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Convolution Equations: Theory, Numerical Methods and Applications
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Convolution Equations: Theory, Numerical Methods and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 5559

Special Issue Editors

Prof. Dr. Ilya Boykov

E-Mail Website
Guest Editor
Department of Higher and Applied Mathematics, Penza State University, 40 Krasnaya Str., 440026 Penza, Russia
Interests: numerical analysis; approximation theory; singular and hypersingular integrals and integral equations; stability theory; dynamical systems; indentification methods; mathematical methods in geophysics and immunology
Dr. Alla Boykova

E-Mail Website
Guest Editor
Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia
Interests: numetical analysis; mathematical methods in geophysics; statistics; data science

Special Issue Information

Dear Colleagues, 

Convolution equations and related Riemann and Hilbert boundary value problems comprise an active and growing field in mathematics. This field involves a large number of independent but closely related disciplines. Convolution equations also include singular integral equations, hypersingular integral equations, Wiener–Hopf equations, integral equations with fractional integrals and derivatives, and many others.

It is not easy to list all areas in physics, engineering, and natural sciences where convolution equations play a key role, but some include: aerodynamics, electrodynamics, elasticity theory, automatic control theory, astrophysics, wave theory, materials science, and others.

Although many subjects known as “Convolution equations” have been known for over a century, they are being actively developed today.

Besides the branches mentioned above, singular integral equations and the related Riemann and Hilbert boundary value problems on fractals have been been actively studied over the last few decades.

In this Special Issue, we will present the current state of these areas and trace the connections between them.

This Special Issue will include works devoted to analytical methods for solving hypersingular integral equations, as well as approximate methods for solving weakly singular convolutional integral equations (Volterra and Fredholm), singular integral equations, hypersingular and polyhypersingular integral equations (linear and nonlinear) defined on various manifolds, integral equations with fractional integrals, Wiener–Hopf equations, Ambartsumian–Chandrasekhar equations, and systems of equations, which play an important role in the theory of light scattering in turbid media.

Numerical methods for solving integral equations rely upon efficient numerical methods for integral evaluation. This Special Issue also presents works dedicated to approximate methods for calculating singular and hypersingular integrals.

Prof. Dr. Ilya Boykov
Dr. Alla Boykova
Guest Editors

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • weakly singular convolutional integral equations (Volterra and Fredholm)
  • singular, hypersingular, and polyhypersingular integral equations
  • approximate methods
  • integral equations with fractional integrals
  • Ambartsumian–Chandrasekhar systems of equations
  • Wiener–Hopf equations

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Published Papers (5 papers)

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Research

18 pages, 8461 KiB  
Article
“Spectral Method” for Determining a Kernel of the Fredholm Integral Equation of the First Kind of Convolution Type and Suppressing the Gibbs Effect
by Valery Sizikov and Nina Rushchenko
Mathematics 2024, 12(1), 13; https://doi.org/10.3390/math12010013 - 20 Dec 2023
Cited by 1 | Viewed by 1091
Abstract
A set of one-dimensional (as well as one two-dimensional) Fredholm integral equations (IEs) of the first kind of convolution type is solved. The task for solving these equations is ill-posed (first of all, unstable); therefore, the Wiener parametric filtering method (WPFM) and the [...] Read more.
A set of one-dimensional (as well as one two-dimensional) Fredholm integral equations (IEs) of the first kind of convolution type is solved. The task for solving these equations is ill-posed (first of all, unstable); therefore, the Wiener parametric filtering method (WPFM) and the Tikhonov regularization method (TRM) are used to solve them. The variant is considered when a kernel of the integral equation (IE) is unknown or known inaccurately, which generates a significant error in the solution of IE. The so-called “spectral method” is being developed to determine the kernel of an integral equation based on the Fourier spectrum, which leads to a decrease of the error in solving the IE and image improvement. Moreover, the authors also propose a method for diffusing the solution edges to suppress the possible Gibbs effect (ringing-type distortions). As applications, the problems for processing distorted (smeared, defocused, noisy, and with the Gibbs effect) images are considered. Numerical examples are given to illustrate the use of the “spectral method” to enhance the accuracy and stability of processing distorted images through their mathematical and computer processing. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
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19 pages, 303 KiB  
Article
Hypersingular Integro-Differential Equation Containing Polynomials and Their Derivatives in Coefficients
by Andrei P. Shilin
Mathematics 2023, 11(24), 4940; https://doi.org/10.3390/math11244940 - 12 Dec 2023
Viewed by 828
Abstract
A new linear integro-differential equation is considered on a closed curve located on the complex plane. The integrals in the equation are understood in the sense of a finite part, according to Hadamard. The order of the equation can be higher than one. [...] Read more.
A new linear integro-differential equation is considered on a closed curve located on the complex plane. The integrals in the equation are understood in the sense of a finite part, according to Hadamard. The order of the equation can be higher than one. The coefficients of the equation have a special structure. A characteristic of the coefficients is that they contain two arbitrary polynomials and their derivatives. Solvability conditions are explicitly stated. Whenever they are satisfied, an exact analytical solution is given. Generalized Sokhotsky formulas, the theory of Riemann boundary value problems, methods for solving linear differential equations, and the properties of analytic functions of a complex variable are used. An example is given. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
20 pages, 667 KiB  
Article
Regular, Singular and Hypersingular Integrals over Fractal Contours
by Ilya Boykov, Vladimir Roudnev and Alla Boykova
Mathematics 2023, 11(23), 4752; https://doi.org/10.3390/math11234752 - 24 Nov 2023
Viewed by 907
Abstract
The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a [...] Read more.
The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a definition of a singular integral over non-rectifiable curves and fractals which generalizes the known one. We define hypersingular integrals over non-rectifiable curves and fractals. We construct quadratures for the calculation of Riemann definite integrals, singular and hypersingular integrals over non-rectifiable curves and fractals and the corresponding error estimates for various classes of functions. Singular and hypersingular integrals are defined up to an additive constant (or a combination of constants) that are subject to a convention that depends on the actual problem being solved. We illustrate our theoretical results with numerical examples for Riemann definite integrals, singular integrals and hypersingular integrals over fractals. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
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30 pages, 382 KiB  
Article
A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind
by H. X. Mamatova, Z. K. Eshkuvatov and Sh. Ismail
Mathematics 2023, 11(20), 4404; https://doi.org/10.3390/math11204404 - 23 Oct 2023
Viewed by 1082
Abstract
In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, [...] Read more.
In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
16 pages, 320 KiB  
Article
Application of Wavelet Transform to Urysohn-Type Equations
by V. Lukianenko, M. Kozlova and V. Belozub
Mathematics 2023, 11(18), 3999; https://doi.org/10.3390/math11183999 - 20 Sep 2023
Viewed by 916
Abstract
This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet [...] Read more.
This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet transform is applied to convolution-type equations (based on the Fourier and wavelet transforms) and to Urysohn equations with unknown shift. The wavelet transform is preferable for the cases with approximated right-hand sides and for type 1 equations. We demonstrated that the application of the wavelet transform to Urysohn-type equations with unknown shift translates into a solution of a nonlinear equation with an oscillating kernel. Depending on the availability of a priori information, a combination of regularization and iterative algorithms with the use of close equations are effective for solving convolution-type equations based on the continuous wavelet transform and Urysohn equation. Full article
(This article belongs to the Special Issue Convolution Equations: Theory, Numerical Methods and Applications)
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