Well-posedness of the initial-boundary value problems for the time-fractional degenerate diffusion equations

Abstract

This paper deals with the solving of initial-boundary value problems for the one-dimensional linear timefractional diffusion equations with time-degenerate diffusive coefficients t^β with β>1-α. The solutions to initial-boundary value problems for the one-dimensional time-fractional degenerate diffusion equations with Riemann-Liouville fractional integral I^1-α_0+,t of order α∈(0,1) and with Riemann-Liouville fractional derivative D^α_0+,t of order α∈(0,1) in the variable, are shown. The solutions to these fractional diffusive equations are presented using the Kilbas-Saigo function E_α,m,l(z). The solution to the problems is discovered by the method of separation of variables, through finding two problems with one variable. Rather, through finding a solution to the fractional problem depending on the parameter t, with the Dirichlet or Neumann boundary conditions. The solution to the Sturm-Liouville problem depends on the variable x with the initial fractional-integral Riemann-Liouville condition. The existence and uniqueness of the solution to the problem are confirmed. The convergence of the solution was evidenced using the estimate for the KilbasSaigo function E_α,m,l(z) from and by Parseval’s identity.