Improved (G'/G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Dumitru Baleanu ◽  
Yavuz Uğurlu ◽  
Mustafa Inc ◽  
Bulent Kilic
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Population Model ◽  
Expansion Method ◽  
Fractional Partial Differential Equations ◽  
Basic Variable ◽  
Hilliard Equation ◽  
Biological Population ◽  
Cahn Hilliard Equation

In this paper, we used improved (G'/G)-expansion method to reach the solutions for some nonlinear time-fractional partial differential equations (fPDE). The fPDE is reduced to an ordinary differential equation (ODE) by means of Riemann–Liouille derivative and a basic variable transformation. Various types of functions are obtained for the time-fractional biological population model (fBPM) and Cahn–Hilliard (fCH) equation.

scholarly journals A Novel Effective Approach for Solving Fractional Nonlinear PDEs

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh ◽  
Hadi Rezazadeh
Keyword(s):  
Partial Differential Equations ◽  
Perturbation Method ◽  
Population Model ◽  
Homotopy Perturbation Method ◽  
Analytical Procedure ◽  
Nonlinear Pdes ◽  
Fractional Partial Differential Equations ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Biological Population

The present work introduces an effective modification of homotopy perturbation method for the solution of nonlinear time-fractional biological population model and a system of three nonlinear time-fractional partial differential equations. In this approach, the solution is considered a series expansion that converges to the nonlinear problem. The new approximate analytical procedure depends only on two iteratives. The analytical approximations to the solution are reliable and confirm the ability of the new homotopy perturbation method as an easy device for computing the solution of nonlinear equations.

scholarly journals Travelling waves solution for fractional-order biological population model

2021 ◽  
Vol 16 ◽  
pp. 32
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
J.F. Gómez-Aguilar ◽  
Shoaib ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Traveling Waves ◽  
Population Model ◽  
Travelling Waves ◽  
Order Problem ◽  
Physical Behavior ◽  
Biological Population ◽  
The Given

In this paper, we implemented the generalized (G′/G) and extended (G′/G) methods to solve fractional-order biological population models. The fractional-order derivatives are represented by the Caputo operator. The solutions of some illustrative examples are presented to show the validity of the proposed method. First, the transformation is used to reduce the given problem into ordinary differential equations. The ordinary differential equation is than solve by using modified (G′/G) method. Different families of traveling waves solutions are constructed to explain the different physical behavior of the targeted problems. Three important solutions, hyperbolic, rational and periodic, are investigated by using the proposed techniques. The obtained solutions within different classes have provided effective information about the targeted physical procedures. In conclusion, the present techniques are considered the best tools to analyze different families of solutions for any fractional-order problem.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

HE–ELZAKI METHOD FOR SPATIAL DIFFUSION OF BIOLOGICAL POPULATION

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950069 ◽  
Author(s):  
JAMSHAID UL RAHMAN ◽  
DIANCHEN LU ◽  
MUHAMMAD SULEMAN ◽  
JI-HUAN HE ◽  
MUHAMMAD RAMZAN
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Population Model ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Numerical Procedure ◽  
Spatial Diffusion ◽  
Homotopy Perturbation ◽  
Biological Population

The foremost purpose of this paper is to present a valuable numerical procedure constructed on Elzaki transform and He’s Homotopy perturbation method (HPM) for nonlinear partial differential equation arising in spatial flow characterizing the general biological population model for animals. The actions are made usually by mature animals driven out by intruders or by young animals just accomplished maturity moving out of their parental region to initiate breeding region of their own. He–Elzaki method is a blend of Elzaki transform and He’s HPM. The results attained are compared with Sumudu decomposition method (SDM). The numerical results attained by suggested method specify that the procedure is easy to implement and precise. These outcomes reveal that the proposed method is computationally very striking.

scholarly journals Comparison Theorem for Nonlinear Path-Dependent Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Falei Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Parabolic Partial Differential Equation ◽  
Basic Variable ◽  
Partial Differential ◽  
Fully Nonlinear ◽  
Path Dependent

We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the pathωton an interval [0,t] becomes the basic variable in the place of classical variablest,x∈[0,T]×ℝd. Then we study the comparison theorem of fully nonlinear PPDE and give some of its applications.

scholarly journals Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Burgers Equation ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Biological Population ◽  
Regularized Long Wave Equation ◽  
Generalized Time

In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3+1)-space-time fractional Zakharov-Kuznetsov equation are obtained, respectively.

scholarly journals Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei Li ◽  
Huizhang Yang ◽  
Bin He
Keyword(s):  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Expansion Method ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Hilliard Equation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Cahn Hilliard Equation

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

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