Zahar ◽  
Mona D. Aljoufi ◽  
Jae Dong Chung
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Hybrid Technique ◽  
Science And Engineering ◽  
Helmholtz Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Suggested Technique ◽  
Present Technique

AbstractIn this article, a hybrid technique called the homotopy perturbation Elzaki transform method has been implemented to solve fractional-order Helmholtz equations. In the hybrid technique, the Elzaki transform method and the homotopy perturbation method are amalgamated. Three problems are solved to validate and demonstrate the efficacy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the results by other techniques. It is shown that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

A FRACTAL VARIATIONAL PRINCIPLE FOR THE TELEGRAPH EQUATION WITH FRACTAL DERIVATIVES

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050058 ◽  
Author(s):  
KANG-LE WANG ◽  
SHAO-WEN Yao ◽  
YAN-PING LIU ◽  
LI-NA ZHANG
Keyword(s):  
Variational Principle ◽  
Perturbation Method ◽  
Inverse Method ◽  
Homotopy Perturbation Method ◽  
Telegraph Equation ◽  
Transform Method ◽  
Homotopy Perturbation

A fractal modification of the telegraph equation with fractal derivatives is given, and its variational principle is established by the semi-inverse method. The two-scale transform method and He’s homotopy perturbation method are successfully adopted to solve the fractal equation.

scholarly journals Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Nonlinear Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Radial Symmetry ◽  
Space Time ◽  
Homotopy Perturbation ◽  
Initial Stage

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

scholarly journals Analytical Solution of Space-Time Fractional Fokker-Planck Equation by Homotopy Perturbation Sumudu Transform Method

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ravi Shanker Dubey ◽  
Badr Saad T. Alkahtani ◽  
Abdon Atangana
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Space Time ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Sumudu Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

An efficient approach based on homotopy perturbation method by using Sumudu transform is proposed to solve some linear and nonlinear space-time fractional Fokker-Planck equations (FPEs) in closed form. The space and time fractional derivatives are considered in Caputo sense. The homotopy perturbation Sumudu transform method (HPSTM) is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. Some examples show that the HPSTM is an effective tool for solving many space time fractional partial differential equations.

Sign in / Sign up

Export Citation Format

Share Document