scholarly journals Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 223
Author(s):  
Pedro González-Rodelas ◽  
Miguel Pasadas ◽  
Abdelouahed Kouibia ◽  
Basim Mustafa
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Volterra Integral Equation ◽  
Computational Cost ◽  
Basis Functions ◽  
Numerical Examples ◽  
Acceptable Accuracy ◽  
Convergence Results ◽  
Radial Basis ◽  
Low Computational Cost

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

Numerical Investigation of Electromagnetic Scattering Problems Based on the Compactly Supported Radial Basis Functions

2016 ◽  
Vol 71 (8) ◽  
pp. 677-690 ◽  
Author(s):  
Hadi Roohani Ghehsareh ◽  
Seyed Kamal Etesami ◽  
Maryam Hajisadeghi Esfahani
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Cross Section ◽  
Computational Cost ◽  
Basis Functions ◽  
Test Problems ◽  
Em Scattering ◽  
Compactly Supported ◽  
Radial Basis ◽  
Field Integral

AbstractIn the current work, the electromagnetic (EM) scattering from infinite perfectly conducting cylinders with arbitrary cross sections in both transverse magnetic (TM) and transverse electric (TE) modes is numerically investigated. The problems of TE and TM EM scattering can be mathematically modelled via the magnetic field integral equation (MFIE) and the electric field integral equation (EFIE), respectively. An efficient technique is performed to approximate the solution of these surface integral equations. In the proposed numerical method, compactly supported radial basis functions (RBFs) are employed as the basis functions. The radial and compactly supported properties of these basis functions substantially reduce the computational cost and improve the efficiency of the method. To show the accuracy of the proposed technique, it has been applied to solve three interesting test problems. Moreover, the method is well used to compute the electric current density and also the radar cross section (RCS) for some practical scatterers with different cross section geometries. The reported numerical results through the tables and figures demonstrate the efficiency and accuracy of the proposed technique.

A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

2021 ◽  
Vol 63 ◽  
pp. 469-492
Author(s):  
Pouria Assari ◽  
Fatemeh Asadi-Mehregan ◽  
Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Integral Equation Method ◽  
Basis Functions ◽  
Computer Memory ◽  
Shape Functions ◽  
Integration Rule ◽  
Radial Basis ◽  
Small Set ◽  
Discrete Galerkin Method

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. doi:10.1017/S1446181121000377

A MESHLESS LOCAL GALERKIN INTEGRAL EQUATION METHOD FOR SOLVING A TYPE OF DARBOUX PROBLEMS BASED ON RADIAL BASIS FUNCTIONS

2021 ◽  
pp. 1-24
Author(s):  
P. ASSARI ◽  
F. ASADI-MEHREGAN ◽  
M. DEHGHAN
Keyword(s):  
Integral Equation ◽  
Radial Basis Functions ◽  
Integral Equation Method ◽  
Basis Functions ◽  
Computer Memory ◽  
Shape Functions ◽  
Integration Rule ◽  
Radial Basis ◽  
Small Set ◽  
Discrete Galerkin Method

Abstract The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions.

DIV-CURL WEIGHTED MINIMIZING SPLINES

2007 ◽  
Vol 05 (02) ◽  
pp. 95-122 ◽  
Author(s):  
M. N. BENBOURHIM ◽  
A. BOUHAMIDI
Keyword(s):  
Vector Field ◽  
Radial Basis Functions ◽  
Thin Plate ◽  
Approximation Problem ◽  
Scattered Data ◽  
Scalar Function ◽  
Basis Functions ◽  
Thin Plate Splines ◽  
Numerical Examples ◽  
Radial Basis

The paper deals with a div-curl approximation problem by weighted minimizing splines. The weighted minimizing splines are an extension of the well-known thin plate splines and are radial basis functions which allow the approximation or the interpolation of a scalar function from given scattered data. In this paper, we show that the theory of the weighted minimizing splines may also be used for the approximation or for the interpolation of a vector field controlled by the divergence and the curl of the vector field. Numerical examples are given to show the efficiency of this method.

A NUMERICAL METHOD FOR 1D TIME-DEPENDENT SCHRÖDINGER EQUATION USING RADIAL BASIS FUNCTIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341010 ◽  
Author(s):  
TONGSONG JIANG ◽  
ZHAOLIN JIANG ◽  
JOSEPH KOLIBAL
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Radial Basis Functions ◽  
Finite Difference Scheme ◽  
Good Accuracy ◽  
Basis Functions ◽  
Time Dependent ◽  
Numerical Examples ◽  
Radial Basis

This paper proposes a new numerical method to solve the 1D time-dependent Schrödinger equations based on the finite difference scheme by means of multiquadrics (MQ) and inverse multiquadrics (IMQ) radial basis functions. The numerical examples are given to confirm the good accuracy of the proposed methods.

scholarly journals Free vibrations and buckling analysis of laminated plates by oscillatory radial basis functions

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. M. A. Neves ◽  
A. J. M. Ferreira
Keyword(s):  
Radial Basis Functions ◽  
Meshless Method ◽  
Free Vibrations ◽  
Buckling Analysis ◽  
Laminated Plates ◽  
Basis Functions ◽  
Numerical Examples ◽  
Transverse Normal Stress ◽  
Radial Basis ◽  
Refined Version

AbstractIn this paper the free vibrations and buckling analysis of laminated plates is performed using a global meshless method. A refined version of Kant’s theorie which accounts for transverse normal stress and through-the-thickness deformation is used. The innovation is the use of oscillatory radial basis functions. Numerical examples are performed and results are presented and compared to available references. Such functions proved to be an alternative to the tradicional nonoscillatory radial basis functions.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

scholarly journals Use of Multiquadric Functions for Multivariable Representation of the Aerodynamic Coefficients of Airfoils

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Filipe Ribeiro ◽  
Pedro Albuquerque ◽  
Pedro Gamboa ◽  
Kouamana Bousson
Keyword(s):  
Radial Basis Functions ◽  
Piecewise Linear ◽  
Research Work ◽  
Computational Cost ◽  
Basis Functions ◽  
Aerodynamic Coefficients ◽  
Independent Variables ◽  
Advantages And Disadvantages ◽  
Radial Basis ◽  
Multivariable Interpolation

Given an array (or matrix) of values for a function of one or more variables, it is often desired to find a value between two given points. Multivariable interpolation and approximation by radial basis functions are important subjects in approximation theory that have many applications in Science and Engineering fields. During the last decades, radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. This research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. The main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor. It also benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. The approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio (cflap), and flap deflection (δflap).

Sign in / Sign up

Export Citation Format

Share Document