Results in Applied Mathematics最新文献

This work is concerned with solving parabolic Volterra partial integro-differential equations (PIDE) considering differentiable and singular kernels. The implicit finite difference scheme is implemented to approximate the differential operator, and the nonlocal term is discretized based on an open-type formula with two distinct time step sizes related to the nature of the time level to guarantee to avoid the singular terms at the endpoints and denominators. The properties of the plied scheme are investigated, more precisely, its stability and consistency. Four detailed examples are implemented to demonstrate the efficiency and reliability of the applied finite difference scheme.

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

This paper presents a new framework for efficient and accurate analysis of transient elastodynamic cracks by using the generalized finite difference method (GFDM). The method first discretizes the solution domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated by using the local Taylor series expansions and moving-least square approximation. The degree of the Taylor series used in the local subdomain is increased automatically in the regions near the crack-tips, in order to appropriately describe the local asymptotic behavior of near-tip displacement and stress fields. The path-independent J-integral and sub-domain technique are adopted to compute the dynamic stress intensity factors (SIFs) of the cracked bodies. Preliminary numerical experiments for dynamic SIFs with both uniform and variable loading conditions are given to show the efficient and accuracy of the present method for transient elastodynamic crack analysis.

In this article a numerical method for numerical modelling of advection diffusion equation is developed. The proposed method is based on Laplace transform (LT) and Chebyshev spectral collocation method (CSCM). The LT is used for time-discretization and the CSCM is used for discretization of spatial derivatives. The LT is used to transform the time variable and avoid the finite difference time stepping method. In time stepping technique the accuracy is achieved for very small time step which results in a very high computational time. The spatial operators are discretized using CSCM to achieve high accuracy as compared to other methods. The method is composed of three primary stages: firstly the given problem is transformed into a corresponding inhomogeneous elliptic problem by using the LT; secondly the CSCM used to solve the transformed problem in LT domain; finally the solution obtained in LT domain is converted to time domain via numerical inverse LT. The inversion of LT is generally an ill-posed problem and due to this reason various numerical inversion methods have been developed. In this article we have utilized the contour integration method which is one of the most efficient methods. The most important feature of this approach is that it handles the time derivative with the Laplace transform rather than the finite difference time stepping approach, avoiding the untoward impact of time steps on stability and accuracy of the method. Five test problems are used to validate the efficiency and accuracy of the proposed numerical scheme.

We solve the existence problem for the minimal positive solutions $u\in L^p(\Omega ,dx)$ to the Dirichlet problems for sublinear elliptic equations of the form $\left(\begin{array}{c}Lu=\sigma u^q+\mu \text{in}\Omega ,\\ \underset{x\to y}{lim\; inf}u\left(x\right)=0y\in {\partial }_{\infty }\Omega ,\end{array}\right)$where $0<q<1$ and $Lu≔-\text{div}(A\left(x\right)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $\sigma$ and data $\mu$ are nonnegative Radon measures on an arbitrary domain $\Omega \subset R^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

In this paper, we first use an $L_2-$discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space $H^1\left(K\right)$ equipped with a reproducing kernel. The concept of stratified sampling under general equal measure partition is introduced into the research. For different sampling modes, we obtain a better convergence order $O\left(N^{-1-\frac{1}{d}}\right)$ for the stratified sampling set than for the Monte Carlo sampling method and the Latin hypercube sampling method. Second, we give several expected uniform integration approximation bounds for functions equipped with boundary conditions in the general Sobolev space $F_{d,q}^{\ast }$, where $\frac{1}{p}+\frac{1}{q}=1$. Probabilistic $L_p-$discrepancy bound under general equal measure partition, including the case of Hilbert space-filling curve-based sampling are employed. All of these give better general results than simple random sampling, and in particular, Hilbert space-filling curve-based sampling gives better results than simple random sampling for the appropriate sample size.

This paper presents and successfully applies an optimized hybrid block technique using a variable stepsize implementation to integrate a type of singularly perturbed parabolic convection–diffusion problems. The problem under consideration is semi-discretized by utilizing the method of lines. A few numerical experiments have been presented to ascertain the proposed error estimation and adaptive stepsize strategy. Furthermore, the comparison of the proposed method with other techniques in the literature is conducted via numerical experiments, and the results show that our method outperforms other existing methods.

Interleaved learning in machine learning algorithms is a biologically inspired training method with promising results. In this short note, we illustrate the interleaving mechanism via a simple statistical and optimization framework based on Kalman Filter for Linear Least Squares.

The construction of the limiter is a critical factor in the traditional Total Variation Diminishing (TVD) scheme. Among the classical limiters, superbee has the lowest numerical dissipation, but it can lead to over-compression in smooth regions and excessive artificial steepening at discontinuities and critical points if the computation time is prolonged. Classical limiters like Minmod, van Leer, van Albada, and MC fail to distinguish between different wave types, and they can even cause numerical oscillations for multi-critical value problems with prolonged computation times. A class of adaptive limiters has been created by combining classical limiters with superbee. This adaptive limiter can achieve second-order accuracy in smoothed regions and effectively reduce over-compression and excessive artificial steepening for long computation times. Analytical and numerical results show that the MUSCL scheme with an adaptive limiter is efficient.