Related papers: On the matrices in B-spline collocation methods fo…
The analysis of the spectral features of a Toeplitz matrix-sequence $\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\in L^1([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided in great detail in the last…
In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional…
Fractional boundary value problems are often used to model complex systems and processes characterized by memory effects and anomalous diffusion. In this paper, we consider fractional boundary value problems involving the Riesz-Caputo…
By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schr\"odinger equations leads to a…
We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the…
It is well known that square matrices with independent and identically distributed (iid) random entries are typically well conditioned. A natural question is whether this favorable behavior persists for random matrices whose entries obey…
The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the…
The high-order numerical analysis for fractional Laplacian via the Riesz fractional derivative, under the low regularity solution, has presented significant challenges in the past decades. To fill in this gap, we design a grid mapping…
In this paper, we develop a fast numerical method for solving the time-dependent Riesz space fractional diffusion equations with a nonlinear source term in the convex domain. An implicit finite difference method is employed to discretize…
The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and…
We consider the task of approximating a matrix function $f(A)$, where $A$ is a matrix in which only a relatively small number of (not necessarily consecutive) sub- and superdiagonals contain nonzero entries. Approximating $f$ by a…
We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our…
In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations…
In this paper, we study a $\tau$-matrix approximation based preconditioner for the linear systems arising from discretization of unsteady state Riesz space fractional diffusion equation with non-separable variable coefficients. The…
In the present study, we consider the numerical method for Toeplitz-like linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). We apply the Crank-Nicolson (CN) technique to discretize the…
We used a collocation method in refinable spline space to solve a linear dynamical system having fractional derivative in time. The method takes advantage of an explicit derivation rule for the B-spline basis that allows us to efficiently…
We study the Dirichlet problem for a class of fractional $p$-Laplacian operators of order $s \in (0,1)$ defined through the Riesz fractional gradient, which differs fundamentally from the standard fractional $p$-Laplacian. Our analysis…
In this work, a new collocation approach using a combination of a wavelet operational matrix method and the exponential spline interpolation is proposed to solve the time-fractional convection-diffusion equation with variable coefficients.…
The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and…
In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the…