Abstract
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.
Highlights
Partial fractional models are a natural generalization of classical multivariable models with arbitrary order derivate that have received great interest in the scientific community due to their diverse applications in engineering, physics, pharmacology, astronomy, and medicine
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]
Considerable attention has been paid to the topic of fractional differential equations (FDEs) and fractional partial differential equations (FPDEs) due to the fast-growing and widespread of their applications in various Journal of Function Spaces science and engineering areas like medicine, chemistry, biology, electrical engineering, and viscoelasticity, and for more details about these applications and others, we refer to [8,9,10,11,12,13,14,15]
Summary
Partial fractional models are a natural generalization of classical multivariable models with arbitrary order derivate that have received great interest in the scientific community due to their diverse applications in engineering, physics, pharmacology, astronomy, and medicine. The aforesaid standard traditional techniques and others need more computational time and computer memory to determine the closed-form solutions for the nonlinear problems To beat these defects, some scholars coupled the standard existing approaches and the Laplace transform approach like Kumar in [34] investigated the solution for the time-fractional Cauchy-reaction diffusion equation by using the homotopy perturbation transform approach. The motivation of this work is to construct an approximate analytical solution for the nonlinear time-fractional S-H Equation (2) by directly applying the Laplace residual power series (LRPS) technique.
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