The Novel Analytical Methods for the Solutions of Generalized-fractional Order Differential Equations Arising in Physical Models

Abstract

Fractional calculus is a branch of calculus that generalizes the derivative of a function to arbitrary order. In contemporary years, fractional calculus has become the focus of curiosity for many researchers in exclusive disciplines of applied science and engineering because its application across diverse disciplines of applied science and engineering for the description of properties of various real physical phenomena.

So, the main objective of this dissertation is to present an extensive study of different semi-analytical and analytical methods for obtaining approximate and exact solutions of numerous nonlinear fractional differential equations appearing in disciplines of science and engineering.

Therefore, in the present dissertation, various semi-analytical methods like homotopy perturbation method (HPM), homotopy perturbation transform method (HPTM), modified homotopy analysis method (MHAM) and modified homotopy analysis method with Fourier transform method (MHAM-FT), successive recursion method have been utilized for getting approximate solutions for nonlinear fractional differential equations. Also, some analytical methods viz., fractional-sub equation method, improved fractional-sub equation method, (G'/G)-expansion method, improved (G'/G)-expansion method, proposed tanh method, modified Kudryashov method, Jacobi elliptic function method, and other new proposed method have been successfully employed to numerous fractional differential equations for getting exact solutions.

Moreover, by applying semi-analytical methods, the approximate solutions of the nonlinear fractional differential equations viz. time-fractional Lotka-Volterra equations of prey predator model, fractional coupled Klein-Gordon-Schrödinger (K-G-S) equations, fractional coupled Klein-Gordon-Zakharov (K-G-Z) equations, fractional coupled sine-Gordon equations, Riesz fractional diffusion equation (RFDE), Riesz fractional advection–dispersion equation (RFADE), Riesz time-fractional Camassa–Holm equation, variable order spring-mass damper systems have been discussed in the present work.

Furthermore, the exact solutions of some nolinear fractional differential equations viz. space-time fractional Zakharov-Kuznetsov (ZK) equation, space-time fractional modified Zakharov-Kuznetsov (mZK) equation, time fractional (3+1) dimensional KdV-Zakharov-Kuznetsov, space-time fractional (3+1) dimensional modified KdV-Zakharov-Kuznetsov equations, time fractional modified Kawahara equation, fractional coupled Jaulent-Miodek (JM) equation, time fractional modified KdV equation, the time fractional Kaup-Kupershmidt equation, time-fractional fifth-order Sawada–Kotera equation (S-K), time-fractional fifth-order modified Sawada–Kotera (mS-K) equation, time-fractional fifth-order Kuramoto-Sivashinsky (K-S) equations, time fractional time-fractional KdV-Burgers equation, time-fractional KdV-mKdV equation, time-fractional coupled Schrödinger–KdV equation, time-fractional coupled Schrödinger–Boussinesq equations and time-fractional coupled Drinfeld–Sokolov–Wilson equations have been presented by using various analytical methods.

Also, it can be established that the semi-analytical and analytical methods provide worthy approximate and exact analytical solutions for fractional order partial differential equations. Also, it is worthwhile to mention that the proposed semi-analytical and analytical methods are promising and powerful methods for solving fractional differential equations in mathematical physics.