Efficient acoustic scalar wave equation modeling in VTI media

We present a new approach for acoustic wave modeling in transversely isotropic media with a vertical axis of symmetry. This approach is based on using a pure acoustic wave equation derived from the basic physical laws – Hooke’s law and the equation of motion. We show that the conventional equation noted as pure quasi-P wave equation computes only one stress component. In our approach, there is no need to approximate the pseudo-differential operator for decomposition purposes. We make a discrete inverse Fourier transform of the desired frequency response contained in the pseudo-differential operator to build the corresponding spatial operator. We then cut off the operator with a window to reduce edge effects. As a result, the obtained spatial operator is applied locally to the wavefield through a simple convolution. Consequently, we derive an explicit numerical scheme for a pure quasi-P wave mode. The most important advantage of our method lies in its locality, which means that our spatial operator can be applied in any selected region separately. Our approach can be combined with classical fast finite-difference methods when media are isotropic or elliptically anisotropic, therefore avoiding spurious fields and reducing the total computational time and memory. The accuracy, stability, and the absence of the residual S-waves of our approach were demonstrated with several numerical examples.

Asymptotics of fundamental solutions for time fractional equations with convolution kernels

The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation with a convolution type operator. In this equation we use a Caputo time derivative of order α ∈ (0, 1), and assume that the convolution kernel of the spatial operator is symmetric, integrable and shows a super-exponential decay at infinity. Under these assumptions we describe the point-wise asymptotic behavior of the fundamental solution in all space-time regions.

Global Asymptotic Stability and Nonlinear Analysis of the Model of the Square Immunopixels Array Based on Delay Lattice Differential Equations

Biosensors and immunosensors show an increasing attractiveness when developing current cheap and fast monitoring and detecting devices. In this work, a model of immunosensor in a class of delayed lattice differential equations is offered and studied. The spatial operator describes symmetric diffusion processes of antigenes between pixels. The main results are devoted to the qualitative research of the model. The conditions of global asymptotic stability, which are constructed with the help of Lyapunov functionals, determine a lower estimate of the time of immune response. Nonlinear analysis of the model is performed with help of a series of numerical characteristics including autocorrelation function, mutual information, embedding, and correlation dimensions, sample entropy, the largest Lyapunov exponents. We consider the influence of both symmetric and unsymmetric diffusion of antigens between pixels on the qualitative behavior of the system. The outcomes are verified with the help of numerical simulation in cases of 4 × 4 - and 16 × 16 - arrays of immunopixels.

Dual Time-Stepping Using Second Derivatives

We present a modified formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiffness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive effects require that an approximation of the square root of the spatial operator is available and inexpensive.

On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice

In the work, a nonlinear reaction-diffusion model in a class of delayed differential equations on the hexagonal lattice is considered. The system includes a spatial operator of diffusion between hexagonal pixels. The main results deal with the qualitative investigation of the model. The conditions of global asymptotic stability, which are based on the Lyapunov function construction, are obtained. An estimate of the upper bound of time delay, which enables stability, is presented. The numerical study is executed with the help of the bifurcation diagram, phase trajectories, and hexagonal tile portraits. It shows the changes in qualitative behavior with respect to the growth of time delay; namely, starting from the stable focus at small delay values, then through Hopf bifurcation to limit cycles, and finally, through period doublings to deterministic chaos.

Minimum Base Attitude Disturbance Planning for a Space Robot During Target Capture

This paper presents a method to minimize the base attitude disturbance of a space robot during target capture. First, a general dynamic model of a free-floating space robot capturing a target is established using spatial operator Algebra, and a simple analytical formula for the base angular velocity change during the impact phase is obtained. Compared with the former models proposed in the literature, this model has a simpler form, a wider range of applications, and O(n) computation complexity. Second, based on the orthogonal projection matrix lemma, we propose the generalized mass Jacobian matrix (GMJM) and find that the base angular velocity change is a constant multiple of the component which the impact impulse projects to the column space of the GMJM. Third, a new concept, the base attitude disturbance ellipsoid (BADE), is proposed to express the relationship between the base attitude disturbance and the impact direction. The impact direction satisfying the minimum base attitude disturbance can be straightforwardly obtained from the BADE. In particular, for a planar space robot, we draw the useful conclusion that the impact direction unchanged base attitude must exist. Furthermore, the average axial length of the BADE is used as a measurement to illustrate the average base attitude disturbance under impact impulses from different directions. With this measurement, the desired pre-impact configuration with minimum average base attitude disturbance can be easily determined. The validity and the efficiency of this method are verified using a three-link planar space robot and a 7DOF space robot.

Numerical Approximation of Space-Time Fractional Parabolic Equations

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator {E(t)} for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function {e_{\alpha,1}} and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator {W(t)} and the forcing function {F(t)}. We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of {\frac{1}{h}}. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves {k^{-2}} nodes and results in an additional {O(\exp(-\frac{c}{k}))} error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of {W_{h}(s)}, (the semi-discrete approximation to {W(s)}) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of {O({\mathcal{N}}\log({\mathcal{N}}))} intervals, the error between the semi-discrete and fully discrete approximation is {O({\mathcal{N}}^{-2}+\log({\mathcal{N}})\exp(-\frac{c}{k}))}. We also report the results of numerical experiments that are in agreement with the theoretical error estimates.