Consistent travelling waves solutions to the non-linear time fractional Klein–Gordon and Sine-Gordon equations through extended tanh-function approach

Abstract

In this study, the extended tanh-function method has been used to find further general travelling wave solutions for space-time fractional nonlinear partial differential equations, namely, the time fractional nonlinear Sine-Gordon equation and Klein–Gordon equation. The mentioned equations are useful for explaining a series of experimental seismic data, modelling of strain waves, and structures correlated to fault dynamics and the subduction slab, as well as slow earthquakes, episodic tremor, slow slip events, and tremor migration patterns. This method is used to attain exact solutions to a variety of nonlinear partial differential equations that were spatially temporal fractional. To get analytical solutions of the travelling wave type of certain nonlinear evolution equations, a power series in tanh was first utilized as an ansatz. These nonlinear fractional partial differential equations (NLFPDEs) are solved on a variety of non-rectangular domains. By using the fractional complex transform and the properties of the confirmable derivative, the two equations are reduced to ordinary differential equations. The solutions are sketched in 3D, and contour patterns, including king type, single soliton, double solitons, multiple solitons, bell shape, and other sorts of solutions. The achieved solutions are very much effective for explaining the above-stated phenomena.