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Stable numerical results to a class of time-space fractional partial differential equations via spectral method.
Journal of Advanced Research ( IF 10.7 ) Pub Date : 2020-06-19 , DOI: 10.1016/j.jare.2020.05.022
Kamal Shah,Fahd Jarad,Thabet Abdeljawad

In this paper, we are concerned with finding numerical solutions to the class of time–space fractional partial differential equations:

Dtpu(t,x)+κDxpu(t,x)+τu(t,x)=g(t,x),1<p<2,(t,x)[0,1]×[0,1],

under the initial conditions.

u(0,x)=θ(x),ut(0,x)=ϕ(x),

and the mixed boundary conditions.

u(t,0)=ux(t,0)=0,

where Dtp is the arbitrary derivative in Caputo sense of order p corresponding to the variable time t. Further, Dxp is the arbitrary derivative in Caputo sense with order p corresponding to the variable space x. Using shifted Jacobin polynomial basis and via some operational matrices of fractional order integration and differentiation, the considered problem is reduced to solve a system of linear equations. The used method doesn’t need discretization. A test problem is presented in order to validate the method. Moreover, it is shown by some numerical tests that the suggested method is stable with respect to a small perturbation of the source data g(t,x). Further the exact and numerical solutions are compared via 3D graphs which shows that both the solutions coincides very well.



中文翻译:

通过频谱方法,对一类时空分数阶偏微分方程具有稳定的数值结果。

在本文中,我们关注的是找到时空分数阶偏微分方程类的数值解:

dŤpüŤX+κdXpüŤX+τüŤX=GŤX1个<p<2ŤX[01个]×[01个]

在初始条件下。

ü0X=θXüŤ0X=ϕX

和混合边界条件。

üŤ0=üXŤ0=0

哪里 dŤp是Caputo阶p对应于可变时间t的任意导数。进一步,dXp是Caputo意义上的任意导数,其阶次p对应于变量空间x。使用移位雅可比多项式基础并通过分数阶积分和微分的一些运算矩阵,可以减少所考虑的问题来求解线性方程组。使用的方法不需要离散化。提出了一个测试问题,以验证该方法。此外,通过一些数值测试表明,所建议的方法相对于源数据的微小扰动是稳定的GŤX。进一步的精确解和数值解通过3D图形进行了比较,表明这两个解非常吻合。

更新日期:2020-08-28
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