Analytical Solutions and Efficient Compact Schemes of the Caputo-Hadamard Time Fractional Model for Nanofluid

In this paper, we consider the Caputo-Hadamard time fractional model for unsteady MHD mixed convection problem of Maxwell fractional nanofluid. Two kinds of boundary conditions are proposed, which are Dirichlet boundary conditions and Robin boundary conditions. The analytical solutions for the equation are derived by using the method of separation of variables and the generalized Laplace transform. The numerical schemes are developed based on the L1 and L1-2 approximations for the Caputo-Hadamard fractional derivatives and the compact finite difference scheme for spatial derivative. Stability and convergence of the obtained numerical scheme are analyzed theoretically. Numerical examples are constructed to demonstrate the accuracy of the numerical scheme as compared to theoretical analysis.