# Resonant Scattering and Generation of WavesNumerical Treatment of the System of Integral Equations

Resonant Scattering and Generation of Waves: Numerical Treatment of the System of Integral Equations [The numerical solution of the system of nonlinear Hammerstein integral equations of second kind (3.17) is based on the so-called Nyström method, where the integrals are approximated by appropriate quadrature rules. As the result of this method, a nonlinear system of complex algebraic equations arises. Analogously to the finite element method described in Sect. 5.1, we consider N∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N},$$\end{document}N≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2,$$\end{document} nodes zj,Nj=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ z_{j,N}\right\} _{j=1}^N$$\end{document} such that -2πδ=:z1,N<z2,N<⋯<zN-1,N<zN,N:=2πδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2\pi \delta =:z_{1,N}<z_{2,N}<\cdots<z_{N-1,N}<z_{N,N}:=2\pi \delta$$\end{document} and the subintervals Ij,N=zj,N,zj+1,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {I}_{j,N}=\left( z_{j,N},z_{j+1,N}\right)$$\end{document} with the lengths hj,N=zj+1,N-zj,N.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{j,N}=z_{j+1,N}-z_{j,N}.$$\end{document} Then, given a continuous function v:Icl→C,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {C},$$\end{document} a numerical integration scheme for the integral I(v):=∫Iv(z)dz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} I(v):=\int _\mathscr {I}v(z)dz \end{aligned}\end{document}can be defined by a quadrature rule IN(v):=∑j=1Nνj,Nv(zj,N),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} I_N(v):=\sum _{j=1}^N \nu _{j,N} v(z_{j,N})\,, \end{aligned}\end{document}where the coefficients νj,N∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{j,N}\in \mathbb {R}$$\end{document} are known.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Resonant Scattering and Generation of WavesNumerical Treatment of the System of Integral Equations

7 pages

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# References (6)

Publisher
Springer International Publishing
© Springer International Publishing AG, part of Springer Nature 2019
ISBN
978-3-319-96300-6
Pages
115 –122
DOI
10.1007/978-3-319-96301-3_6
Publisher site
See Chapter on Publisher Site

### Abstract

[The numerical solution of the system of nonlinear Hammerstein integral equations of second kind (3.17) is based on the so-called Nyström method, where the integrals are approximated by appropriate quadrature rules. As the result of this method, a nonlinear system of complex algebraic equations arises. Analogously to the finite element method described in Sect. 5.1, we consider N∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N},$$\end{document}N≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2,$$\end{document} nodes zj,Nj=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ z_{j,N}\right\} _{j=1}^N$$\end{document} such that -2πδ=:z1,N<z2,N<⋯<zN-1,N<zN,N:=2πδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2\pi \delta =:z_{1,N}<z_{2,N}<\cdots<z_{N-1,N}<z_{N,N}:=2\pi \delta$$\end{document} and the subintervals Ij,N=zj,N,zj+1,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {I}_{j,N}=\left( z_{j,N},z_{j+1,N}\right)$$\end{document} with the lengths hj,N=zj+1,N-zj,N.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{j,N}=z_{j+1,N}-z_{j,N}.$$\end{document} Then, given a continuous function v:Icl→C,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {C},$$\end{document} a numerical integration scheme for the integral I(v):=∫Iv(z)dz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} I(v):=\int _\mathscr {I}v(z)dz \end{aligned}\end{document}can be defined by a quadrature rule IN(v):=∑j=1Nνj,Nv(zj,N),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned} I_N(v):=\sum _{j=1}^N \nu _{j,N} v(z_{j,N})\,, \end{aligned}\end{document}where the coefficients νj,N∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{j,N}\in \mathbb {R}$$\end{document} are known.]

Published: Jul 27, 2018

Keywords: Nonlinear Hammerstein Integral Equations; Quadrature Rule; Implemented Numerical Algorithms; Numerical Spectral Analysis; Finding Nontrivial Solutions