Journal ArticleDOI
Fractional Sturm-Liouville eigen-problems
TLDR
It is proved that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal, hence completing the whole family of the Jacobi poly-fractonomials.About:
This article is published in Journal of Computational Physics.The article was published on 2013-11-01. It has received 281 citations till now. The article focuses on the topics: Fractional calculus & Sturm–Liouville theory.read more
Citations
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Journal ArticleDOI
Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
TL;DR: A new deep neural network called DeepONet can lean various mathematical operators with small generalization error and can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations.
Journal ArticleDOI
DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
TL;DR: This work proposes deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset, and demonstrates that DeepONet significantly reduces the generalization error compared to the fully-connected networks.
Journal ArticleDOI
Fractional spectral collocation method
TL;DR: A new family of interpolants are introduced, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points and are developed as an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs).
Journal ArticleDOI
Generalized Jacobi functions and their applications to fractional differential equations
Sheng Chen,Jie Shen,Li-Lian Wang +2 more
TL;DR: In this article, a new class of generalized Jacobi functions (GJFs) is defined, which are intrinsically related to fractional calculus and can serve as natural basis functions for properly de- signed spectral methods for fractional dif- ferential equations (FDEs).
Journal ArticleDOI
A novel high order space-time spectral method for the time fractional fokker-planck equation ∗
TL;DR: In this article, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-planck initial-boundary value problem, which employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretisation.
References
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Book
Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Fractional Calculus in Bioengineering
TL;DR: Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems, which is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research.
BookDOI
Fractals and fractional calculus in continuum mechanics
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.