On The QTT Method For Numerical Solution Of The Fractional Diffusion-wave Equation

Fractional differential equations are widely used in the field of scientific com-puting,and many phenomena in physics can be described by fractional differential equations.In particular,fractional diffusion-wave equation can accurately de-scribe anomalous diffusion phenomena.The QTT method for differential equations based on finite difference has been studied,but it is an integer differential equa-tion.The QTT method of fractional differential equations has not been studied at present.The key to solve this problem lies in the construction of QTT decomposi-tion of Caputo fractional derivative operator and the QTT decomposition of space compact format operator.In this paper,we study the QTT numerical algorithm of fractional diffusion wave equation.Firstly,the QTT decomposition of the Hankel matrix and the QTT decom-position of the inverse matrix of the Toeplitz matrix and the Hankel matrix are obtained on the basis of the QTT decomposition of the Toeplitz matrix.And the relationship between the QTT decomposition of the Toeplitz matrix and the QTT decomposition of the Hankel matrix is pointed out,and the two can be easily transformed into each other.Secondly,the discrete format of fractional diffusion-wave equation is Ca-puto fractional derivative operator,Laplace operator and space compact difference scheme operator.In this paper,we show the low rank QTT representation of the Caputo fractional derivative operator and the space compact difference scheme operator.Based on the particularity of the matrix structure of the fractional diffusion-wave equation,we obtain the low rank QTT representation of the equa-tion.Finally,the DMRG method is used to solve the fractional diffusion-wave equa-tion,and the fast numerical algorithm is obtained.Numerical experiments show that the QTT decomposition method is a powerful tool to solve this kind of equa-tion.