SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].

THREE-SPHERE INEQUALITIES FOR SECOND ORDER SINGULAR PARTIAL DIFFERENTIAL EQUATIONS

In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div（A∨u） - Vu =0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation Эtu-div（A∨u） ＋ Vu = 0, where the singular potential V belonging to the Kato-Fefferman- Phong＇s class. Some applications are also discussed.

Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense

The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.

A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation

In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

G-Function Solutions for Schrodinger Equation in Cylindrical Coordinates System

In this paper, the Schr?dinger equation is solved by Modified separation of variables (MSV) method suggested by Pishkoo and Darus. Using this method, Meijer’s G-function solutions are derived in cylindrical coordinate system for quantum particle in cylindrical can. All elementary functions and most of the special functions which are the solution of extensive problems in physics and engineering are special cases of Meijer’s G-functions.

An Alternative Method for Solving Lagrange＇s First-Order Partial Differential Equation with Linear Function Coefficients

An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.

Adomian's Method Applied to Solve Ordinary and Partial Fractional Differential Equations

This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian's Method. In this paper, we firstly use the convergence of Adomian's Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations.We can see the advantage of this method to deal with fractional differential equations.

一类带交易费用含红利的B-SPDE差分方法

对于一类特殊的带交易费用并含红利的B-S偏微分方程,假设标的股票有交易费用以及红利支付,用有限差分及模拟技术求解了期权定价的近似解,并给出了具体实例验证了方法的有效性.

A Comparative Study of Adomain Decompostion Method and He-Laplace Method

In this paper, we present a comparative study between the He-Laplace and Adomain decomposition method. The study outlines the significant features of two methods. We use the two methods to solve the nonlinear Ordinary and Partial differential equations. Laplace transformation with the homotopy method is called He-Laplace method. A comparison is made among Adomain decomposition method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easy handled by the use He’s polynomials and provides better results.

A Comparative Study of Variational Iteration Method and He-Laplace Method

In this paper, variational iteration method and He-Laplace method are used to solve the nonlinear ordinary and partial differential equations. Laplace transformation with the homotopy perturbation method is called He-Laplace method. A comparison is made among variational iteration method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easily handled by the use of He’s polynomials and provides better results.

The Mathematical and Physical Theory of Rational Human Intelligence: Complete Empirical-Digital Properties;Full Electrochemical-Mechanical Model (Part I: Mathematical Foundations)

The design of this paper is to present the first installment of a complete and final theory of rational human intelligence. The theory is mathematical in the strictest possible sense. The mathematics involved is strictly digital—not quantitative in the manner that what is usually thought of as mathematics is quantitative. It is anticipated at this time that the exclusively digital nature of rational human intelligence exhibits four flavors of digitality, apparently no more, and that each flavor will require a lengthy study in its own right. (For more information,please refer to the PDF.)

On Some Bending Problems of Prismatic Shell with the Thickness Vanishing at Infinity

The present work is devoted to the bending problems of prismatic shell with the thickness vanishing at infinity as an exponential function. The bending equation in the zero approximation of Vekua＇s hierarchical models is considered. The problem is reduced to the Dirichlet boundary value problem for elliptic type partial differential equations on half-plane. The solution of the problem under consideration is constructed in the integral form.

基于Runge-Kutta的自回归物理信息神经网络求解偏微分方程

物理信息神经网络离散时间模型(PINN-RK)是深度学习技术与龙格库塔方法相结合的产物,在求解偏微分方程时具有非常出色的稳定性和较高的求解精度.但是,受到龙格库塔算法本身的限制,PINN-RK模型仅能实现单步时间预测,且计算效率较低.因此,为了实现多时间步长预测和提高模型的计算效率,提出了一种基于龙格库塔法的自回归物理信息神经网络模型(SR-PINN-RK).该模型基于自回归时间步进机制,改进了神经网络的训练流程和网络结构,相比PINN-RK模型,大幅减少了神经网络的训练参数,提高了模型的计算效率.此外,在自回归机制的作用下,该模型通过对标签数据的动态更新,成功实现了对偏微分方程解的多时间步长预测.为了验证文中模型的求解精度和计算效率,分别求解了Allen-Cahn方程和Burgers方程,并与文献中的基准解进行了对比.结果表明,模型预测解与基准解之间具有很高的一致性,求解Allen-Cahn方程和Burgers方程的最大相对误差均低于0.009.

Talagrand's T_2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction- Diffusion equations are provided.

STOCHASTIC HEAT EQUATIONS WITH RANDOM INITIAL CONDITIONS

In this paper the author constructs a solution of parabolic stochastic partial differential equation with random initial conditions by Kolmogorov's criterion.

The double auxiliary equations method and its application to space-time fractional nonlinear equations

This paper reflects the execution of a reliable technique which we proposed as a new method called the double auxiliary equations method for constructing new traveling wave solutions of nonlinear fractional differential equation.The proposed scheme has been successfully applied on two very important evolution equations,the space-time fractional differential equation governing wave propagation in low-pass electrical transmission lines equation and the time fractional Burger’s equation.The obtained results show that the proposed method is more powerful,promising and convenient for solving nonlinear fractional differential equations(NFPDEs).To our knowledge,the solutions obtained by the proposed method have not been reported in former literature.

一种基于偏微分方程的车辆加速度信号自适应降噪方法

提出一种针对MEMS加速度计信号的基于偏微分方程的自适应降噪方法,该方法不仅能有效克服由于传感器本身原因及车载环境振动噪声带来的影响,获得准确的加速度信号,而且实现容易、实时性好。通过对车辆加速度信号进行建模并叠加真实加速度噪声作为仿真信号,将该方法与选用db6小波基、heursure自适应阈值、4层分解的最佳小波进行降噪性能对比,证明在车辆正常行驶的加速度幅值下,该方法不仅能够取得和小波近似的降噪性能,而且很大程度上减少了运算时间。最后通过对实际车载加速度信号的降噪处理和倾角测量中的应用,再次证明该方法在滤除噪声的同时能够较好体现细节信息,很适合应用在对实时性和准确性要求高的实际工程中。

偏微分方程(组)守恒律的再扩充

在共轭方程(组)方法、微分形式吴方法和在Nether定理的基础上,利用对称变换作用于已知守恒律产生新守恒律方法确定非变分对称对应的新守恒律,达到了再扩充守恒律的重要目的.

基于P.Perona和J.Malik方程的图像放大

偏微分方程在图像去噪、重建等图像处理领域中都已经有了较多较好的应用,基于一种全新的图像放大方法,即基于偏微分方程的图像放大法,该方法主要是利用热传导原理,建立热传导过程的数学模型,实现图像放大的目的.提出了一种基于P.Perona和J.M alik方程的图像放大的算法.数值离散采用直观的有限差分格式,实验在M atlab环境下进行.实验结果表明该法在放大过程中保持了图像的特征,也说明了该算法实现图像放大的有效性.