A novel distributed order time fractional model for heat conduction, anomalous diffusion, and viscoelastic flow problems

Abstract

A novel distributed order time fractional model is constructed to solve heat conduction, anomalous diffusion and viscoelastic flow problems. Solutions of the formulated governing equation are obtained by the numerical discretization method that the mid-point quadrature rule is applied to approach the distributed order integrals and the L1-formula and L2-formula are used to discrete the time fractional derivatives. The stability and convergence of difference scheme with zero boundary conditions are studied and proven. In the end, three numerical examples are given. By employing an optimization algorithm, we obtain a good fitting result between the computational data with our model and heat conduction experimental data, which validates the effectiveness of the proposed model. Based on the background of the anomalous diffusion of volatile organic compound (VOC) adsorbed on building materials, the second example analyses the error and the convergence order by comparing the discrete solution with the exact solution, which validates the accuracy of the numerical difference method. Based on the solutions by using the difference method, the last example studies the unidirectional flows of generalized Oldroyd-B fluid driven by the moving plate or the pressure gradient and the effects of involved parameters on the velocity distribution are discussed graphically.