Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations
We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today,hypersingular integral equations of this t...
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doaj-f405fdccc6b44b73899914a5743254382020-11-25T03:40:41ZengMDPI AGAxioms2075-16802020-07-019747410.3390/axioms9030074Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral EquationsIlya Boykov0Vladimir Roudnev1Alla Boykova2Department of Mathematics, The Penza State University, 40, Krasnaya Str., 440026 Penza, RussiaDepartment of Computational Physics, Saint Petersburg State University, 7/9 Universitetskaya Emb., 199034 Saint Petersburg, RussiaDepartment of Mathematics, The Penza State University, 40, Krasnaya Str., 440026 Penza, RussiaWe propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today,hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method.https://www.mdpi.com/2075-1680/9/3/74hypersingular integral equationsiterative projection methodLyapunov stability theory |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Ilya Boykov Vladimir Roudnev Alla Boykova |
spellingShingle |
Ilya Boykov Vladimir Roudnev Alla Boykova Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations Axioms hypersingular integral equations iterative projection method Lyapunov stability theory |
author_facet |
Ilya Boykov Vladimir Roudnev Alla Boykova |
author_sort |
Ilya Boykov |
title |
Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations |
title_short |
Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations |
title_full |
Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations |
title_fullStr |
Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations |
title_full_unstemmed |
Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations |
title_sort |
approximate methods for solving linear and nonlinear hypersingular integral equations |
publisher |
MDPI AG |
series |
Axioms |
issn |
2075-1680 |
publishDate |
2020-07-01 |
description |
We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today,hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method. |
topic |
hypersingular integral equations iterative projection method Lyapunov stability theory |
url |
https://www.mdpi.com/2075-1680/9/3/74 |
work_keys_str_mv |
AT ilyaboykov approximatemethodsforsolvinglinearandnonlinearhypersingularintegralequations AT vladimirroudnev approximatemethodsforsolvinglinearandnonlinearhypersingularintegralequations AT allaboykova approximatemethodsforsolvinglinearandnonlinearhypersingularintegralequations |
_version_ |
1724533412071473152 |