Results in Applied Mathematics最新文献

This paper aims to study a numerical method to solve a transient eddy current problem involving velocity terms in a bounded domain including conductor and insulator regions. For this purpose, we show that the formulation admits a well-posed saddle point structure given by the curl-free condition for the magnetic field in the insulator domain. We propose a full discretization based on a backward Euler method in time variable and finite element method in space variable. Then, we use Nédélec edge element on the tetrahedral meshes, for which we obtain error estimates. For numerical purposes we used a block-Krylov method to solve the linear system of equations obtained in the fully discretization. Finally, we present some numerical results to validate the theoretical findings obtained.

The aim of this article is to probe the convergence analysis of an efficient scheme, developed by Jentzen et al. (2011), for the stochastic Burgers’ equation (SBE) with term of additive noise. Although, the same scheme was used by Blomker et al. (2013) to carry out the full discretization of the SBE. But therein, Taylor series was not applied. In this work, Taylor series in integral form with remainder after one term is applied. As a consequence, minimum convergence order in time is updated to $3\theta$ from $\theta$, where $\theta \in (0,\frac{1}{2})$. Although, minimum temporal convergence order is proved to be as $2\theta$ by Khan (2021) using the higher order scheme. But the proposed scheme is simple in a manner that former uses two linear functionals of noise, whereas later employs single linear functional of noise. Finally, run time of the existing and the proposed scheme are compared to justify the analytical outcomes.

The complete elliptic integral of the first kind (CEI-1) plays a significant role in mathematics, physics and engineering. There is no simple formula for its computation, thus numerical algorithms are essential for coping with the practical problems involved. The commercial implementations for the numerical solutions, such as the functions ellipticK and EllipticK provided by MATLAB and Mathematica respectively, are based on $K_{\mathrm{cs}}\left(m\right)$ instead of the usual form $K\left(k\right)$ such that $K_{\mathrm{cs}}\left(k^2\right)=K\left(k\right)$ and $m=k^2$. It is necessary to develop open source implementations for the computation of the CEI-1 in order to avoid potential risks of using commercial software and possible limitations due to the unknown factors. In this paper, the infinite series method, arithmetic-geometric mean (AGM) method, Gauss–Chebyshev method and Gauss–Legendre methods are discussed in details with a top-down strategy. The four key algorithms for computing the CEI-1 are designed, verified, validated and tested, which can be utilized in R& D and be reused properly. Numerical results show that our open source implementations based on $K\left(k\right)$ are equivalent to the commercial implementation based on $K_{\mathrm{cs}}\left(m\right)$. The general algorithms for computing orthogonal polynomials developed are valuable for the STEM education and scientific computation.

Schistosomiasis is classified by WHO as a neglected tropical disease. Recent research works have shown that large-scale development projects involving massive population displacement and water irrigation, such as the construction of dams, lakes, and the development of agricultural areas, favour the proliferation of bilharzia. These observations motivate us to propose a reaction–diffusion model to assess the role of the displacements of humans, snails, cercaria, miracidia in the transmission dynamics of Schistosomiasis. The model incorporates a general non-linear contact functions and density-dependent parameters. The aim is to better understanding the role of spatial interactions on the spread of Schistosomiasis, in order to propose appropriate recommendations for the control of that silent threat. We characterize the basic reproduction number $R_0$ of the model. The uniform persistence theory, the maximum principle are used to conduct an in-depth analysis of both the homogeneous and heterogeneous models. Theoretical results are illustrated through numerical simulations.

The Volterra integral equations (VIEs) with oscillatory kernels arise in several applied problems and need to be treated with a computational method have multiple characteristics. In the literature (Zaheer-ud-Din et al., 2022; Li et al., 2012), the Levin method combined with multiquadric radial basis functions (MQ-RBFs) and Chebyshev polynomials are well-known techniques for treating oscillatory integrals and integral equations with oscillatory kernels. The numerical experiments show that the Levin method with MQ-RBFs and Chebyshev polynomials produces dense and ill-conditioned matrices, specifically in the case of large data and high frequency. Therefore, the main task in this study is to combine the Levin method with compactly supported radial basis functions (CS-RBFs), which produce sparse and well-conditioned matrices, and subsequently obtain a stable, efficient, and accurate algorithm to treat VIEs. The theoretical error bounds of the method are derived and verified numerically. Although the error bounds obtained are not improved significantly, alternatively, a stable and efficient algorithm is obtained. Several numerical experiments are performed to validate the capabilities of the proposed method and compare it with counterpart methods (Zaheer-ud-Din et al., 2022; Li et al., 2012).

Computer graphics is a dynamic field that heavily relies on mathematical techniques. For instance, subdivision scheme is used to create smooth and visually appealing curves and surfaces of arbitrary topology. The primary focus of this study is to transform two 5-point binary subdivision schemes into a single 7-point binary subdivision scheme with shape control. We have merged two binary approximating schemes that were constructed using the uniform B-spline basis function and the Lagrange basis function into a new subdivision scheme. It has been demonstrated that, for fixed values of the global shape control parameters, the curves generated by the proposed 7-point binary subdivision scheme maintain $C^6$ continuity everywhere. Furthermore, a brief discussion on the analysis of the Gibbs phenomenon in the new subdivision scheme has been presented. This is also a reminder of the challenges and intricacies involved in computer graphics and geometric modeling.

The stochastic FitzHugh–Nagumo (FHN) neural information transduction model has been widely used in different fields, but there are few numerical studies on this model. In this paper, the stochastic FHN model driven by multiplicative noise is studied based on the spectral Galerkin method. The model is firstly discreted by semi-implicit Euler–Maruyama scheme in time and spectral Galerkin method in space. The error estimation and convergence order are then analyzed. Finally, the one-dimensional and two-dimensional stochastic FHN models are numerically calculated and the convergence order is verified. Moreover, this study promotes the understanding of the information transmission law of neural information transmission model under the influence of stochastic factors.

This work deal with global existence and general decay of solutions of a wave equation with acoustic and fractional boundary conditions coupling by source and delay terms. Under some hypotheses, we study the global existence of the solution and by suitable Lyapunov function the general decay result is proved.

Coastal erosion describes the displacement of sand caused by the movement induced by tides, waves or currents. Some of its wave phenomena are modelled by Helmholtz-type equations. Our purposes, in this paper are, first, to study optimal shapes obstacles to mitigate sand transport under the constraint of the Helmholtz equation. And the second side of this work is related to Dirichlet and Neumann spectral problems. We show the existence of optimal shapes in a general admissible set of quasi open sets. And necessary optimality conditions of first order are given in a regular framework using both shape and topological optimization. Some numerical simulations are given to represent optimal domains.

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of $V_n≔\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{n}\varphi \left(n\right)}$ under some reasonable assumptions, where ${\left\{X_i\right\}}_{i=1}^{\infty }$ is a sequence of random variables and $\varphi$ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).