Homotopy analysis method for the fractional nonlinear equations

Ghazanfari, Bahman; Veisi, F

doi:10.1016/j.jksus.2010.07.019

Cited by 16 publications

(13 citation statements)

References 9 publications

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“…Utilizing equations (20) and (21) in equations (14) and (15), and then equating the coefficients of corresponding power of p on both sides, we have…”

Section: General Procedures Of Generalized Least Square Homotopy Pertumentioning

confidence: 99%

Generalized least square homotopy perturbation solution of fractional telegraph equations

2019

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In this paper, generalized aspects of least square homotopy perturbations are explored to treat the system of non-linear fractional partial differential equations and the method is called as generalized least square homotopy perturbations (GLSHP).The concept of partial fractional Wronskian is introduced to detect the linear independence of functions depending on more than one variable through Caputo fractional calculus. General theorem related to Wronskian is also proved. It is found that solutions converge more rapidly through GLSHP in comparison to classical fractional homotopy perturbations. Results of this generalization are validated by taking examples from nonlinear fractional wave equations.Mathematics Subject Classification. 35Qxx, 65Mxx, 65Zxx.In above equations, φ (x,t), ψ(x,t), f i (t) and g i (t) for i = 1, 2 are given functions. Wei [16] analyzed a local finite difference scheme for the diffusion-wave equation of fractional order. A fourth order compact scheme which preserve energy was devised by Diaz et al. [17] for the solution of fractional nonlinear wave equations. Weak solutions for time fractional diffusion equations were examined by Yamamoto [18]. Other schemes which prescribe the information regarding the solution of fractional wave equations can be studied from [19]-[27].In previous years, scientists and applied mathematicians have been attracted towards modifications of classical homotopy perturbations in order to achieve accelerated accuracy. Some modifications can be seen in [28] and [29]. A variational form for homotopy perturbation iteration method was developed for fractional diffusion equation by Guo et al. [30]. Recently, Constantin and Caruntu [31] presented least square homotopy perturbations for non-linear ordinary differential equations.But this technique was not suitable to handle fractional partial differential equations as convergent solutions are not expected.Therefore, our main objective is to generalize this idea of the coupling of fractional homotopy perturbations and least square approximations, and to propose fractional partial Wronskian.The paper is organized in the following sequence: In section 2, basic definitions of fractional calculus are presented.The definition and theory of fractional partial Wronskian is also developed. In section 3, basic theory of generalized least square homotopy perturbations are proposed together with necessary definitions and lemma. Section 4 deals with numerical examples. Lastly conclusion is derived. Fundamentals of fractional calculusWe provide some basic definitions, properties and theorem related to fractional calculus which will be used at later stage C(0, ∞), and is said to be in the space C n µ if and only if f n ∈ C µ , n ∈ N .Definition 2.2. The (left sided) Riemann-Liouville fractional integral operator of order α 0 for a function f ∈ C µ , µ −1 is defined as:α > 0, t > 0, β 2 are linear operators, N , M are non-linear operators and u(x,t), v(x,t) are unknown functions. Whereas B 1 , B 2 are boundary operators, f (x,t) and g...

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“…Utilizing equations (20) and (21) in equations (14) and (15), and then equating the coefficients of corresponding power of p on both sides, we have…”

Section: General Procedures Of Generalized Least Square Homotopy Pertumentioning

confidence: 99%

Generalized least square homotopy perturbation solution of fractional telegraph equations

2019

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“…Hereby, a considerable effort has been expended to construct numerical and analytical methods for solving FPDEs, in recent years. Some of them are, homotopy analysis method (Ghazanfari and Veisi 2011;Song and Zhang 2007), fractional variational iteration method [Guo and Mei 2011;Wu and Lee 2010], Adomian decomposition method (Jafari and Daftardar 2006;Momani and Shawagfeh;Song and Wang 2013), fractional differential transform method (Arikoglu and Ozkol 2009;El-Sayed et al 2014;Momani et al 2007) and perturbationiteration algorithm (Şenol and Dolapci 2016;Şenol et al 2018).…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of the Time-Fractional Kadomtsev-Petviashvili Equation with Conformable Derivative

Durur

Şenol

Kurt

et al. 2019

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this study, residual power series method, namely RPSM, is applied to solve time-fractional Kadomtsev-Petviashvili (K-P) differential equation. In the solution procedure, the fractional derivatives are explained in the conformable sense. The model is solved approximately and the obtained results are compared with exact solutions obtained by the sub-equation method. The results reveal that the present method is accurate, dependable, simple to apply and a good alternative for seeking solutions of nonlinear fractional partial differential equations.

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“…After the identification of the FDEs, various numerical methods are developed to work on the approximate solutions of FDEs by many authors. Some of these known numerical methods are Finite Difference Method [7], Variational Iteration Method [8][9], Adomian Decomposition Method [10][11], Homotopy Perturbation Method [12], Homotopy Analysis Method [13], Wavelets [14][15], Spectral Methods [16] and Sinc-Collocation Method [17][18][19][20]. If we take a brief look at the applications of these methods: In [7], Li and Zeng investigated the stability and convergence of some FDEs using the fractional Euler, fractional Adams and high order methods which are related to Finite Difference Method.…”

Section: Introductionmentioning

confidence: 99%

“…Then Homotopy Analysis Method is considered to obtain the solution of the equation. In the work of Ghazanfari and Veisi [13], an extended Homotopy Analysis Method is considered to investigate the nonlinear fractional wave equation. Results approve the accurancy and efficiency of the method.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solutions of Fractional Boundary Value Problems Based on Beta Fractional Derivative

Daş

2018

Marmara Fen Bilimleri Dergisi

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Bu çalışmada, farklı sınır koşulları için kesirli mertebeden bir diferansiyel denklem sınıfını ele aldık. Bu kesirli mertebeden sınır değer problemlerinin(KSDP) yaklaşık çözümlerini elde etmek için Sinc Sıralama Yöntemi (SSY) uygulanmıştır. Kesirli türevler için Beta türevi alınmıştır. Ayrıca, bir takım test örnekleri sayısal simulasyonlarla birlikte verilmiştir. Yakınsaklık analizi SSY 'nin tutarlı ve etkin bir yöntem olduğunu göstermiştir.

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Homotopy analysis method for the fractional nonlinear equations

Cited by 16 publications

References 9 publications

Generalized least square homotopy perturbation solution of fractional telegraph equations

Generalized least square homotopy perturbation solution of fractional telegraph equations

Approximate Solutions of the Time-Fractional Kadomtsev-Petviashvili Equation with Conformable Derivative

Approximate Solutions of Fractional Boundary Value Problems Based on Beta Fractional Derivative

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