A New Matrix Approach For Solving Second-Order Linear Matrix Partial Differential Equations

The basic aim of this article is to present a novel efficient matrix approach for solving the second-order linear matrix partial differential equations (MPDEs) under given initial conditions. For imposing the given initial conditions to the main MPDEs, the associated matrix integro-differential equations (MIDEs) with partial derivatives are obtained from direct integration with regard to the spatial variable x and time variable t. Hence, operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials are used to reduce the obtained MIDEs to the corresponding algebraic Sylvester equations. Using two well-known subspace Krylov iterative methods (i.e., GMRES(10) and Bi-CGSTAB) we provide two algorithms for solving the mentioned Sylvester equations. A numerical example is provided to show the efficiency and accuracy of the presented approach.

     

一类中立型时滞抛物偏微分方程的强迫振动性

研究了一类中立型时滞抛物偏微分方程:t(u(x,t)-pu(x,t-τ))-∑rk=1ak(t)Δu(x,t-ρk(t))+∑mj=1qj(t)u(x,t-σj(t))=e(x,t),的强迫振动性(其中(x,t)∈Ω×[0,∞)≡G,Ω是n维欧几里得空间Rn中带有逐段光滑边界Ω的有界区域,Δ是Rn中带有三类不同边值条件的拉普拉斯算子,强迫项e(x,t)是定义在G上的一个振荡函数),给出了一些新的振动性判据,这些结果推广了已知的一些结论.     

New Oscillation Criteria for a Class of Linear Delay Differential Equations

Oscillation criteria for the first-order linear delay differentialequation are established.Our results improve the known results in the literature. 1991Mathematics Subject Classification 34K15.     

一类具时滞的高阶中立型偏微分方程解的振动性

研究一类具时滞的偶数阶非线性中立型偏微分方程解的振动性,获得了一些新的充分判据,所得结果推广和包含了文[1]及[2]的相应结果.     

一类具连续分布滞量的高阶偏微分方程的振动性

研究一类具有连续分布滞量中立项的非线性偏微分方程解的振动性,得到该方程在给定边值条件下振动和非振动的一些充分条件.     

Existence of Solution for Delay Fractional Differential Equations

This paper is devoted to the existence of solution for a class of delay fractional differential equations. First we prove some singular integral inequalities. Then using a fixed point theorem of nonlinear alternative Leray–Schauder type, the existence of at least one solution is proved. To illustrate the main result some examples are given.     

Correction Procedure for Solving Partial Differential Equations

The correction procedure has been discussed by L. Fox and V. Pereyra for accelerating the convergence of a certain approximate solution. Its theoretical basis is the existence of an asymptotic expansion for the error of discretization proved by Filippov and Rybinskii and Stetter: $u-u_h=h^2 v+O(h^4)$, where $u$ is the solution of the original differential equation, $u_h$ the solution of the approximate finite difference equation with parameter $h$ and $v$ the solution of a correction differential equation independent of $h$. Stetter et al. used the extrapolation procedure to eliminate the auxiliary function $v$ while Pereyra et al. used some special procedure to solve v approximately. In the present paper we will present a difference procedure for solving $v$ easily.     

Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations

We investigate mathematically a nonlinear approximation type approach recently introduced in Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) to solve high-dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied, e.g., in DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996). On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the nonvariational version of the approach, consisting of simply solving the first-order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.     

脉冲时滞抛物型偏微分方程组解的振动性

研究一类脉冲时滞抛物型偏微分方程组解的振动性,利用一阶脉冲时滞微分不等式获得了该类方程组在两类不同边值条件下所有解振动的若干充分条件.所得结果充分反映了脉冲和时滞在振动中的影响作用.     

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

In this article, we will prove the existence, uniqueness and Hölder regularity of the solution to the fractional stochastic partial differential equation of the form

$$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$

where \(\mathfrak {D}(x,D)\) denotes the Markovian generator of stable-like Feller process, \(f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is a measurable function, and \(\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)\) is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

     

一类具连续分布滞量的非线性中立型偏微分方程解的振动性

In this paper,some sufficient conditions are obtained for the oscillation of solutions for a class of second order nonlinear neutral partial differential equations with continuous distribution delay under Robin and Dirichlet's boundary value conditions.     

Solving Inhomogeneous Linear Partial Differential Equations

Lagrange＇s variation-of-constants method for solving linear inhomogeneous ordinary differential equations （ode＇s） is replaced by a method based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for partial differential equations （pde＇s）. It is applied to equations of second order in two independent variables, and to a certain system of third-order pde＇s. Therewith all possible linear inhomogeneous pde＇s are covered that may occur when third-order linear homogeneous pde＇s in two independent variables are solved.     

一类线性随机微分方程的解法

本文关注一类线性随机微分方程的解法,先求解伪齐次随机微分方程,变易对应解的常数,再带回原方程求解.这区别于以往求解随机微分方程所对应的齐次微分方程的常数变易法.多个例子证明本文的方法更简明.     

一类分数阶微分方程解的性质

本文主要讨论了一类非线性分数阶微分方程的初值问题的解的存在唯一性,平衡点的稳定性。最后,运用微分变换的方法对它的解析进行了估计.     

一类带有分数阶积分边值条件的分数阶微分方程的可解性(英文)

本文研究下面一类带有分数阶积分边值条件的分数阶微分方程cDα0+u(t)=f(t,u(t),cDβ0+u(t)),0相似文献   

Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Potential Analysis - Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$ partial^{beta}_{t}u_{t}(x)=-nu(-{Delta})^{alpha/2}...     

一类时间分数阶偏微分方程的解

考虑一类时间分数阶偏微分方程,该方程包含几种特殊情况:时间分数阶扩散方程、时间分数阶反应-扩散方程、时间分数阶对流-扩散方程以及它们各自相对应的整数阶偏微分方程． 通过Laplace-Fourier变换及其逆变换,该方程在空间全平面和半平面内的基本解可以求出,但其表达式则是通过适当的变形来求．另外,对于有限域上的初边值问题,则可由Sine(Cosine)-Laplace变换导出该方程的一种级数形式的解,并通过两个数值例子来说明该方法的有效性．     

Particle Approximations for a Class of Stochastic Partial Differential Equations

The paper presents a particle approximation for a class of nonlinear stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The new results permit the treatment of filtering problems where the signal noise is no longer independent of the observation noise.