On the solutions of fractional Swift Hohenberg equation with dispersion

Vishal, K.; Das, S.; Ong, S. H.; Ghosh, Pradyumna

doi:10.1016/j.amc.2012.12.032

Cited by 14 publications

(8 citation statements)

References 25 publications

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“…Consider fractional S‐H equation with dispersion of the form

D_{t}^{μ} v ((), x, t) + \frac{\partial^{4} v ((), x, t)}{\partial x^{4}} + 2 \frac{\partial^{2} v ((), x, t)}{\partial x^{2}} - η \frac{\partial^{3} v ((), x, t)}{\partial x^{3}} - italicαv ((), x, t) - 2 v^{2} ((), x, t) + v^{3} ((), x, t) = 0,

with initial condition

v ((), x, 0) = \frac{1}{10} italicsin ((), \frac{italicπx}{l}) .

…”

Section: Solution For Fractional S‐h Equationmentioning

confidence: 99%

“…The solution for S‐H equation is studied by many scholars using distinct techniques like Vishal et al employed homotopy analysis method (HAM) and HPTM, Merdan studied with the help of variational iteration technique using Riemann‐Liouville derivative, Khan et al used differential transform technique, and others . Motivated by the above work, we find the approximated analytical solution for proposed problem with the aid of a novel computational method called q ‐HATM and also present the convergence analysis for the considered problem.…”

Section: Introductionmentioning

confidence: 97%

“…In many physical phenomena including the study of laser, hydrodynamics, liquid crystals, flame dynamics, and statistical mechanics, the portrayal of S‐H equation is very essential. Now, consider the time‐fractional non‐linear S‐H equation as

D_{t}^{µ} v ((), x, t) + ((), \partial^{4} v ((), x, t)) / ((), \partial x^{4}) + 2 ((), \partial^{2} v ((), x, t)) / ((), \partial x^{2}) + ((), 1 - µ) v ((), x, t) + v^{3} ((), x, t) = 0, 0 < µ \leq 1, t > 0,

and in presence of dispersive term as

D_{t}^{µ} v ((), x, t) + ((), \partial^{4} v ((), x, t)) / ((), \partial x^{4}) + 2 ((), \partial^{2} v ((), x, t)) / ((), \partial x^{2}) - \partial ((), \partial^{3} v ((), x, t)) / ((), \partial x^{3}) - \partial v ((), x, t) - 2 v^{2} ((), x, t) + v^{3} ((), x, t) = 0,

where v ( x , t ) is probability density function and η and α are, respectively, dispersive and bifurcation parameters.…”

Section: Introductionmentioning

confidence: 98%

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Analysis of fractional Swift‐Hohenberg equation using a novel computational technique

Veeresha

Prakasha

Băleanu

2019

Math Methods in App Sciences

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In this paper, the approximated analytical solution for fractional Swift-Hohenberg (S-H) equation is found with the aid of novel technique called qhomotopy analysis transform method (q-HATM). To ensure the applicability and efficiency of the proposed algorithm, we consider non-linear arbitraryorder S-H equation in presence and absence of dispersive term. The convergence analysis for the projected problem is presented, and the numerical simulations have been conducted to verify the future scheme is reliable and accurate. Further, the effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are presented through plots. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyse the complex problems that arose in science and technology. K E Y W O R D SFractional differential equations, fractional Swift-Hohenberg equation, Laplace transform, q-Homotopy analysis transform method J E L C L A S S I F I C A T I O N26A33; 34E10; 41A58 | INTRODUCTIONDerivatives of arbitrary order are invented by Leibnitz soon after the integer-order derivatives, and lately, it magnetized the attention of many researchers. It was shortly revealed that fractional calculus (FC) is more suitable for modelling real-world problems as compared with classical calculus. The theory of FC interprets reality of nature in an excellent and systematic manner. Recently, it has magnetized many researchers because of its ability to provide an exact explanation to non-linear complex systems. The advantage of using fractional models of differential equations in any models is their nonlocal property. Fractional-order derivative is nonlocal while integer-order derivative is local in nature. It shows that the upcoming state of physical system is also dependent on all of its historical states in addition to its present state. Hence, the fractional models are more realistic as compared with modelled with the aid integer-order derivatives.Moreover, many senior scholars are defining revolutionary definitions of FC and laid the foundations for FC. 1-6 With the swift growth of digital computer knowledge, many researchers begin to work on the theory and applications of the FC. The theory of fractional-order calculus has been related to practical projects, and it has been applied to analyse and study different phenomena including chaos theory, 7 financial models, 8 human

