Numerical methods for nonlocal and fractional models
Partial differential equations (PDEs) are used with huge success to model phenomena
across all scientific and engineering disciplines. However, across an equally wide swath …
across all scientific and engineering disciplines. However, across an equally wide swath …
nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications
Physics-informed neural networks (PINNs) are effective in solving inverse problems based
on differential and integro-differential equations with sparse, noisy, unstructured, and multi …
on differential and integro-differential equations with sparse, noisy, unstructured, and multi …
Towards a unified theory of fractional and nonlocal vector calculus
Nonlocal and fractional-order models capture effects that classical partial differential
equations cannot describe; for this reason, they are suitable for a broad class of engineering …
equations cannot describe; for this reason, they are suitable for a broad class of engineering …
Fractional operators applied to geophysical electromagnetics
CJ Weiss, BG van Bloemen Waanders… - Geophysical Journal …, 2020 - academic.oup.com
SUMMARY A growing body of applied mathematics literature in recent years has focused on
the application of fractional calculus to problems of anomalous transport. In these analyses …
the application of fractional calculus to problems of anomalous transport. In these analyses …
External optimal control of nonlocal PDEs
Abstract Very recently Warma (2019 SIAM J. Control Optim. to appear) has shown that for
nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability …
nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability …
An optimization-based approach to parameter learning for fractional type nonlocal models
Nonlocal operators of fractional type are a popular modeling choice for applications that do
not adhere to classical diffusive behavior; however, one major challenge in nonlocal …
not adhere to classical diffusive behavior; however, one major challenge in nonlocal …
A priori error estimates for the optimal control of the integral fractional Laplacian
We design and analyze solution techniques for a linear-quadratic optimal control problem
involving the integral fractional Laplacian. We derive existence and uniqueness results, first …
involving the integral fractional Laplacian. We derive existence and uniqueness results, first …
Towards a unified theory of fractional and nonlocal vector calculus
Nonlocal and fractional-order models capture effects that classical partial differential
equations cannot describe; for this reason, they are suitable for a broad class of engineering …
equations cannot describe; for this reason, they are suitable for a broad class of engineering …
A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems
Fractional equations have become the model of choice in several applications where
heterogeneities at the microstructure result in anomalous diffusive behavior at the …
heterogeneities at the microstructure result in anomalous diffusive behavior at the …
Error estimates for the optimal control of a parabolic fractional PDE
We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic
optimal control problem for the so-called fractional heat equation; control constraints are …
optimal control problem for the so-called fractional heat equation; control constraints are …