LIE SYMMETRY ANALYSIS AND EXACT SOLUTIONS FOR CONFORMABLE TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

In this paper, the Cauchy problems of a class of conformable time fractional nonlinear partial differential equations (PDEs) are investigated and the exact solutions are constructed with the help of Hodograph transformations and the fundamental solutions of the linear PDEs. Specifically, the Hodograph transformations are introduced first to convert the considered time fractional nonlinear PDEs into their corresponding linear PDEs, and the connection between the Cauchy problems of the nonlinear PDEs before and after the transformations is verified. In addition, it is demonstrated that the generalized Laplace transforms or Fourier transforms of the fundamental solutions can be generated based on the nontrivial Lie symmetries admitted by the considered fractional linear PDEs. Then, the fundamental solutions to the fractional linear PDEs are established by inverting the obtained transforms. Finally, by means of the fundamental solutions mentioned above and Hodograph transformations, the solutions to the Cauchy problems of nonlinear PDEs are formulated explicitly.