Communications on Applied Mathematics and Computation

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T_M-Eigenvalues of Odd-Order Tensors

M. Pakmanesh, Hamidreza Afshin

Communications on Applied Mathematics and Computation 2022, 4 (4): 1258-1279. DOI: 10.1007/s42967-021-00172-z

Abstract （7491）

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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.

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Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

Hong Xiong, Maoning Tang, Qingxin Meng

Communications on Applied Mathematics and Computation 2022, 4 (4): 1386-1415. DOI: 10.1007/s42967-021-00181-y

Abstract （3657）

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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.

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Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence

Mohammed Homod Hashim, Akil J. Harfash

Communications on Applied Mathematics and Computation 2022, 4 (3): 1011-1056. DOI: 10.1007/s42967-021-00124-7

Abstract （3600）

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In this paper, a finite element scheme for the attraction-repulsion chemotaxis model is analyzed. We introduce a regularized problem of the truncated system. Then we obtain some a priori estimates of the regularized functions, independent of the regularization parameter, via deriving a well-defined entropy inequality of the regularized problem. Also, we propose a practical fully discrete finite element approximation of the regularized problem. Next, we use a fixed point theorem to show the existence of the approximate solutions. Moreover, a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived. In addition, the uniqueness of the fully discrete approximations is preformed. Finally, we discuss the convergence to the fully discrete problem.

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Perturbation Analysis for t-Product-Based Tensor Inverse, Moore-Penrose Inverse and Tensor System

Zhengbang Cao, Pengpeng Xie

Communications on Applied Mathematics and Computation 2022, 4 (4): 1441-1456. DOI: 10.1007/s42967-022-00186-1

Abstract （3556）

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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.

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Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu

Communications on Applied Mathematics and Computation 2022, 4 (4): 1509-1530. DOI: 10.1007/s42967-022-00193-2

Abstract （3532）

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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(h⁴ + τ²) with mesh-size h and time-step τ. 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.

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Space-Fractional Diffusion with Variable Order and Diffusivity:Discretization and Direct Solution Strategies

Hasnaa Alzahrani, George Turkiyyah, Omar Knio, David Keyes

Communications on Applied Mathematics and Computation 2022, 4 (4): 1416-1440. DOI: 10.1007/s42967-021-00184-9

Abstract （3495）

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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(N²)) 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.

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Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations

Xiaoying Han, Habib N. Najm

Communications on Applied Mathematics and Computation 2022, 4 (4): 1457-1493. DOI: 10.1007/s42967-022-00188-z

Abstract （3495）

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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.

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Parametric Regression Approach for Gompertz Survival Times with Competing Risks

H. Rehman, N. Chandra

Communications on Applied Mathematics and Computation 2022, 4 (4): 1175-1190. DOI: 10.1007/s42967-021-00154-1

Abstract （3447）

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Regression models play a vital role in the study of data regarding survival of subjects. The Cox proportional hazards model for regression analysis has been frequently used in survival modelling. In survival studies, it is also possible that survival time may occur with multiple occurrences of event or competing risks. The situation of competing risks arises when there are more than one mutually exclusive causes of death (or failure) for the person (or subject). In this paper, we developed a parametric regression model using Gompertz distribution via the Cox proportional hazards model with competing risks. We discussed point and interval estimation of unknown parameters and cumulative cause-specific hazard function with maximum-likelihood method and Bayesian method of estimation. The Bayes estimates are obtained based on non-informative priors and symmetric as well as asymmetric loss functions. To observe the finite sample behaviour of the proposed model under both estimation procedures, we carried out a Monte Carlo simulation analysis. To demonstrate our methodology, we also included real data analysis.

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Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Oke Davies Adeyemo, Chaudry Masood Khalique

Communications on Applied Mathematics and Computation 2022, 4 (4): 1531-1582. DOI: 10.1007/s42967-022-00195-0

Abstract （3444）

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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.

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Dual Quaternions and Dual Quaternion Vectors

Liqun Qi, Chen Ling, Hong Yan

Communications on Applied Mathematics and Computation 2022, 4 (4): 1494-1508. DOI: 10.1007/s42967-022-00189-y

Abstract （3442）

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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, ∞-norm, and 2-norm to dual quaternion vectors.

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A Mixed Finite-Element Method on Polytopal Mesh

Yanping Lin, Xiu Ye, Shangyou Zhang

Communications on Applied Mathematics and Computation 2022, 4 (4): 1374-1385. DOI: 10.1007/s42967-021-00180-z

Abstract （3317）

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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.

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Reconstruction of a Heat Equation from One Point Observations

H. Al Attas, A. Boumenir

Communications on Applied Mathematics and Computation 2022, 4 (4): 1280-1292. DOI: 10.1007/s42967-021-00174-x

Abstract （3225）

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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.

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Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson

Communications on Applied Mathematics and Computation 2022, 4 (4): 1191-1228. DOI: 10.1007/s42967-021-00159-w

Abstract （3217）

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We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

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An Alternative to the Marshall-Olkin Family of Distributions:Bootstrap, Regression and Applications

Christophe Chesneau, Kadir Karakaya, Hassan S. Bakouch, Coşkun Kuş

Communications on Applied Mathematics and Computation 2022, 4 (4): 1229-1257. DOI: 10.1007/s42967-021-00167-w

Abstract （3198）

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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.

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Some Numerical Extrapolation Methods for the Fractional Sub-diffusion Equation and Fractional Wave Equation Based on the L1 Formula

Ren-jun Qi, Zhi-zhong Sun

Communications on Applied Mathematics and Computation 2022, 4 (4): 1313-1350. DOI: 10.1007/s42967-021-00177-8

Abstract （3190）

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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_∞-norm are proved to be 2, 3-α, and 4-2α, respectively, where 0 < α < 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-γ and 6-2γ order in L_∞-norm, respectively, where 1 < γ < 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.

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P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience

F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi

Communications on Applied Mathematics and Computation 2022, 4 (4): 1293-1312. DOI: 10.1007/s42967-021-00176-9

Abstract （3178）

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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.

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A Stable Numerical Scheme Based on the Hybridized Discontinuous Galerkin Method for the Ito-Type Coupled KdV System

Shima Baharlouei, Reza Mokhtari, Nabi Chegini

Communications on Applied Mathematics and Computation 2022, 4 (4): 1351-1373. DOI: 10.1007/s42967-021-00178-7

Abstract （3122）

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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 L₂ 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.

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Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

Jiawei Sun, Shusen Xie, Yulong Xing

Communications on Applied Mathematics and Computation 2022, 4 (2): 381-416. DOI: 10.1007/s42967-021-00119-4

Abstract （2355）

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Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, the BBM-BBM system, the Bona-Smith system, etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

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A General Fractional Pollution Model for Lakes

Babak Shiri, Dumitru Baleanu

Communications on Applied Mathematics and Computation 2022, 4 (3): 1105-1130. DOI: 10.1007/s42967-021-00135-4

Abstract （2010）

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A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.

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High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu

Communications on Applied Mathematics and Computation 2023, 5 (1): 485-528. DOI: 10.1007/s42967-021-00183-w

Abstract （1990）

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A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. The exact C-property is investigated, and comparison with the standard finite difference WENO scheme is made. Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems. The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems, indicating smaller errors compared with the Lax-Friedrichs solver. In addition, we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

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