Fractional modelling arising in unidirectional propagation of long waves in dispersive media

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
Sunil Kumar ◽  
Devendra Kumar ◽  
Jagdev Singh
Keyword(s):  
Numerical Solutions ◽  
Simple Algorithm ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Dispersive Media ◽  
Transform Method ◽  
Adomian Polynomials ◽  
Modified Method ◽  
Homotopy Analysis ◽  
The Laplace Transform

AbstractThe purpose of this paper is to propose a modified and simple algorithm for fractional modelling arising in unidirectional propagation of long wave in dispersive media by using the fractional homotopy analysis transform method (FHATM). This modified method is an innovative adjustment in the Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. This modified method yields an analytical and approximate solution in terms of a rapidly convergent series with easily computable terms. The numerical solutions obtained by the proposed algorithm indicate that the approach is easy to implement and computationally very attractive. Comparing our solution with the existing ones, we note an excellent agreement.

scholarly journals Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Laplace Transform ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Space Time ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Promising Tool ◽  
The Laplace Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

scholarly journals A new analytical modelling for fractional telegraph equation via Elzaki transform

2015 ◽  
Vol 11 (9) ◽  
pp. 5617-5625
Author(s):  
Huan Li
Keyword(s):  
Numerical Solutions ◽  
Simple Algorithm ◽  
Telegraph Equation ◽  
Analytical Modelling ◽  
Transform Method ◽  
Fractional Telegraph Equation ◽  
Homotopy Analysis ◽  
Accuracy And Stability

The main aim of this paper is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (FHATM). The fractional homotopy analysis transform method is an innovative adjustment in Elzaki transform algorithm (ETA) and makes the calculation much sim-pler. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical ex-amples are given to illustrate the accuracy and stability of this method.

A reliable algorithm for KdV equations arising in warm plasma

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Amit Goswami ◽  
Jagdev Singh ◽  
Devendra Kumar
Keyword(s):  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Convergent Series ◽  
Science And Engineering ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Kdv Equations ◽  
The Laplace Transform ◽  
Warm Plasma

AbstractThe aim of the present work is to propose a simple and reliable algorithm namely homotopy perturbation transform method (HPTM) for KdV equations in warm plasma. The homotopy perturbation transform method is a combined form of the Laplace transform method with the homotopy perturbation method. In this method, the solution is calculated in the form of a convergent series with an easily computable compact. To illustrate the simplicity and reliability of the method, several examples are provided. In this paper, the homotopy perturbation transform method has been applied to obtain the solution of the KdV, mKdV, K(2, 2) and K(3,3) equations. We compared our solutions with the exact solutions. Results illustrate the applicability, efficiency and accuracy of HPTM to solve nonlinear equations despite needlessness to any linearization or perturbation. It is predicted that the proposed algorithm can be widely applied to other nonlinear problems in science and engineering.

scholarly journals Aboodh Transform Iterative Method for Spatial Diffusion of a Biological Population with Fractional-Order

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 155
Author(s):  
Gbenga O. Ojo ◽  
Nazim I. Mahmudov
Keyword(s):  
Iterative Method ◽  
Fractional Order ◽  
Population Model ◽  
Numerical Solutions ◽  
Computational Effort ◽  
Spatial Diffusion ◽  
Transform Method ◽  
Biological Population ◽  
The Laplace Transform ◽  
New Iterative Method

In this paper, a new approximate analytical method is proposed for solving the fractional biological population model, the fractional derivative is described in the Caputo sense. This method is based upon the Aboodh transform method and the new iterative method, the Aboodh transform is a modification of the Laplace transform. Illustrative cases are considered and the comparison between exact solutions and numerical solutions are considered for different values of alpha. Furthermore, the surface plots are provided in order to understand the effect of the fractional order. The advantage of this method is that it is efficient, precise, and easy to implement with less computational effort.

scholarly journals Homotopy Analysis Method for Solving Foam Drainage Equation with Space- and Time-Fractional Derivatives

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Hadi Hosseini Fadravi ◽  
Hassan Saberi Nik ◽  
Reza Buzhabadi
Keyword(s):  
Analytical Solution ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Foam Drainage Equation ◽  
Foam Drainage

The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained.

scholarly journals A study on the mixed convection boundary layer flow and heat transfer over a vertical slender cylinder

2014 ◽  
Vol 18 (4) ◽  
pp. 1247-1258 ◽  
Author(s):  
Rahmat Ellahi ◽  
Arshad Riaz ◽  
Saeid Abbasbandy ◽  
Tasawar Hayat ◽  
Kambiz Vafai
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Nonlinear Problems ◽  
Convection Boundary Layer ◽  
Series Solutions ◽  
Convection Boundary ◽  
Homotopy Analysis ◽  
Layer Flow

In this investigation, the series solutions of mixed convection boundary layer flow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method (HAM) and Pade technique which have been proven to be successful in tackling nonlinear problems. We offer various verifications of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding flow) but also for small values (opposing flow) of ? and the dual solutions do not obtain in both cases.

scholarly journals Homotopy Perturbation Algorithm Using Laplace Transform for Gas Dynamics Equation

2012 ◽  
Vol 8 (1) ◽  
pp. 55-61 ◽  
Author(s):  
Jagdev Singh ◽  
Devendra Kumar ◽  
Keyword(s):  
Laplace Transform ◽  
Gas Dynamics ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
The Laplace Transform ◽  
Gas Dynamics Equation ◽  
Adomian’S Polynomials

Homotopy Perturbation Algorithm Using Laplace Transform for Gas Dynamics EquationIn this paper, we apply a combined form of the Laplace transform method with the homotopy perturbation method to obtain the solution of nonlinear gas dynamics equation. This method is called the homotopy perturbation transform method (HPTM). This technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that this scheme solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method. The results reveal that the homotopy perturbation transform method (HPTM) is very efficient, simple and can be applied to other nonlinear problems.

scholarly journals New Modified Variational Iteration Laplace Transform Method Compares Laplace Adomian Decomposition Method for Solution Time-Partial Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mohamed Z. Mohamed ◽  
Tarig M. Elzaki ◽  
Mohamed S. Algolam ◽  
Eltaib M. Abd Elmohmoud ◽  
Amjad E. Hamza
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Transform Method ◽  
Variational Iteration ◽  
Modified Method ◽  
Adomian Decomposition ◽  
Fractional Differential

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.

scholarly journals A Reliable Iterative Transform Method for Solving an Epidemic Model

2021 ◽  
pp. 4839-4846
Author(s):  
Reem Waleed Huisen ◽  
Sinan H. Abd Almjeed ◽  
Areej Salah Mohammed
Keyword(s):  
Epidemic Model ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Absolute Error ◽  
Maximum Error ◽  
Disease Spread ◽  
Approximate Analytic Solution ◽  
Transform Method ◽  
Epidemic Period ◽  
The Laplace Transform

    The main purpose of the work is to apply a new method, so-called LTAM, which couples the Tamimi and Ansari iterative method (TAM) with the Laplace transform (LT). This method involves solving a problem of non-fatal disease spread in a society that is assumed to have a fixed size during the epidemic period. We apply the method to give an approximate analytic solution to the nonlinear system of the intended model. Moreover, the absolute error resulting from the numerical solutions and the ten iterations of LTAM approximations of the epidemic model, along with the maximum error remainder, were calculated by using MATHEMATICA® 11.3 program to illustrate the effectiveness of the method.

Sign in / Sign up

Export Citation Format

Share Document