Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method.

Impact of Wall Waviness on the Convection Patterns Inside a Horizontal Porous Annulus

A numerical simulation of natural convection in a horizontal porous annulus with an internal wavy surface has been carried out. Governing equations include the heat equation and the hydrodynamics equations under Darcy law and Boussinesq approximation. The mathematical model formulated with the temperature-stream function is solved numerically using the finite difference method with the alternating direction implicit scheme. Results show that the heat transfer and the convective instability depend strongly on the waviness of the inner cylinder. Last, streamlines and isotherms are presented for different waviness parameters of the internal wall for better understanding of the influence of amplitude and number of undulations on the thermo-convective instabilities.

The Solution of 2D Parabolic Equations Based on the Alternating Direction Implicit Scheme

The solution of 2D parabolic equations based on the alternating direction implicit scheme is one of the important methods to solve high dimension problems.A alternating direction implicit scheme is presented in this paper, the stability and convergence of the alternating direction implicit scheme is to be proved, and the scheme of the errors are analyzed. Through the numerical experiment, the result shows that the method has good stability and high precision.