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 Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sami Aljhani ◽  
Mohd Salmi Md Noorani ◽  
Khaled M. Saad ◽  
A. K. Alomari
Keyword(s):  
Exact Solution ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Accurate Solution ◽  
Caputo Fractional Derivatives ◽  
Average Residual ◽  
Homotopy Analysis

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏ u , v -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

A robust technique based solution of time-fractional seventh-order Sawada–Kotera and Lax’s KdV equations

2021 ◽  
pp. 2150265
Author(s):  
Rajarama Mohan Jena ◽  
Snehashish Chakraverty ◽  
Dumitru Baleanu ◽  
Waleed Adel ◽  
Hadi Rezazadeh
Keyword(s):  
Convergence Analysis ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Absolute Error ◽  
Transform Method ◽  
Cpu Time ◽  
Kdv Equations ◽  
Differential Transform ◽  
Seventh Order

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax’s KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, [Formula: see text] revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values [Formula: see text] are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated.

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 Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Muhammad Jamil ◽  
Asmat Ara
Keyword(s):  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Higher Dimensions ◽  
Analysis Method ◽  
Transform Method ◽  
Coupled Equations ◽  
Fractional Schrödinger Equations ◽  
Differential Transform ◽  
Homotopy Analysis

We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

Approximate solution of a piecewise linear–nonlinear oscillator using the homotopy analysis method

2017 ◽  
Vol 24 (19) ◽  
pp. 4551-4562 ◽  
Author(s):  
Jixiong Fei ◽  
Bin Lin ◽  
Shuai Yan ◽  
Xiaofeng Zhang
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Nonlinear Damping ◽  
Analysis Method ◽  
Bifurcation Diagrams ◽  
Homotopy Analysis

Most of the piecewise oscillators in engineering fields include nonlinear damping or stiffness and the contained damping or stiffness is strongly nonlinear, but to the authors’ knowledge little attention has been paid to those systems. Thus, in the present paper, a sinusoidal excited piecewise linear–nonlinear oscillator is analyzed. The mathematical model of the oscillator is described by a combination of a linear and a nonlinear differential equation which contains strong nonlinear terms of stiffness. An approximate solution for the oscillator is proposed by using the homotopy analysis method and matching method. The validity of the proposed solution is verified by comparing it with the exact solution. It is found that the approximate solution is in good agreement with the exact solution. The influence of some system parameters on the dynamical behavior of the oscillator is also investigated by the bifurcation diagrams of these parameters. From these bifurcation diagrams, one can observe the motion of the oscillator directly.

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).

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.

scholarly journals Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 125 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Order ◽  
Traveling Wave ◽  
Fractional Derivatives ◽  
Traveling Wave Solution ◽  
Order System ◽  
Transform Method ◽  
Homotopy Analysis ◽  
Definition Of ◽  
Analytical Approaches ◽  
Natural Decomposition

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.

scholarly journals Numerical Solution of Fractional-Order HIV Model Using Homotopy Method

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Sami Aljhani ◽  
Mohd Salmi Md Noorani ◽  
A. K. Alomari
Keyword(s):  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Approximate Analytic Solution ◽  
Convergence Region ◽  
Transform Method ◽  
Hiv Model ◽  
Homotopy Analysis ◽  
Convergent Algorithm ◽  
Present Algorithm ◽  
Good Agreement

In this study, we construct a convergent algorithm for generating an approximate analytic solution for the fractional HIV infection of CD4+ T cells with Atangana–Baleanu fractional derivatives in the Caputo sense. We compute the solution by utilizing the fractional homotopy analysis transform method (FHATM) and achieved a convergence region of the solution by employing an auxiliary parameter. Moreover, we apply a numerical scheme proposed by Toufik and Atangana for solving this kind of problem and compared with our results. A good agreement between the new algorithm and the numerical scheme is remarkable. The solution via the present algorithm can be obtained without any linearization or discretization which makes it reliable and easy to apply.

scholarly journals Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense

2018 ◽  
Vol 64 (5) ◽  
pp. 539 ◽  
Author(s):  
Francisco Gomez ◽  
Khaled Saad
Keyword(s):  
Fractional Derivatives ◽  
Error Function ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Chemical System ◽  
Transform Method ◽  
Diffusion Waves ◽  
Homotopy Analysis ◽  
Optimal Values ◽  
Coupled Reaction

In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

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