An Iterative Method for Time-Fractional Swift-Hohenberg Equation

Li, Wenjin; Pang, Yanni

doi:10.1155/2018/2405432

Cited by 14 publications

(8 citation statements)

References 36 publications

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“…w is a scaler function of x and t defined on the line or the plane. The Swift-Hohenberg (S-H) equation is a mathematical model that has a great role in modeling the pattern formulation theory which includes the chosen of pattern, the impacts of noise on bifurcations, the dynamics of defects, and spatiotemporal chaos [2][3][4]. Also, it describes numerous models in engineering and thermal physics including the pattern formulation theory in fluid layers, hydrodynamics, lasers, flame dynamics, and statistical mechanics [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…The most common of these methods to determine approximate analytical solutions for FPDEs are the Adomian decomposition method, variational iteration method, homotopy perturbation method, and reproducing kernel method [21][22][23][24][25]. In this work, we consider the nonlinear time-fractional S-H equation with bifurcation [4].…”

Section: Introductionmentioning

confidence: 99%

“…Particularly, if β = 1, the fractional sense (2) reduces to the standard case S-H Equation (1). Equation (2) has been solved by numerous scholars, remarkably, Atangana and Kılıçman [3], Li and Pang [4], Prakasha et al [5], and others. Solving nonlinear FDEs and nonlinear FPDEs is not an easy matter.…”

Section: Introductionmentioning

confidence: 99%

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An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Alaroud

Tahat

Al‐Omari

et al. 2021

Journal of Function Spaces

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Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Alaroud

Tahat

Al‐Omari

et al. 2021

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

“…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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“…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: 99%

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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An Iterative Method for Time-Fractional Swift-Hohenberg Equation

Cited by 14 publications

References 36 publications

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Residual Power Series Method for Fractional Swift–Hohenberg Equation

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

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