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Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods [article]

Siavash Jafarzadeh, Adam Larios, Florin Bobaru
2019 arXiv   pre-print
We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions.  ...  To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem.  ...  Acknowledgments This work has been supported by the AFOSR MURI Center for Materials Failure Prediction through Peridynamics (program managers Jaimie Tiley, David Stargel, Ali Sayir, Fariba Fahroo, and  ... 
arXiv:1905.03875v1 fatcat:aqnmqxhaxncqth4rerxbg4pjca

Efficient Solutions for Nonlocal Diffusion Problems Via Boundary-Adapted Spectral Methods

Siavash Jafarzadeh, Adam Larios, Florin Bobaru
2020 Journal of Peridynamics and Nonlocal Modeling  
We introduce an efficient boundary-adapted spectral method for peridynamic transient diffusion problems with arbitrary boundary conditions.  ...  To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem.  ...  Boundary-adapted spectral method implementation for PD diffusion in MATLAB Here the MATLAB implantation for boundary-adapted spectral method with volume penalization (BASM-VP) is provided.  ... 
doi:10.1007/s42102-019-00026-6 fatcat:ykqkg6vm5rdy7ez4libbafurfq

Page 4333 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
The diffusion is spectrally discretized and integrated exactly in time. The reaction term is integrated via an explicit Runge-Kutta scheme of order 4.  ...  order and more simple method but also an adaptive one.” 96g:65108 65N25 Lui, S.  ... 

Recent Advance in Function Spaces and Their Applications in Fractional Differential Equations

Xinguang Zhang, Lishan Liu, Yonghong Wu, Liguang Wang
2019 Journal of Function Spaces  
for a class of nonlinear time fractional partial differential equations with delay by using the theory of solution operator and the general Banach contraction mapping principle.  ...  In the paper titled "Existence Results for a Class of Semilinear Fractional Partial Differential Equations with Delay in Banach Spaces", the authors consider the existence and uniqueness of the mild solutions  ...  for integral boundary value problems of nonlinear Hadamard fractional differential equations by using fixed point methods.  ... 
doi:10.1155/2019/5719808 fatcat:mwdi7rybjvabjot56zpesc6f7a

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Raimondas Čiegis, Remigijus Čiegis, Ignas Dapšys
2021 Mathematics  
The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators.  ...  A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem  ...  Nonuniform and Adaptive Time Meshes It is well known that adaptive time meshes efficiently resolve fast and slow dynamics of the solution, including stiff problems.  ... 
doi:10.3390/math9121344 fatcat:tocmyvjx3bfftfk2yx3gphfvom

Numerical methods for nonlocal and fractional models [article]

Marta D'Elia, Qiang Du, Christian Glusa, Max Gunzburger, Xiaochuan Tian, Zhi Zhou
2020 arXiv   pre-print
We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered  ...  In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples.  ...  SNL is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the US Department  ... 
arXiv:2002.01401v2 fatcat:jmqemyhsgrefrm5kuhg4ramoca

The diffusion equation with nonlocal data [article]

Peter D. Miller, David A. Smith
2017 arXiv   pre-print
Despite arising naturally from the Fokas transform method, the uniqueness argument appears to be novel even for initial-boundary value problems.  ...  The solution and well-posedness results rely upon an extension of the Fokas (or unified) transform method to initial-nonlocal value problems for linear equations; the necessary extensions are described  ...  The authors wish to thank David Stewart for providing a Chemist's perspective on the proposed experimental method.  ... 
arXiv:1708.00972v1 fatcat:3uvpmcxeuzf67myj2hj4tftcfe

Page 1290 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
By using the boundary element method and Fourier transform, the author solves initial-boundary value problems for a class of two- dimensional reaction-diffusion equations with different diffusion coefficients  ...  This paper discusses a coupled finite element and boundary ele- ment method for solving exterior Dirichlet problems.  ... 

