共查询到20条相似文献,搜索用时 15 毫秒
1.
《数学季刊》2016,(1):69-81
Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper. 相似文献
2.
61.IntroductionThesemilinearwaveequationareoftenappearinthestudyofphysical,chemical,mechani-cal,biologicalandotherproblems,onespecialexampleisthewell-knownKlein-Gordonequa-tlonutt-u..=f(u)(l)whichtakesanimportantrolesinthestudyofSoliton.Choicedifferentnonlineartermf,thevariousstandardequations['jaregotsuchastakingf(u)=sin(u),then(l)asaSine-Gordonequationandtakingf(u)=u-u3lthen(1)asagi-Wequationwhichisanimportantmodell.insolidphysicsandhighenergyphysics['j.Moreover,takingf(u)=f[sin(u) isin(7)… 相似文献
3.
4.
Many physical processes appear to exhibit fractional order
behavior that may vary with time or space. The continuum of order in
the fractional calculus allows the order of the fractional operator
to be considered as a variable. Numerical methods and analysis of
stability and convergence of numerical scheme for the variable
fractional order partial differential equations are quite limited
and difficult to derive.
This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time
variable fractional order diffusion equation on a finite domain. It
is worth mentioning that here we use the Coimbra-definition variable time
fractional derivative
which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems.
An implicit Euler approximation is proposed and then
the stability
and convergence of the numerical scheme are investigated.
Finally, numerical examples are provided to show that the implicit Euler approximation is computationally
efficient. 相似文献
5.
An Chen 《Numerical Functional Analysis & Optimization》2016,37(1):19-39
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. 相似文献
6.
In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017 相似文献
7.
一类非线性反应-扩散方程有限差分格式的稳定性研究 总被引:1,自引:0,他引:1
In the article,the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established.Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space.The approach used is of a simple characteristic in gaining the stability condition of the scheme. 相似文献
8.
Zheng Dayi Lu Xuanzhu 《Annals of Differential Equations》2005,21(3):518-524
In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given. 相似文献
9.
非线性Sobolev-Galpern方程有限差分格式在t>0时的长时间收敛性和稳定性 总被引:1,自引:0,他引:1
本文讨论了非线性Sobolev-Galpern初边值问题,给出了Sobolev-Galpern方程的有限差分格式在t>0时的长时间收敛性和稳定性的证明. 相似文献
10.
Hossein Fazli HongGuang Sun Juan J. Nieto 《Mathematical Methods in the Applied Sciences》2022,45(1):197-205
We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory. 相似文献
11.
时间分数阶扩散方程的数值解法 总被引:1,自引:0,他引:1
马亮亮 《数学的实践与认识》2013,43(10)
分数阶微分方程在许多应用科学上比整数阶微分方程更能准确地模拟自然现象.考虑时间分数阶扩散方程,将一阶的时间导数用分数阶导数α(0<α<1)替换,给出了一种计算有效的隐式差分格式,并证明了这个隐式差分格式是无条件稳定和无条件收敛的,最后用数值例子说明差分格式是有效的. 相似文献
12.
本文对一类非线性Sine-Gordon方程的初边值问题提出了两个隐式差分格式.两个隐式差分格式的精度均为O(τ~2 h~2).我们用离散泛函分析的方法证明了格式的收敛性和稳定性,并证明了求解格式的追赶迭代法的收敛性,最后给出了数值结果.结果表明本文的格式是有效的和可靠的. 相似文献
13.
Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative 下载免费PDF全文
Qinwu Xu & Zhoushun Zheng 《数学研究》2014,47(2):173-189
In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme. 相似文献
14.
利用待定参数法,对一维抛物型方程构造出了一个截断误差为O(△x^4+△x^4)的隐式差分格式,格式的稳定性条件为r=a△t/△x^2≤1/√2,可用追赶法求解。 相似文献
15.
Chen Chunhua Lu Xuanzhu 《Annals of Differential Equations》2005,21(3):250-255
In this paper, a time fractional advection-dispersion equation is considered. From the relationship between the Caputo derivative and the Grunwald derivative, the Caputo derivative is approximated by using the Griinwald derivative. An implicit difference approximation for this equation is proposed. We prove that this approximation is unconditionally stable and convergent. Finally, numerical examples are given. 相似文献
16.
17.
Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation 下载免费PDF全文
Tingchun Wang & Boling Guo 《偏微分方程(英文版)》2013,26(2):107-121
A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis. 相似文献
18.
New numerical techniques are presented for the solution of the
two-dimensional time fractional evolution equation in the unit
square. In these methods, Galerkin finite element is used for the
spatial discretization, and, for the time stepping, new alternating
direction implicit (ADI) method based on the backward Euler method
combined with the first order convolution quadrature approximating
the integral term are considered. The ADI Galerkin finite element
method is proved to be convergent in time and in the $L^2$ norm in
space. The convergence order is$\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is
the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to
support our theoretical analysis. 相似文献
19.
对广义Rosenau-Burgers方程的初边值问题进行了数值研究,提出了新的两层隐式差分格式,得到了差分解的存在唯一性,并利用能量方法分析了该格式的二阶收敛性与无条件稳定性,并且给出数值算例进行验证. 相似文献