Results in Applied Mathematics最新文献

In this paper, we study the application of Itô-vector projection [1] to the optimal filtering problem. The algorithm projects one SDE to another, possibly lower dimensional, SDE by minimizing an Itô–Taylor expansion of the local projection error’s $L^2$ norm. We explicitly derive the projection filter equation for a general class of parametric densities, and then specifically apply it to exponential families. We demonstrate that for the case where the measurement drift function is in the span of the natural statistics, the Itô-vector projection filter (IVPF) coincides with the Stratonovich-projection filter (SPF) [2]. We then compare the performance of the IVPF against the SPF (with both being implemented using the Gaussian bijection proposed in [3] and the sparse Gauss–Patterson numerical integration) for two-dimensional optimal filtering problem to show the effectiveness of the proposed algorithm. We vary the measurement drift function to four different functions that are not in the span of natural statistics. Based on one hundred Monte Carlo simulations for each measurement drift, we found that their performances are comparable, with the IVPF potentially offering a slightly more robust performance. However, in our current numerical implementation, the SPF consistently outperforms the IVPF in terms of speed.

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $\Omega$ of $R^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving “mimetic” Poincaré inequalities giving the existence and continuity of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

In this study, the $\psi$-Caputo fractional derivative (as a generalization of the classical Caputo derivative where the fractional derivative is defined with respect to the function $\psi$) is considered to introduce a class of multi-term time fractional 2D diffusion equations. A numerical method based on the Chebyshev cardinal polynomials (CCPs) is proposed to solve this problem. In this way, a new operational matrix for the $\psi$-Caputo fractional derivative of the CCPs is provided. By approximating the solution of the problem by a finite series of the CCPs (with some unknown coefficients) and employing the derived fractional matrix, an algebraic system of equations is generated, which by solving it the expressed coefficients, and consequently, the problem’s solution are identified. The validity of the established method is investigated by solving some numerical examples.

In the traditional heterogeneous agent model, investors are assumed to be risk averse, and the wealth expected utility function maximization principle is used to form the optimal asset quantity demand. In such models, the risk aversion coefficient of investors is often assumed to be constant. This paper considers that the risk aversion coefficient of investors is time-varying and changes with the change of wealth, and establishes an endogenous evolutionary mechanism model formed by fundamental analysts, technical analysts, and market makers. Compared with the fixed risk aversion coefficient model, this paper analyzes the investor’s behavior, the interaction between investor behaviors, and the influence of different types of investors on the stability of the market. At the same time, we test asset price and asset behavior and conclude that investor behavior affects the stability of the system model. The numerical simulation of the corresponding stochastic model shows that the model can simulate the basic characteristics of financial time series, such as the partial peak and thick tail of asset return series, and the long memory of fluctuations.

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.