Approximate analytical solutions of strongly nonlinear fractional BBM-Burger’s equations with dissipative term

Mukiawa, Soh Edwin; Enyi, Cyril Dennis; Iyiola, Olaniyi S.; Audu, Johnson D.

doi:10.12988/ams.2014.49754

Cited by 10 publications

(8 citation statements)

References 9 publications

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“…Also the power series solution in subfigure (F) and Equation (10) when n → ∞ converge to the exact solution. In Table 4, we find that the numerical results resulting from the presented method were compared with the numerical results from homotopy analysis method (HAM), 41 q-homotopy analysis method (q-HAM), 42 modified residual power series method (RPSM), 19 variational iteration method (VIM) 40 and homotopy perturbation method (HPM), 40 which show the superiority of the proposed method over other methods in obtaining a lower error rate and thus a better approximation to the exact solution. Table 2 shows a comparison of the convergence analysis resulting from the proposed method with HAM, q-HAM, modified RPSM, VIM, and HPM, which shows the advantage of the proposed method.…”

Section: Applications With Convergence Analysismentioning

confidence: 99%

A fractional Temimi‐Ansari method (FTAM) with convergence analysis for solving physical equations

Arafa

El‐Sayed

Hagag

2021

Math Methods in App Sciences

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The aim of this paper is to describe the new fractional Temimi-Ansari method (FTAM) for the KdV-Burgers equation and the Benjamin-Bona-Mahoney-Burger equation (BBMB) with time fractional order. Convergence of the presented method has been successfully achieved. This method does not need any assumptions for nonlinear terms. FTAM with time fractional order is demonstrated to be a very simple and effective approach to solving nonlinear fractional problems. The accuracy and efficiency of FTAM has been demonstrated by studying the convergence of this technique.

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Section: Applications With Convergence Analysismentioning

confidence: 99%

A fractional Temimi‐Ansari method (FTAM) with convergence analysis for solving physical equations

Arafa

El‐Sayed

Hagag

2021

Math Methods in App Sciences

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“…As is shown in Table 3, we present the absolute error in (48) between different values of x and t when α � 1, 0 ≤ t ≤ 1, − 2 ≤ x ≤ 2. And the least-squares residual power series method (LSRPSM) and the q-homotopy analysis method (q-HAM) [32] with u i (x, t) when i � 2 are compared with the classic residual power series method with u i (x, t) when i � 2, as shown in Table 3. e approximate solutions of the q-HAM can be written as 8 Complexity…”

Section: Illustrative Examplesmentioning

confidence: 99%

Least‐Squares Residual Power Series Method for the Time‐Fractional Differential Equations

Zhang

Wei

et al. 2019

Complexity

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In this study, an applicable and effective method, which is based on a least-squares residual power series method (LSRPSM), is proposed to solve the time-fractional differential equations. The least-squares residual power series method combines the residual power series method with the least-squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least-squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least-squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.

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“…The search for a better way to expand the convergence region led to the modification of HAM, called q-HAM, more of a general method than HAM [48]. Many authors have taken advantage of q-HAM and used it to solve nonlinear fractional partial differential equations [49][50][51][52][53][54][55]. The q-HATM was proposed by Singh et al [56] and did not require any form of discretization, linearization, or perturbation as compared to other methods.…”

Section: Introductionmentioning

confidence: 99%

A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations

Akinyemi

Iyiola

2020

Adv Differ Equ

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This paper employs an efficient technique, namely q-homotopy analysis transform method, to study a nonlinear coupled system of equations with Caputo fractional-time derivative. The nonlinear fractional coupled systems studied in this present investigation are the generalized Hirota-Satsuma coupled with KdV, the coupled KdV, and the modified coupled KdV equations which are used as a model in nonlinear physical phenomena arising in biology, chemistry, physics, and engineering. The series solution obtained using this method is proved to be reliable and accurate with minimal computations. Several numerical comparisons are made with well-known analytical methods and the exact solutions when α = 1. It is evident from the results obtained that the proposed method outperformed other methods in handling the coupled systems considered in this paper. The effect of the fractional order on the problem considered is investigated, and the error estimate when compared with exact solution is presented.Here, we present some useful definitions, properties, and notations that will be used in this work. Definition 2.1 The Riemann-Liouville (R-L) fractional integral of order α (α ≥ 0) of a function Q(x, t) ∈ C m , m ≥ -1, is given as [29, 63-65] J α Q(x, t) = 1 Γ (α) t 0

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Approximate analytical solutions of strongly nonlinear fractional BBM-Burger’s equations with dissipative term

Cited by 10 publications

References 9 publications

A fractional Temimi‐Ansari method (FTAM) with convergence analysis for solving physical equations

A fractional Temimi‐Ansari method (FTAM) with convergence analysis for solving physical equations

Least‐Squares Residual Power Series Method for the Time‐Fractional Differential Equations

A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations

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