Results in Applied Mathematics最新文献

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series solution methods, outperforms others, exhibiting superior performance with higher accuracy orders compared to those obtained from Chebyshev spectral methods, Bernoulli wavelet method, and Spectral Tau method.

The Chebyshev polynomial approximation is a useful tool to approximate smooth and non-smooth functions. In fact, for a sufficiently smooth function, the partial sum of Chebyshev series expansion provides optimal polynomial approximation. Moreover, because the construction of these polynomial approximations is computational efficient, they are widely used in numerical schemes for solving partial deferential equations. Significant efforts have been devoted to establishing decay bounds for series coefficients, including Chebyshev, Jacobi, and Legendre series, for both smooth and non-smooth univariate functions. However, the literature lacks similar estimates for bivariate functions. This paper aims to address this gap by examining the decay estimates of bivariate Chebyshev coefficients, contributing both theoretically and practically to the understanding and application of Chebyshev series expansions, especially concerning functions with limited smoothness. Additionally, we derive $L^1$-error estimates for the partial sum of Chebyshev series expansions of functions with bounded Vitali variation. Furthermore, we provide an estimate for the discrepancy between exact and approximated Chebyshev coefficients, leveraging a quadrature formula. This analysis leads to the deduction of an asymptotic $L^1$-approximation error for finite partial sums of Chebyshev series with approximated coefficients.

In this paper we address the connections between the computational models of coupled flow and mechanical deformation in soils at the Darcy-scale and pore-scale. At the Darcy scale the Biot model requires data including permeability which is traditionally provided by experiments and empirical measurements. At the pore-scale we consider the Discrete Element Method (DEM) to generate physically realistic assemblies of the particles, and we follow up with the Stokes flow model. Next we apply upscaling to obtain the permeabilities which we find dependent on the deformation. We outline the workflow with its challenges and methods, and present results which show, e.g., hysteretic dependence of the permeability and porosity on the load. We also show how to incorporate the deformation dependent permeability in a nonlinear Biot model, and illustrate with computational results.

The objective of this study is to examine the dynamic components of option pricing in the European put option market by utilizing the two-dimensional time fractional-order Black–Scholes equation. To enhance the classical Black–Scholes equation, we utilize the Caputo type of the Katugampola fractional derivative. The Reconstruction of Variational Iteration Method is employed as a powerful tool for analyzing option price behavior in the European-style market. In our investigation, we utilize this method to obtain an exact solution for fractional Black–Scholes with two assets. Moreover, the findings demonstrate the impressive effectiveness of the Reconstruction of Variational Iteration Method in addressing two-dimensional fractional-order differential equations, thereby highlighting its potential as a valuable numerical solution technique.

A truncated form of a matrix variate gamma distribution is introduced and a number of properties of this distribution such as cumulative distribution function, orthogonal invariance, moment generating function, marginal distribution of block matrices, and moments are derived. Some results on distribution of random quadratic forms are also derived.

This paper investigates a fully discrete characteristic finite element approximation of bilinear unsteady convection–diffusion optimal control problems. The characteristic line method is used to treat the convection term and the finite element method is adopted to treat the diffusion term. The state and adjoint state are discretized by piecewise linear functions, the control is approximated by piecewise constant functions. A priori error estimates are derived for the state, adjoint state and control variables. Some numerical examples are provided to confirm our theoretical findings.

This paper presents a difference interior penalty discontinuous Galerkin method for the 3D elliptic boundary-value problem. The main idea of this method is to combine the finite difference discretization in the $z$-direction with the interior penalty discontinuous Galerkin discretization in the $(x,y)$-plane. One of the advantages of this method is that the solution of 3D problem is transformed into a series of 2D problems, thereby overcoming the computational complexity of traditional interior penalty discontinuous Galerkin method for solving high-dimensional problems and allowing for code reuse. Additionally, we use the interior penalty discontinuous Galerkin method to solve each 2D problem, therefore, this method retains the advantage of the interior penalty discontinuous Galerkin method in dealing with non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees. Then, the error estimates are given for difference interior penalty discontinuous Galerkin method. Finally, numerical experiments demonstrate the accuracy and effectiveness of the difference interior penalty discontinuous Galerkin method.

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.