An analytical study for Fisher type equations by using homotopy perturbation method

Agirseven, Deniz; Öziş, Turgut

doi:10.1016/j.camwa.2010.05.006

Cited by 38 publications

(37 citation statements)

References 51 publications

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“…For the nonlinear space-time generalized fractional Fisher equation, we have successfully recovered the previously known solution (110) that has been found in [31], but to the best of our knowledge the solutions (86), (94), and (102) have not been obtained previously in the literature.…”

Section: Case 4 Whenmentioning

confidence: 86%

“…We will show that, for the analytical solutions of the ZKBBM fractional partial differential equation, one of the advantages of Feng's first integral method is the evaluation of the constants involved in the analytical solutions that have not been evaluated within the subequation method [29]. Furthermore for the analytical solutions of the space-time fractional generalized Fisher equation, by applying Feng's first integral method, we will obtain three new analytical solutions that have not been obtained in previous works [30,31,33].…”

Section: Introductionmentioning

confidence: 87%

“…The second nonlinear PDE mentioned before, the generalized Fisher equation [30][31][32], can be obtained as a special case of the reaction-diffusion equation. The reactiondiffusion equation plays a fundamental role in a great number of various models of heat and reaction-diffusion processes, mathematical biology, chemistry, genetics, and many other fields.…”

Section: Introductionmentioning

confidence: 99%

“…The above equation has been introduced as a model for the propagation of a mutant gene in mathematical biology Journal of Applied Mathematics 3 [30][31][32] and can be extended to fractional order derivatives obtaining the generalized Fisher space-time fractional equation:…”

Section: Introductionmentioning

confidence: 99%

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Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

Yépez-Martínez

Sosa

Reyes

2015

Journal of Applied Mathematics

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The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng's first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

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Section: Case 4 Whenmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

Yépez-Martínez

Sosa

Reyes

2015

Journal of Applied Mathematics

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“…This problem has been studied by several authors via different methods. Table 1 and Table 2 display the 3-terms approximation obtained by Adomian decomposition method (ADM) [29], variational iteration method (VIM) [31] , homotopy perturbation method (HPM) [32], PIA-1 with 2-terms and 3-terms approximation of PIA-2 for different λ's. It is clear that PIA-2 gives better results than other techniques even for n=2.…”

Section: Pia-2 Solutionmentioning

confidence: 99%

A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations

Bıldık

Deni̇z

2017

ITM Web Conf.

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Abstract. In this article, a framework is developed to get more approximate solutions to nonlinear partial differential equations by applying perturbation iteration technique. This technique is modified and improved to solve nonlinear diffusion equations of the Fisher type. Some problems are investigated to illustrate the efficiency of the method. Comparisons between the new results and the solutions obtained by other techniques prove that this technique is highly effective and accurate in solving nonlinear problems. Convergence analysis and error estimate are also provided by using some related theorems. The basic ideas indicated in this work are anticipated to be further developed to handle nonlinear models.

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Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

Loyinmi

Akinfe

2020

Engineering Reports

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In this research work, we propose an unprecedented hybrid algorithm which involves the coupling of a new integral transform, namely, the Elzaki transform and the well‐known homotopy perturbation method called the Elzaki homotopy transformation perturbation method (EHTPM) to solve for the exact solution of three distinct types of Fisher's equation, namely, the Fisher's equation of two cases, the sixth‐order Fisher's equation and the nonlinear diffusion equation of the Fisher type. These equations are prominent in mathematical biology and highly applicable in genetic propagation, population dynamics, stochastic processes, combustion theory, as well as a prototype model for a spreading flame, and so on. The efficacy and authenticity of this method was established via convergence and error analysis, and shows that the solutions obtained from EHTPM are unique and convergent. The results of EHTPM when compared with the exact results, homotopy results, and results from the other existing literature via table of comparison, 3D plots, and the convergence plots validates that EHTPM is an efficient and reliable tool of providing exact solutions to a wider class of nonlinear partial differential equations in a simple and straightforward manner, with no discretization, linearization, computation of Adomian polynomials, and devoid of errors.

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An analytical study for Fisher type equations by using homotopy perturbation method

Cited by 38 publications

References 51 publications

Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations

Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

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