Fractional order differentiation by integration with Jacobi polynomials
King Abdullah university of science and technology（阿卜杜拉国王科技大学）；Arts et Métiers ParisTech（巴黎高等工艺学院）；École Centrale de Lille（里尔中央理工学院）；INRIA Lille-Nord Europe（法国国家信息与自动化研究所北欧分部）
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises.
Differentiation by integration with Jacobi polynomials
INRIA Lille-Nord Europe（法国国家信息与自动化研究所里尔北欧分部）；Université de Lille 1（里尔第一大学）；Arts et Métiers ParisTech centre de Lille（巴黎国立高等工艺美术学院里尔中心）；École Centrale de Lille（里尔中央理工学院）
In this paper, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.
BIFOLD – Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；LVMT, ENPC, Institut Polytechnique de Paris, Univ Gustave Eiffel（法国国立路桥学校，巴黎理工学院，古斯塔夫·埃菲尔大学）；Machine Learning Group, Technical University of Berlin（柏林工业大学机器学习组）
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator (FSO). In the problem, in order to get reliable gaps distribution statistics, we have to calculate accurately and efficiently a very large number of eigenvalues, e.g. up to thousands or even millions eigenvalues, of an eigenvalue problem related to the FSO. The proposed Jacobi spectral method is extremely suitable and demanded for the discretization of an eigenvalue problem when a large number of eigenvalues need to be calculated. The…
In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the ba…
Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations
Beijing Computational Sciences and Research Center（北京计算科学研究中心）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-μ},0<μ<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to deri…
Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- φ(t,x))= f(t,x).$$ For the scheme building on the second order BDF formula (BDF2), we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis, an equivalence of the obstacle equation with a Hamilton-Jacobi-Bellman equation…
A nonstandard fractional Jacobi pseudospectral method for weakly regular right-sided Caputo terminal problems
Al-Azhar University（爱资哈尔大学）；Beni-Suef University（贝尼苏韦夫大学）；Mississippi State University（密西西比州立大学）；Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊玛目穆罕默德·本·沙特伊斯兰大学）
