scholarly journals Efficacious Analytical Technique Applied to Fractional Fornberg–Whitham Model and Two-Dimensional Fractional Population Model

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1976
Author(s):  
Cyril D. Enyi
Keyword(s):  
Biological Sciences ◽  
Exact Solution ◽  
Population Model ◽  
Two Dimensional ◽  
Transform Method ◽  
One Dimensional ◽  
Analytical Technique ◽  
Extensive Analysis ◽  
Homotopy Analysis ◽  
Fractional Differential

This paper presents an efficacious analytical and numerical method for solution of fractional differential equations. This technique, here in named q-HATM (q-homotopy analysis transform method) is applied to a one-dimensional fractional Fornberg–Whitham model and a two-dimensional fractional population model emanating from biological sciences. The overwhelming agreement of our analytical solution by the q-HATM technique with the exact solution indeed establishes the efficacy of q-HATM to solve the fractional Fornberg–Whitham model and the two-dimensional fractional population model. Furthermore, comparisons by means of extensive analysis using numerics, graphs and error analysis are presented to affirm the preference of q-HATM technique over other methods. A variant of the q-HATM using symmetry can also be considered to solve these problems.

Numerical computation of fractional Lotka-Volterra equation arising in biological systems

2015 ◽  
Vol 4 (2) ◽  
Author(s):  
Ramswroop ◽  
Jagdev Singh ◽  
Devendra Kumar
Keyword(s):  
Exact Solution ◽  
Difference Equations ◽  
Volterra Equation ◽  
Auxiliary Function ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Effective Role ◽  
Fractional Differential ◽  
And Control

AbstractIn this paper, we present the homotopy analysis transform method (HATM) to solve fractional Lotka- Volterra equation, which describes the long term servival of species. The HATM solutions, denotes less error compare with their respective exact solution for alpha = 1. In addition to non-proposed techniques, HATMis valid for both small and large parameters, it also provides us with a simpleway to adjust and control the parameter hbar and auxiliary function H(t), which play effective role for convergence solutions of fractional differential-difference equations (FDDEs).

On a new modified fractional analysis of Nagumo equation

2019 ◽  
Vol 12 (03) ◽  
pp. 1950034 ◽  
Author(s):  
Khaled M. Saad ◽  
Si̇nan Deni̇z ◽  
Dumi̇tru Baleanu
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Fractional Derivatives ◽  
Transform Method ◽  
Average Residual ◽  
Homotopy Analysis ◽  
Fractional Analysis ◽  
Nagumo Equation ◽  
Modified Equation

In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called [Formula: see text]-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly.

scholarly journals On the Application of Homotopy Analysis Method to Fractional Differential Equations

2021 ◽  
pp. 1-18
Author(s):  
Khalid Suliman Aboodh ◽  
Abu baker Ahmed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Fractional Differential ◽  
Convergence Control Parameter

In this paper, an attempt has been made to obtain the solution of linear and nonlinear fractional differential equations by applying an analytic technique, namely the homotopy analysis method (HAM). The fractional derivatives are described by Caputo’s sense. By this method, the solution considered as the sum of an infinite series, which converges rapidly to exact solution with the help of the nonzero convergence control parameter ℏ. Some examples are given to show the efficiently and accurate of this method. The solutions obtained by this method has been compared with exact solution. Also our graphical represented of the solutions have been given by using MATLAB software.

scholarly journals New Classes of Solutions of Dynamical Problems of Plasticity

2020 ◽  
pp. 792-796
Author(s):  
Sergei I. Senashov ◽  
Olga V. Gomonova ◽  
Irina L. Savostyanova ◽  
Olga N. Cherepanova
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Science And Engineering ◽  
Spatial Variables ◽  
One Dimensional ◽  
New Exact Solutions

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

Toward a New Algorithm for Nonlinear Fractional Differential Equations

2013 ◽  
Vol 5 (2) ◽  
pp. 222-234
Author(s):  
Fadi Awawdeh ◽  
S. Abbasbandy
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Homotopy Analysis Method ◽  
Analytical Techniques ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Transform Method ◽  
Nonlinear Fractional Differential Equations ◽  
Homotopy Analysis ◽  
Fractional Differential

AbstractThis paper is concerned with the development of an efficient algorithm for the analytic solutions of nonlinear fractional differential equations. The proposed algorithm Laplace homotopy analysis method (LHAM) is a combined form of the Laplace transform method with the homotopy analysis method. The biggest advantage the LHAM has over the existing standard analytical techniques is that it overcomes the difficulty arising in calculating complicated terms. Moreover, the solution procedure is easier, more effective and straightforward. Numerical examples are examined to demonstrate the accuracy and efficiency of the proposed algorithm.

scholarly journals A reliable analytical technique for fractional Caudrey-Dodd-Gibbon equation with Mittag-Leffler kernel

2020 ◽  
Vol 9 (1) ◽  
pp. 319-328 ◽  
Author(s):  
P. Veeresha ◽  
D. G. Prakasha
Keyword(s):  
Fractional Order ◽  
Nonlinear Differential Equations ◽  
Order Model ◽  
Science And Engineering ◽  
Transform Method ◽  
Analytical Technique ◽  
Fractional Order Model ◽  
Analysis Scheme ◽  
Laplace Transform Technique ◽  
Homotopy Analysis

AbstractThe pivotal aim of the present work is to find the solution for fractional Caudrey-Dodd-Gibbon (CDG) equation using q-homotopy analysis transform method (q-HATM). The considered technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the projected fractional-order model. In order to illustrate and validate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The obtained results elucidate that, the considered algorithm is easy to implement, highly methodical as well as accurate and very effective to examine the nature of nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

scholarly journals A mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Sina Etemad ◽  
Hakimeh Mohammadi
Keyword(s):  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Disease Model ◽  
Time Interval ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Sumudu Transform ◽  
Existence Criterion ◽  
Fractional Differential ◽  
The Stability

Abstract We study a fractional-order model for the anthrax disease between animals based on the Caputo–Fabrizio derivative. First, we derive an existence criterion of solutions for the proposed fractional $\mathcal {CF}$ CF -system of the anthrax disease model by utilizing the Picard–Lindelof technique. By obtaining the basic reproduction number $\mathcal{R}_{0}$ R 0 of the fractional $\mathcal{CF}$ CF -system we compute two disease-free and endemic equilibrium points and check the asymptotic stability property. Moreover, by applying an iterative approach based on the Sumudu transform we investigate the stability of the fractional $\mathcal{CF}$ CF -system. We obtain approximate series solutions of this system by means of the homotopy analysis transform method, in which we invoke the linear Laplace transform. Finally, after the convergence analysis of the numerical method HATM, we present a numerical simulation of the $\mathcal{CF}$ CF -fractional anthrax disease model and review the dynamical behavior of the solutions of this $\mathcal {CF}$ CF -system during a time interval.

Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

2019 ◽  
Vol 25 (1) ◽  
pp. 1-11
Author(s):  
Seyed Mousa Torabi ◽  
Abolfazl Tari ◽  
Sedaghat Shahmorad
Keyword(s):  
Stability Analysis ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Two Dimensional ◽  
Transform Method ◽  
One Dimensional ◽  
Convergence And Stability ◽  
Differential Transform ◽  
Convergence And Stability Analysis

Abstract In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.

Sign in / Sign up

Export Citation Format

Share Document