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In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative ... (0,1), and the second-order space derivatives with the fractional Laplacian ... Traditional approximation of ... requires diagonalization of A, which is very time-consuming for large sparse matrices. The novelty of the authors' proposed numerical schemes is that, using either the finite difference method or the Laplace transform to handle the Caputo time fractional derivative, the solution of the TSFDE-2D is written in terms of a matrix function vector product f(A)b at each time step, where b is a suitably defined vector. They give error bounds for the new methods and illustrate their roles in solving the TSFDE-2D. They also derive the analytical solution of the TSFDE-2D in terms of the Mittag--Leffler function.(ProQuest: ... denotes formulae/symbols omitted.)

In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

This paper presents a method for determining the equivalent circuit parameters of magnetically coupled air-core coils used in wireless power transfer (WPT) systems. The proposed approach enables fast and accurate modeling of inductively coupled energy transfer structures, which is essential for the design and optimization of high-efficiency wireless energy systems. The equivalent circuit of the analyzed system was developed using Cauer circuits, while a two-dimensional (2D) axisymmetric electromagnetic field model was employed to derive the equations. The model was implemented in proprietary software based on the edge-element finite element method (FEM) using the A–V formulation. The A–V formulation combines the magnetic vector potential A and the electric scalar potential V, enabling simultaneous representation of magnetic field distribution and current flow in conducting regions. The eddy currents in the conductors were considered in the electromagnetic field analysis. Simulations were carried out for two operating states: short-circuit and idle. The results were used to determine the parameters of the horizontal and magnetizing branches of the equivalent circuit of considered system and to analyze the frequency dependence of the resistances and inductances of the coupled coil system. The proposed modeling approach provides an effective and energy-oriented tool for the design of wireless power transfer systems with improved efficiency and reduced computational cost. The proposed method reproduces impedance characteristics with an accuracy of 0.2 × 10−3% in the idle state and 1.4 × 10−3% in the short-circuit state compared to the full FEM model, while significantly reducing the computation time.

When using the Arnoldi method for approximating $f(A){\\mathbf b}$, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which has been universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments. [PUBLICATION ABSTRACT]

In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.