Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques.

Basically enzymes are biological catalysts that increase the speed of a chemical reaction without undergoing any permanent chemical change. With the application of Euler’s forward scheme, a discrete-time enzyme model is presented. Further investigation related to its qualitative behaviour revealed that discrete-time model shows rich dynamics as compared to its continuous counterpart. It is investigated that discrete-time model has a unique trivial equilibrium point. The local asymptotic behaviour of equilibrium is discussed for discrete-time enzyme model. Furthermore, with the help of the bifurcation theory and centre manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two existing chaos control methods, that is, Ott, Grebogi and Yorke feedback control and hybrid control strategy, are implemented. In particular, a novel chaos control technique, based on state feedback control is introduced for controlling chaos under the influence of flip and Hopf bifurcations in discrete-time enzyme model. Numerical simulations are provided to illustrate theoretical discussion and effectiveness of newly introduced chaos control method.

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of $$\exp \,\left( { - \varphi \left( \eta \right)} \right)$$ exp - φ η -expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.