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 On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

Robotnov function based operator for biological population model of biology

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sunil Kumar ◽  
Surath Ghosh ◽  
Shaher Momani ◽  
S. Hadid
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Derivative ◽  
Analytical Methods ◽  
Population Model ◽  
Nonlinear Partial Differential Equation ◽  
Spreading Rate ◽  
Fractional Operator ◽  
Content Type ◽  
Biological Population

Purpose The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense. Design/methodology/approach This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. Findings This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results. Research limitations/implications The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results. Practical implications The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results. Social implications The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results. Originality/value The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

scholarly journals A Reliable Computational Algorithm for Solving Fractional Biological Population Model

2021 ◽  
Vol 13 (1) ◽  
pp. 59-71
Author(s):  
A. Devi ◽  
M. Jakhar
Keyword(s):  
Perturbation Method ◽  
Population Model ◽  
Homotopy Perturbation Method ◽  
Computational Algorithm ◽  
Spatial Diffusion ◽  
Fractional Partial Differential Equations ◽  
Homotopy Perturbation ◽  
Biological Population ◽  
Fractional Orders ◽  
Real World Problems

In this work, authors obtained the series solution of nonlinear fractional partial differential equations, which is emerging in a spatial diffusion of biological population model using Elzaki transform homotopy perturbation method (ETHPM). The Elzaki transform homotopy perturbation method is a combined form of the Elzaki transform and homotopy perturbation method. Three test illustrations are used to show the proficiency and accuracy of the projected method. It has been observed that the proposed technique can be widely employed to examine other real world problems. The results obtained with the help of the proposed technique are plotted for different fractional orders.

scholarly journals Solution of the Kadomtsev-Petviashvili equation using an improved homotopy perturbation method

2021 ◽  
Vol 2070 (1) ◽  
pp. 012065
Author(s):  
C F Sagar Zephania ◽  
P C Harisankar ◽  
Tapas Sil
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Higher Dimension ◽  
Kp Equation ◽  
Homotopy Perturbation ◽  
Petviashvili Equation

Abstract An improved homotopy perturbation method (LH) applied to find approximate solution of KP equation. The results obtained ensure that LH is capable for solving the strongly higher dimension nonlinear partial differential equation such as KP equation. The approximated solution obtained by LH is compared with exact solution.

scholarly journals A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsaid ◽  
D. Hammad
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Gordon Equation ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Klein Gordon ◽  
Fractional Orders ◽  
Klein Gordon Equation

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

2016 ◽  
Vol 9 (1) ◽  
pp. 144-156 ◽  
Author(s):  
Majid Ghadiri ◽  
Mohsen Safi
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Simply Supported ◽  
Homotopy Perturbation ◽  
Functionally Graded Nanobeam

AbstractIn this paper, He's homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen's nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin's method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

Laplace Decomposition Method to Study Solitary Wave Solutions of Coupled Nonlinear Partial Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Arun Kumar ◽  
Ram Dayal Pankaj
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Weak Nonlinearity ◽  
Partial Differential ◽  
New Approach ◽  
Laplace Decomposition Method

Analytical and numerical solutions are obtained for coupled nonlinear partial differential equation by the well-known Laplace decomposition method. We combined Laplace transform and Adomain decomposition method and present a new approach for solving coupled Schrödinger-Korteweg-de Vries (Sch-KdV) equation. The method does not need linearization, weak nonlinearity assumptions, or perturbation theory. We compared the numerical solutions with corresponding analytical solutions.

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