This paper introduces a new rich family of distributions based on mixtures and the so-called Marshall-Olkin family of distributions. It includes a wide variety of well-established mixture distributions, ensuring a high ability for data fitting. Some distributional properties are derived for the general family. The Weibull distribution is then considered as the baseline, exhibiting a pliant four-parameter lifetime distribution. Five estimation methods for the related parameters are discussed. Bootstrap confidence intervals are also considered for these parameters. The distribution is reparametrized with location-scale parameters and it is used for a lifetime regression analysis. An extensive simulation is carried out on the estimation methods for distribution parameters and regression model parameters. Applications are given to two practical data sets to illustrate the applicability of the new family.
In this paper, we propose a definition for eigenvalues of odd-order tensors based on some operators. Also, we define the Schur form and the Jordan canonical form of such tensors, and discuss commuting families of tensors. Furthermore, we prove some eigenvalue inequalities for Hermitian tensors. Finally, we introduce characteristic polynomials of odd-order tensors.
We are concerned with the reconstruction of the heat sink coefficient in a one-dimensional heat equation from the observations of solutions at the same point. This direct method which is based on spectral estimation and asymptotics techniques provides a fast algorithm and also an alternative to the Gelfand-Levitan theory or minimization procedures.
This study aims to determine the phenomenological bifurcation (P-bifurcation) occurring in the van der Pol (VDP) neuronal model of burst neurons with a random signal. We observe the P-bifurcation under an intense noise stimulus which would become chaotic transitions. Bursting and spiking simulations are used to describe the causes of chaotic transitions between two periodic phases that are the reason for the neuronal activities. Randomness plays a crucial role in detecting the P-bifurcation. To determine the equilibrium points, stability or instability of the stochastic VDP equation, and bifurcation, we use the stochastic averaging method and some related theorems. Apart from theoretical methods, we also examine numerical simulations in the particular case of that stochastic equation that illustrates the outcome of theorems for various noise types. The most striking part of our theoretical findings is that these results are also valid for the Izhikevich-FitzHugh model, Bonhoeffer-van der Pol oscillator in dynamical systems of neuroscience. Finally, we will discuss some applications of the VDP equation in neuronal activity.
With the help of the asymptotic expansion for the classic L1 formula and based on the L1-type compact difference scheme, we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation. Three extrapolation formulas are presented, whose temporal convergence orders in $L_\infty$-norm are proved to be 2, $3-\alpha$, and $4-2\alpha$, respectively, where $0<\alpha <1$. Similarly, by the method of order reduction, an extrapolation method is constructed for the fractional wave equation including two extrapolation formulas, which achieve temporal $4-\gamma$ and $6-2\gamma$ order in $L_\infty$-norm, respectively, where $1<\gamma <2$. Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation, the fast extrapolation methods are obtained which reduce the computational complexity significantly while keeping the accuracy. Several numerical experiments confirm the theoretical results.
The purpose of this paper is to develop a hybridized discontinuous Galerkin (HDG) method for solving the Ito-type coupled KdV system. In fact, we use the HDG method for discretizing the space variable and the backward Euler explicit method for the time variable. To linearize the system, the time-lagging approach is also applied. The numerical stability of the method in the sense of the $L_2$ norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet boundary conditions. Numerical experiments confirm that the HDG method is capable of solving the system efficiently. It is observed that the best possible rate of convergence is achieved by the HDG method. Also, it is being illustrated numerically that the corresponding conservation laws are satisfied for the approximate solutions of the Ito-type coupled KdV system. Thanks to the numerical experiments, it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.
In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.
This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels martingales associated with Lévy processes. In either case, we obtain the optimality system for the optimal controls in open-loop form, and by means of a decoupling technique, we obtain the optimal controls in closed-loop form which can be represented by two Riccati differential equations. Moreover, the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.
We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due to the kernel singularity in the fractional integral operator and the resulting dense discretized operators, which quickly become prohibitively expensive to handle because of their memory and arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. We also present a block low rank representation to handle the dense matrix representations, by exploiting the ability to approximate blocks of the resulting formally dense matrix by low rank factorizations. A Cholesky factorization solver operates directly on this representation using the low rank blocks as its atomic computational tiles, and achieves high performance on multicore hardware. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. They also show that considerable savings are obtained in storage ($O(N^{1.5})$) and computational cost ($O(N^2)$) compared to dense factorizations. This translates to orders-of-magnitude savings in memory and time on multidimensional problems, and shows that the proposed methods offer practical tools for tackling large nonlocal fractional diffusion simulations.
This paper establishes some perturbation analysis for the tensor inverse, the tensor Moore-Penrose inverse, and the tensor system based on the t-product. In the settings of structured perturbations, we generalize the Sherman-Morrison-Woodbury (SMW) formula to the t-product tensor scenarios. The SMW formula can be used to perform the sensitivity analysis for a multilinear system of equations.
Numerical algorithms for stiff stochastic differential equations are developed using linear approximations of the fast diffusion processes, under the assumption of decoupling between fast and slow processes. Three numerical schemes are proposed, all of which are based on the linearized formulation albeit with different degrees of approximation. The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems. Convergence analysis is conducted for one of the schemes, that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1. Approximations arriving at the other two schemes are discussed. Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.
We introduce a total order and an absolute value function for dual numbers. The absolute value function of dual numbers takes dual number values, and has properties similar to those of the absolute value function of real numbers. We define the magnitude of a dual quaternion, as a dual number. Based upon these, we extend 1-norm, $\infty$-norm, and 2-norm to dual quaternion vectors.
In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of $O(h^4+\tau ^2)$ with mesh-size h and time-step $\tau $. In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties.
Having realized various significant roles that higher-dimensional nonlinear partial differential equations (NLPDEs) play in engineering, we analytically investigate in this paper, a higher-dimensional soliton equation, with applications particularly in ocean physics and mechatronics (electrical electronics and mechanical) engineering. Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differential equations. In addition, we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved. Further, a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation. This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations (ODEs). On solving the achieved nonlinear differential equations, we secure various solitonic solutions. In consequence, these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures, ranging from periodic, kink and kink-shaped nanopteron, soliton (bright and dark) to breather waves with extensive wave collisions depicted. We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions, two dimensions and density plots. Moreover, the gained group-invariant solutions involved several arbitrary functions, thus exhibiting rich physical structures. We also implore the power series technique to solve part of the complicated differential equations and give valid comments on their results. Later, we outline some applications of our results in ocean physics and mechatronics engineering.