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