Travelling waves solution for fractional-order biological population model

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.

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4029254 ◽

2015 ◽

Vol 10 (5) ◽

Cited By ~ 14

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.

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On biological population model of fractional order

International Journal of Biomathematics ◽

10.1142/s1793524516500704 ◽

2016 ◽

Vol 09 (05) ◽

pp. 1650070 ◽

Cited By ~ 12

Author(s):

Syed Tauseef Mohyud-Din ◽

Ayyaz Ali ◽

Bandar Bin-Mohsin

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Exact Solutions ◽

Fractional Order ◽

Population Model ◽

Numerical Data ◽

Population Changes ◽

Partial Differential ◽

Biological Population ◽

Fractional Pdes

In this paper, we extensively studied a mathematical model of biology. It helps us to understand the dynamical procedure of population changes in biological population model and provides valuable predictions. In this model, we establish a variety of exact solutions. To study the exact solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into corresponding partial differential equation and modified exp-function method is implemented to investigate the nonlinear equation. Graphical demonstrations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, unfailing, well-organized and pragmatic for fractional PDEs and could be protracted to further physical happenings.

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

Mathematics ◽

10.3390/math9020155 ◽

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.

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HE–ELZAKI METHOD FOR SPATIAL DIFFUSION OF BIOLOGICAL POPULATION

Fractals ◽

10.1142/s0218348x19500695 ◽

2019 ◽

Vol 27 (05) ◽

pp. 1950069 ◽

Cited By ~ 10

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.

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A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

Journal of Applied Mathematics ◽

10.1155/2019/3456848 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7 ◽

Cited By ~ 11

Author(s):

Mohamed R. Ali

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Power Series ◽

Fractional Order ◽

Nonlinear Ordinary Differential Equation ◽

Lie Symmetry ◽

Series Method ◽

Lie Symmetry Analysis ◽

Power Series Method ◽

Bo Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

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Resonance and non-resonance in terms of average values. II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001359 ◽

2001 ◽

Vol 131 (5) ◽

pp. 1217-1235

Author(s):

M. N. Nkashama ◽

S. B. Robinson

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Nonlinear Term ◽

Spectral Gap ◽

Elliptic Boundary ◽

Sufficient Conditions ◽

Existence Results ◽

Elliptic Boundary Value ◽

Semilinear Elliptic ◽

The Given

We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.

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Travelling waves in radially symmetric parabolic systems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014281 ◽

1985 ◽

Vol 99 (3-4) ◽

pp. 269-275 ◽

Cited By ~ 1

Author(s):

Peter W. Bates

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Plane Wave ◽

Travelling Waves ◽

Parabolic Systems ◽

Physical Significance ◽

Physical Sciences ◽

Wave Solutions ◽

The Stability ◽

Symmetric Systems

SynopsisIt is shown that all travelling or standing plane-wave solutions to certain radially symmetric parabolic systems may be found by solving a related scalar ordinary differential equation (ODE). The radially symmetric systems considered here are those whose reaction term is radially directed and points inward near infinity. The stability of these waves is also discussed. Many systems arising in the physical sciences are included in the class studied and so the classification and stability of the travelling waves has physical significance.

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On the Approximate Solutions of Heat-Conduction Problems

Journal of Heat Transfer ◽

10.1115/1.3686066 ◽

1963 ◽

Vol 85 (3) ◽

pp. 203-207 ◽

Cited By ~ 3

Author(s):

Fazil Erdogan

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Partial Differential Equation ◽

Heat Conduction ◽

Approximate Solutions ◽

Integral Transforms ◽

Partial Differential ◽

The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

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Study on Approximate Solution of Fractional Order Biological Population Model

Archives of Current Research International ◽

10.9734/acri/2018/46155 ◽

2019 ◽

Vol 15 (4) ◽

pp. 1-11

Author(s):

Yunqi Xu ◽

Dianchen Lu

Keyword(s):

Approximate Solution ◽

Fractional Order ◽

Population Model ◽

Biological Population

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Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method

International Journal of Biomathematics ◽

10.1142/s1793524516500819 ◽

2016 ◽

Vol 09 (06) ◽

pp. 1650081 ◽

Cited By ~ 11

Author(s):

S. Sarwar ◽

M. A. Zahid ◽

S. Iqbal

Keyword(s):

Fractional Order ◽

Asymptotic Method ◽

Population Model ◽

Approximate Solutions ◽

High Accuracy ◽

Population Models ◽

Mathematical Study ◽

Third Order ◽

Biological Population ◽

Optimal Homotopy Asymptotic Method

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

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