On a nonlocal moving frame approximation of traveling waves

Jose M. Arrieta, Maria López-Fernández, Enrique Zuazua
2011 Comptes rendus. Mathematique  
In this new formulation both the profile and the propagation speed of the traveling waves emerge as asymptotic limits of solutions of a nonlocal reaction-diffusion problem when time goes to infinity.  ...  The profiles of traveling wave solutions of a 1-d reaction-diffusion parabolic equation are transformed into equilibria of a nonlocal equation, by means of an appropriate nonlocal change of variables.  ...  Asymptotic stability of the stationary solutions of the nonlocal problem Our last main result ensures the stability of the stationary solutions of the nonlocal problem.  ... 
doi:10.1016/j.crma.2011.07.001 fatcat:7vs7fkeisvcqvkrztwxijfu5zu

An a posteriori error estimator for the spectral fractional power of the Laplacian [article]

Raphaël Bulle, Olga Barrera, Stéphane P. A. Bordas, Franz Chouly, Jack S. Hale
2022 arXiv   pre-print
used to compute the solution to the original fractional problem.  ...  We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian.  ...  Efficient methods for solving fractional problems typically rely on a combination of different discretization methods.  ... 
arXiv:2202.05810v1 fatcat:ilx4ouzts5h27pqcu5atuk2v5q

What Is the Fractional Laplacian? [article]

Anna Lischke, Guofei Pang, Mamikon Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth, George Em Karniadakis
2019 arXiv   pre-print
The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its  ...  in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions.  ...  ., for the U.S.  ... 
arXiv:1801.09767v3 fatcat:nmzykkn2vzaqnpf272cgq6lypq

An MBO Scheme on Graphs for Classification and Image Processing

Ekaterina Merkurjev, Tijana Kostić, Andrea L. Bertozzi
2013 SIAM Journal of Imaging Sciences  
We are particularly interested in nonlocal total variation methods, as they are a link between spectral graph theory and diffuse interface models and thus can be used as a motivation for our algorithm.  ...  The results show that our method is multiple times more efficient than other well-known nonlocal models. Abstract.  ...  The authors would like to thank Yanina Landa for providing a MAT-LAB version of the code of the algorithm in [8] , and Chris Anderson for providing a code for the Raleigh-Chebyshev procedure of [1] .  ... 
doi:10.1137/120886935 fatcat:sbf543ubebf2ho6fqcwtvmmiri

Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains [article]

Mingtao Xia, Sihong Shao, Tom Chou
2020 arXiv   pre-print
We apply our methods to a number of different models, including diffusive and moving Fermi-Dirac distributions and nonlinear Dirac solitary waves, and demonstrate recovery of spectral convergence for time-dependent  ...  to the diffusive behavior of the solution.  ...  Spectral methods incorporating both scaling and moving. For problems that involve both translation and diffusion in unbounded domains, we need to incorporate both the moving and scaling procedures.  ... 
arXiv:2009.13170v1 fatcat:v3v5xqswtvhdzoejk4kjiju6oy

A space-fractional Monodomain model for cardiac electrophysiology combining anisotropy and heterogeneity on realistic geometries

N. Cusimano, L. Gerardo-Giorda
2018 Journal of Computational Physics  
The numerical method for the solution of the fractional Monodomain relies on an integral representation of the nonlocal operator combined with a finite element discretisation in space, allowing to handle  ...  We consider a modification of the Monodomain model obtained by replacing the diffusive term of the classical formulation with a fractional power of the operator, defined in the spectral sense.  ...  [17] analysed in detail three methods for the solution of fractional-in-space reaction-diffusion equations involving fractional powers of a suitable matrix linked to the finite element mass and stiffness  ... 
doi:10.1016/ fatcat:wdj362zpovf2bj7aclrotkhcfy


Yong Zhou, Vasily E. Tarasov, Juan J. Trujillo, Roberto Garrappa
2013 The European Physical Journal Special Topics  
In paper [16] the theory of Hausdorff measure of noncompactness and fixed point theorems are applied to study the abstract nonlocal Cauchy problem of a class a  ...  New theories and methods are thus required to be specifically developed for FPDEs, whose investigation becomes more challenging.  ...  Wang for their support and help.  ... 
doi:10.1140/epjst/e2013-01960-6 fatcat:w56ojdosfzecre3czom3teegl4
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