A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Urban growth management for sustainable urbanization : examples from Asia-Pacific city regions

During the past three decades cities in the Asia-Pacific region have undergone massive transformations, characterised by rapid population growth and urbanisation. The rapid pace of globalisation and economic restructuring has resulted in these cities receiving the full impact of urbanisation pressures. In attempting to ease these pressures, major cities have advocated growth management approaches that give particular interest to sustainable urbanization and emphasise compact and optimum development of urban forms. This paper seeks to provide an insight into sustainable urbanisation practice, particularly on the promotion of compact urbanisation within Asia-Pacific’s fastest growing regions. The finding shows that within the context of resource constraints, sustainable urbanisation has been a key factor in the adoption of urban growth management initiatives promoting viable use of scarce resources for urban expansion.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Vitronectin modulates human mesenchymal stem cell response to insulin-like growth factor-I and transforming growth factor beta 1 in a serum-free environment

The use of animal sera for the culture of therapeutically important cells impedes the clinical use of the cells. We sought to characterize the functional response of human mesenchymal stem cells (hMSCs) to specific proteins known to exist in bone tissue with a view to eliminating the requirement of animal sera. Insulin-like growth factor-I (IGF-I), via IGF binding protein-3 or -5 (IGFBP-3 or -5) and transforming growth factor-beta 1 (TGF-beta(1)) are known to associate with the extracellular matrix (ECM) protein vitronectin (VN) and elicit functional responses in a range of cell types in vitro. We found that specific combinations of VN, IGFBP-3 or -5, and IGF-I or TGF-beta(1) could stimulate initial functional responses in hMSCs and that IGF-I or TGF-beta(1) induced hMSC aggregation, but VN concentration modulated this effect. We speculated that the aggregation effect may be due to endogenous protease activity, although we found that neither IGF-I nor TGF-beta(1) affected the functional expression of matrix metalloprotease-2 or -9, two common proteases expressed by hMSCs. In summary, combinations of the ECM and growth factors described herein may form the basis of defined cell culture media supplements, although the effect of endogenous protease expression on the function of such proteins requires investigation.