Asymptotically optimal approximation in fractional Sobolev spaces and the numerical solution of differential equations

1974 ◽  
Vol 22 (2) ◽  
pp. 75-87 ◽  
Author(s):  
C. A. Micchelli ◽  
W. L. Miranker
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Sobolev Spaces ◽  
Optimal Approximation ◽  
Fractional Sobolev Spaces ◽  
Asymptotically Optimal

scholarly journals On a new class of fractional partial differential equations

2015 ◽  
Vol 8 (4) ◽  
Author(s):  
Tien-Tsan Shieh ◽  
Daniel E. Spector
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Variational Problems ◽  
Fractional Sobolev Spaces ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Class ◽  
Natural Way

AbstractIn this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

scholarly journals Anisotropic Sobolev Capacity with Fractional Order

2017 ◽  
Vol 69 (4) ◽  
pp. 873-889 ◽  
Author(s):  
Jie Xiao ◽  
Deping Ye
Keyword(s):  
Fractional Order ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Sobolev Class ◽  
Minkowski Inequality ◽  
Fractional Sobolev Spaces ◽  
Geometric Quantity ◽  
Asymptotically Optimal ◽  
Order Estimation ◽  
Sobolev Capacity

AbstractIn this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure μ that naturally induces an embedding of the anisotropic fractional Sobolev class into the μ-based-Lebesgue-space with 0 < β ≤ n. Also, we investigate the anisotropic fractional α-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as α →0+, will be provided.

Numerical solution of fractal-fractional Mittag–Leffler differential equations with variable-order using artificial neural networks

2021 ◽  
Author(s):  
C. J. Zúñiga-Aguilar ◽  
J. F. Gómez-Aguilar ◽  
H. M. Romero-Ugalde ◽  
R. F. Escobar-Jiménez ◽  
G. Fernández-Anaya ◽  
...  
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Differential Equations ◽  
Numerical Solution ◽  
Variable Order ◽  
Artificial Neural
Sign in / Sign up

Export Citation Format

Share Document