Novel approaches for nonlinear time-fractional Telegraph equation through stability analysis

In recent years, many researchers have explored the use of fractional partial differential equations (FPDEs) to model real-world phenomena through transport of porous media in engineering and applied sciences. Investigating nonlinear time-fractional Telegraph equation with quadratic nonlinearity, we transform the partial differential equation (PDE) into ordinary differential equation (ODE) by using the stability analysis. The obtained solutions con- tain different ways of exponential functions. Our method uses stability analysis to solve the complex equations easily and their applications in physics, mathematics and engineering. This indicates how system behaves near an equilibrium point and underlying the models are stable and unstable using the concept of stability. Signifficant outcomes are obtained from the various solution families. To examine the impact of several gained parameters, the propagation behavior of the obtained data is shown in 2D, 3D, and contour representations. The signifficance of physical situations will be better understood by the researchers to these findings. The nonlinear time-fractional Telegraph equations play a central role in many areas of electrical engineering for transport in porous media. This paper reports that our research contributes to the existing litera- ture, offering a fresh perspective on these equations and concrete the way for future research for transport in porous media. Our approach provides stability analysis with a high accuracy for two-dimensional plane autonomous systems.