Numerical Simulation of the Variable Order Fractional Integro-Differential Equation via Chebyshev Polynomials

Mathematical Notes - In this paper, the Chebyshev polynomials method is applied to solve a space-time variable fractional order integro-differential equation. Using operational matrices of...     

Numerical Solution of Fractional Diffusion-Wave Equation

In this article, a novel compact finite difference scheme is mboxconstructed to solve the fractional diffusion-wave equation based on its equivalent integro-differential equation. In the temporal direction, the product trapezoidal scheme is employed to treat the fractional integral term. The convergence and stability of the scheme are proved. Numerical examples are also provided to verify the theoretical analysis.     

非线性反应扩散方程的数值解

本文采用Petrov-Galerkin有限元法构造了非线性反应扩散方程的数值格式,既适用于全场性的初值,也适用于局部性的初值,利用全场性初值求得的进波数值解与精确解高度吻合,证明本方法与其它数值方法比较,有更高的精度和稳定性,利用各种局部性初值所算出的数值解表明,任何局部的扰动都会得到充分的发展,且当时间充分长后,演变为向左、右传播的行波,其波前的形状及传播速度完全由系统本身所决定,而与初值的类型无关.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

非线性分数微分方程边值问题单调正解的存在性（英文）

In this paper, we consider a class of nonlinear fractional differential equation boundary value problem. The existence of monotone positive solution is derived by the iterative technique.     

基于切比雪夫小波基与年龄相关种群模型的数值解

基于切比雪夫小波基给出与年龄相关种群模型的数值解.利用切比雪夫小波基的性质使得所求偏微分方程转化为矩阵方程,从而简化了数值解的求解过程.最后通过数值例子验证其理论结果.     

Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with which the unconditional stability and convergence are proved. The numerical results demonstrate the effectiveness of the theoretical analysis, and the method and technique can also be applied to other kinds of time/space fractional diffusion equations.     

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).

     

Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem

Differential Equations - We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the...     

一非线性双曲型方程解的爆破

本文应用压缩映射原理证明一非线性双曲型方程的初边值问题存在唯一局部广义解，并给出此问题的广义解爆破的充分条件。     

Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements

Mediterranean Journal of Mathematics - We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions...     

非线性分数阶微分方程的奇摄动

研究了—类奇摄动非线性分数阶微分方程Cauchy问题．在适当的条件下,首先求出了原问题的外部解,然后利用伸长变量、合成展开法和幂级数展开理论构造出解的初始层项,并由此得到解的形式渐近展开式．最后利用微分不等式理论,讨论了问题解的渐近性态,得到了原问题解的一致有效的渐近估计式．     

Ramm积分方程的数值解（Ⅱ）

本文提出了解如下三维的Ramm积分方程的一种新的数值方法:采用带滤子的奇异值分解方法计算上述方程的近似解,并取得了令人满意的数值结果。此外,本文还讨论了解这类积分方程的吉洪诺夫正则方法.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

一类非线性的偏微分方程的解

给出了一类带有时滞的偏微分方程.该方程描述得是含有非局部和时滞边界条件的分布参数系统.运用泛函分析和积分方程的理论,证明了方程解的存在唯一性,得到解的解析表达式.     

一个非线性积分微分方程的数值研究

本文采用Ｆｏｒｎｂｅｒｇ和Ｗｈｉｔｈａｍ的拟谱方法，数值研究了一个非线性积分微分方程的初值问题：Ａｔ＋６ＡＡｘ＋１２｜ｌｎε｜∫＋∞－∞Ａ（ｘ′，ｔ）（ｘ′－ｘ）２＋ε２｛｝１／２ｄｘ′＝０发现当ε很小时，其解与ＫｄＶ方程的解接近·较大的ε和初始条件对解的影响是很大的·