Unique solutions, stability and travelling waves for some generalized fractional differential problems

Abstract

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann–Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau–Ginzburg–Higgs equation and the Sine–Gordon one. Some numerical simulations are plotted and a conclusion is given at the end.

Keywords

• caputo derivative
• fixed point
• duffing equation
• conformable derivative
• tanh method
• travelling wave