Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

Abstract

Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch–Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch–Torrey equation are still limited. In this paper, the space and time fractional Bloch–Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

“…Recently, fractional order calculus has been used to examine the connection between fractional order dynamics and diffusion by solving the Bloch-Torrey equation [22][23][24][25]. It was pointed out that a fractional diffusion model could be successfully applied to analyzing diffusion images of human brain tissues and provides new insights into further investigations of other tissue structures and the micro-environment.…”

“…Zhou et al [31] applied the results from [30] to analyze diffusion images of healthy human brain tissues in vivo successfully at high values up to 4700 / 2 . Yu et al [23] derived an analytical solution and an effective implicit numerical method for solving equation (1), and also considered the stability and convergence properties of the implicit numerical method. However, due to computational overheads necessary to perform the simulations for ST-FBTE in three dimensions, Yu et al [23] presented a preliminary study based on a two-dimensional example to confirm their theoretical analysis.…”

“…Yu et al [23] derived an analytical solution and an effective implicit numerical method for solving equation (1), and also considered the stability and convergence properties of the implicit numerical method. However, due to computational overheads necessary to perform the simulations for ST-FBTE in three dimensions, Yu et al [23] presented a preliminary study based on a two-dimensional example to confirm their theoretical analysis. Yu et al [22] proposed a fractional alternating direction implicit scheme to overcome this problem, they also proved the stability and convergence of the proposed method with order of convergence one in space.…”

“…Yu et al [22] proposed a fractional alternating direction implicit scheme to overcome this problem, they also proved the stability and convergence of the proposed method with order of convergence one in space. For the Riesz fractional formulation, the Gr¨ nwaldLetnikov derivative approximation scheme of order one can be used [22,23,[30][31][32]. However, in order to better approximate the Riesz fractional derivative, Ortigueira [33] defined a 'fractional centered derivative' and proved that the Riesz fractional derivative of an analytic function can be represented by the fractional centered derivative.…”

Fractional calculus and its applications

“…The rest of the papers can be roughly grouped into three parts: three papers for fundamental theories of fractional calculus [7][8][9], five papers for fractional modelling with applications [10][11][12][13][14] and four papers for numerical approaches [15][16][17][18].…”