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“…Consider fractional S‐H equation with dispersion of the form

D_{t}^{μ} v ((), x, t) + \frac{\partial^{4} v ((), x, t)}{\partial x^{4}} + 2 \frac{\partial^{2} v ((), x, t)}{\partial x^{2}} - η \frac{\partial^{3} v ((), x, t)}{\partial x^{3}} - italicαv ((), x, t) - 2 v^{2} ((), x, t) + v^{3} ((), x, t) = 0,

with initial condition

v ((), x, 0) = \frac{1}{10} italicsin ((), \frac{italicπx}{l}) .

…”

Section: Solution For Fractional S‐h Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

D_{t}^{µ} v ((), x, t) + ((), \partial^{4} v ((), x, t)) / ((), \partial x^{4}) + 2 ((), \partial^{2} v ((), x, t)) / ((), \partial x^{2}) + ((), 1 - µ) v ((), x, t) + v^{3} ((), x, t) = 0, 0 < µ \leq 1, t > 0,

and in presence of dispersive term as

D_{t}^{µ} v ((), x, t) + ((), \partial^{4} v ((), x, t)) / ((), \partial x^{4}) + 2 ((), \partial^{2} v ((), x, t)) / ((), \partial x^{2}) - \partial ((), \partial^{3} v ((), x, t)) / ((), \partial x^{3}) - \partial v ((), x, t) - 2 v^{2} ((), x, t) + v^{3} ((), x, t) = 0,

where v ( x , t ) is probability density function and η and α are, respectively, dispersive and bifurcation parameters.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Analysis of fractional Swift‐Hohenberg equation using a novel computational technique

Veeresha

Prakasha

Băleanu

2019

Math Methods in App Sciences

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“…Later, Vishal et al in [37] constructed the approximate analytic solution with respect initial value ( , 0) = 1/10 sin( / ) using the homotopy analysis method. Furthermore, Vishal et al considered the time-fractional S-H equation with dispersion [38]…”

Section: Linear Swift-hohenberg Equationmentioning

confidence: 99%

An Iterative Method for Time-Fractional Swift-Hohenberg Equation

Pang

2018

Advances in Mathematical Physics

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We study a type of iterative method and apply it to time-fractional Swift-Hohenberg equation with initial value. Using this iterative method, we obtain the approximate analytic solutions with numerical figures to initial value problems, which indicates that such iterative method is effective and simple in constructing approximate solutions to Cauchy problems of time-fractional differential equations.

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“…The solution for the S-H equation is studied by many scholars using distinct techniques like Vishal et al who employed Homotopy Analysis Method (HAM) [24] and Homotopy Perturbation Transform Method (HPTM) [26], Merdan who studied with the help of the variational iteration technique using the Riemann-Liouville derivative [34], Khan et al who used the differential transform technique [25], and others [35][36][37][38]. Motivated by the above work, we find the approximated analytical solution for proposed problem by using RPSM.…”

Section: Introductionmentioning

confidence: 99%

Residual Power Series Method for Fractional Swift–Hohenberg Equation

2019

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In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.

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On the solutions of fractional Swift Hohenberg equation with dispersion

Cited by 14 publications

References 25 publications

Analysis of fractional Swift‐Hohenberg equation using a novel computational technique

Analysis of fractional Swift‐Hohenberg equation using a novel computational technique

An Iterative Method for Time-Fractional Swift-Hohenberg Equation

Residual Power Series Method for Fractional Swift–Hohenberg Equation

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