Advances in Numerical Analysis and Meshless Methods

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 20 July 2024 | Viewed by 1826

Special Issue Editor


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Guest Editor
Department of Data Science and Big Data Analytics, Providence University Taiwan, Sha-Lu, Taiwan
Interests: radial basis functions (RBFs); collocation method; high dimensional problems

Special Issue Information

Dear Colleagues,

In recent years, meshless methods have attracted the attention from many scientists due to their efficiency and accuracy. In numerical Partial Differential Equations (PDEs) this kind of methods is competing with Finite Element Method (FEM), Finite-Difference Methods (FDM), Finite Volume Method (FVM), and Boundary Element Method (BEM). The fast growing amount of relevant papers shows that meshless methods apply to many scientific fields very well. 

This Special Issue aims at promoting the current meshless methods both in theory and practice. Consequently, both theoretical and experimental works are welcome.

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but not limited to) the following: 

  • Radial Basis Functions(RBF);
  • Collocation Method;
  • Numerical Partial Differential Equations;
  • Approximation Theory;
  • Mathematical Modelling. 

We look forward to receiving your contributions. 

Prof. Dr. Lintian Luh
Guest Editor

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Keywords

  • meshless method
  • radial basis function
  • colocation
  • radial basis functions(RBF)
  • collocation method
  • numerical partial differential equations
  • approximation theory
  • mathematical modeling

Published Papers (1 paper)

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Research

17 pages, 5222 KiB  
Article
Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation
by Miglena N. Koleva and Lubin G. Vulkov
Axioms 2024, 13(1), 64; https://doi.org/10.3390/axioms13010064 - 19 Jan 2024
Viewed by 915
Abstract
In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical [...] Read more.
In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis and Meshless Methods)
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