Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems

Tariq, Hira; Akram, Ghazala

doi:10.1007/s12190-016-1056-1

Cited by 28 publications

(15 citation statements)

References 18 publications

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“…For the Kawahara equation, recently Cui et al [63] derived the local well-posedness with−1 < s < 0 when = 0. Moreover, the time-fractional B-L equation is analyzed with various methods including, Variational iteration methods and Homotopy perturbation [60], Residual power series method [64], Reduced differential transform method [61], and others [14,17,24,[65][66][67][68][69][70][71][72].…”

Section: Introductionmentioning

confidence: 99%

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

Gao

Veeresha

Prakasha

et al. 2020

Numerical Methods Partial

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The pivotal aim of the present work is to find the numerical solution for fractional Benney-Lin equation by using two efficient methods, called q-homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three-dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.

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Section: Introductionmentioning

confidence: 99%

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

Gao

Veeresha

Prakasha

et al. 2020

Numerical Methods Partial

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“…Our primary aim is to apply the RPSM [22–25] for solving the fractional VE with uncertainty.

\frac{\partial^{α} \tilde{ψ}}{\partial t^{α}} = {\overset{c}{true}}^{2} (\frac{\partial^{2} \tilde{ψ}}{\partial r^{2}} + \frac{1}{r} \frac{\partial \tilde{ψ}}{\partial r}),

with fuzzy initial conditions (IC)

\tilde{ψ} (r, 0) = (0.5, 1, 1.5) ζ (r),

{\overset{ψ}{true}}_{t} (r, 0) = \tilde{c} ξ (r),

where

\tilde{ψ} false(r, t false)

and

\overset{c}{true}

represent the displacement and wave velocity of free vibration in uncertainty environment and

ϕ (r)

ξ (r)

are the functions of the radius of the membrane ( r ).…”

Section: Introductionmentioning

confidence: 99%

Analysis of time‐fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method

Jena

Sedighi

2020

Z Angew Math Mech

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The study of the time‐fractional vibration equation (VE) of large membranes is vital due to its widespread applications. Various researchers have investigated the titled problem in which the variables and parameters have been considered as crisp/exact. However, in actual practice, this may contain uncertainty because of errors in observations, maintenance induced error, and other errors. In this investigation, we have considered these uncertainties as interval/fuzzy, and a method, namely double parametric form of fuzzy numbers are used to solve the uncertain fractional vibration model of order α(1<α≤2). The titled problem has been solved under various cases using the Residual power series method (RPSM). The performance of the present method is well in terms of computational efficiency. The consistency of the method is displayed by providing numerical solutions for various cases. Obtained results are depicted in terms of figures and are also compared with individual cases.

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“…In [13], an approach was applied to nd the exact solutions of fractional-order time-dependent Schrödinger equations. e residual power series method was also used for many other problems, such as the gas dynamic equation [14], the fractional initial Emden-Fowler equation [15], the generalized Berger-Fisher equation [16], and the nonlinear time-space-fractional Benney-Lin equation [17].…”

Section: Introductionmentioning

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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Residual power series method for solving time-space-fractional Benney-Lin equation arising in falling film problems

Cited by 28 publications

References 18 publications

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

Analysis of time‐fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method

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

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