131 research outputs found
MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling
We propose, Monte Carlo Nonlocal physics-informed neural networks
(MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in
\cite{guo2022monte}, for solving general nonlocal models such as integral
equations and nonlocal PDEs. Similar as in MC-fPINNs, our MC-Nonlocal-PINNs
handle the nonlocal operators in a Monte Carlo way, resulting in a very stable
approach for high dimensional problems. We present a variety of test problems,
including high dimensional Volterra type integral equations, hypersingular
integral equations and nonlocal PDEs, to demonstrate the effectiveness of our
approach.Comment: 23pages, 13figure
A NUMERICAL METHOD FOR SOLVING SYSTEMS OF HYPERSINGULAR INTEGRO-DIFFERENTIAL EQUATIONS
This paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallee Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Holder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy
Modified homotopy perturbation method for solving hypersingular integral equations of the first kind
Resonant behaviour of an oscillating wave energy converter in a channel
A mathematical model is developed to study the behaviour of an oscillating
wave energy converter in a channel. During recent laboratory tests in a wave
tank, peaks in the hydrodynamic actions on the converter occurred at certain
frequencies of the incident waves. This resonant mechanism is known to be
generated by the transverse sloshing modes of the channel. Here the influence
of the channel sloshing modes on the performance of the device is further
investigated. Within the framework of a linear inviscid potential-flow theory,
application of the Green theorem yields a hypersingular integral equation for
the velocity potential in the fluid domain. The solution is found in terms of a
fast-converging series of Chebyshev polynomials of the second kind. The
physical behaviour of the system is then analysed, showing sensitivity of the
resonant sloshing modes to the geometry of the device, that concurs in
increasing the maximum efficiency. Analytical results are validated with
available numerical and experimental data.Comment: Accepted for publicatio
Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay
An elastic continuum model with long-range forces is addressed in this study within the context of approximate analytical methods Such a model stems from a mechanically-based approach to non-local theory where long-range central forces are introduced between non-adjacent volume elements Specifically, long-range forces depend on the relative displacement on the volume product between interacting elements and they are proportional to a proper, material-dependent, distance-decaying function Smooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-decay functions lead to a fractional differential governing equation of Marchaud type In this paper the Galerkin and the Rayleigh-Ritz method are used to build approximate solutions to the integro-differential and the fractional differential governing equations Numerical applications show the accuracy of the proposed approximate solutions as compared to the finite difference approximation and to the fractional finite difference approximatio
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight times a ``regular´´ unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the ``regular´´ unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babu{s}ka and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nystr"om numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Borthagaray, Juan Pablo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Bruno, Oscar Ricardo. California Institute Of Technology; Estados UnidosFil: Maas, MartÃn Daniel. Consejo Nacional de Investigaciónes CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de AstronomÃa y FÃsica del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de AstronomÃa y FÃsica del Espacio; Argentin
Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions
The paper deals with the approximate solution of integro-differential
equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange
interpolation processes are proposed and conditions under which they are stable
and convergent in suitable weighted spaces of continuous functions are proved.
The efficiency of the method has been tested by some numerical experiments,
some of them including comparisons with other numerical procedures. In
particular, as an application, we have implemented the method for solving
Prandtl's equation governing the circulation air flow along the contour of a
plane wing profile, in the case of elliptic or rectangular wing-shape.Comment: 34 page
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