The well-posedness and exact solution of fractional magnetohydrodynamic equations

Abstract

For numerous fluids between elastic and viscous materials, the fractional magnetohydrodynamic models have an advantage over the integer-order models. We study the fractional magnetohydrodynamic equations with external force. On the basis of conformable fractional derivative and the respective useful properties, we study the well-posedness of the incompressible time fractional magnetohydrodynamic equations. In the qualitative study, the complicate nonlinear terms in time fractional magnetohydrodynamic equations are technical handled. The existence of the solution to the fractional incompressible magnetohydrodynamic equations is obtained. Moreover, the uniqueness and continuous dependence of the solution on the initial data are proved. In the quantitative study, using some mathematical transformation with practical physical significance, the exact solutions and the respective figures of the fractional magnetohydrodynamic equations are presented. The time fractional derivative system possesses property of time memory. The fractional derivative influences the control effect in complex systems with fluids between different elastic and viscous materials. The results are still valid if the time order is integer (integer-order magnetohydrodynamic equations case), or the magnetic field is zero (Navier–Stokes equations case).