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Symmetry
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Integral Equations: Theories, Approximations and Applications
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Integral Equations: Theories, Approximations and Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 28401

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Samad Noeiaghdam

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Guest Editor
1. Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
2. Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia
Interests: numerical analysis; solving integral equations; solving ODEs and PDEs; solving ill-posed problems; fuzzy mathematics; stochastic arithmetic; CADNA library; CESTAC method; solving biomathematical models; iterative methods; numerical methods
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Denis N. Sidorov

E-Mail Website
Guest Editor
Energy Systems Institute, Siberian Branch of Russian Academy of Sciences, 664033 Irkutsk, Russia
Interests: integral equations; machine learning; power systems; inverse problems; MPC; cyber physical systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We invite you to submit a research paper in the area of integral equations to this Special Issue, entitled “Integral Equations: Theories, Applications, and Approximations”, of the journal Symmetry. We seek studies on new and innovative approaches to exactly or approximately solving the first and second kinds of integral equations in linear and nonlinear forms. We also seek to cover high-dimensional and systems of integral equations. We welcome submissions presenting new theoretical results, structural investigations, new models and algorithmic approaches, and new applications of integral equations.

Prof. Dr. Samad Noeiaghdam
Prof. Dr. Denis N. Sidorov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 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

  • first-kind integral equations
  • second-kind integral equations
  • Volterra integral equations
  • Fredholm integral equations
  • linear and nonlinear problems
  • singular problems
  • ill-posed problems
  • systems of integral equations
  • high-dimensional integral equations
  • convergence analysis
  • error analysis
  • numerical methods
  • analytical methods
  • semi-analytical methods

Published Papers (12 papers)

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Editorial

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4 pages, 173 KiB  
Editorial
Integral Equations: Theories, Approximations, and Applications
by Samad Noeiaghdam and Denis Sidorov
Symmetry 2021, 13(8), 1402; https://doi.org/10.3390/sym13081402 - 02 Aug 2021
Cited by 3 | Viewed by 2255
Abstract
Linear and nonlinear integral equations of the first and second kinds have many applications in engineering and real life problems [...] Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)

Research

Jump to: Editorial

12 pages, 293 KiB  
Article
Numerical Solution of Two-Dimensional Fredholm–Volterra Integral Equations of the Second Kind
by Sanda Micula
Symmetry 2021, 13(8), 1326; https://doi.org/10.3390/sym13081326 - 23 Jul 2021
Cited by 5 | Viewed by 1884
Abstract
The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their [...] Read more.
The paper presents an iterative numerical method for approximating solutions of two-dimensional Fredholm–Volterra integral equations of the second kind. As these equations arise in many applications, there is a constant need for accurate, but fast and simple to use numerical approximations to their solutions. The method proposed here uses successive approximations of the Mann type and a suitable cubature formula. Mann’s procedure is known to converge faster than the classical Picard iteration given by the contraction principle, thus yielding a better numerical method. The existence and uniqueness of the solution is derived under certain conditions. The convergence of the method is proved, and error estimates for the approximations obtained are given. At the end, several numerical examples are analyzed, showing the applicability of the proposed method and good approximation results. In the last section, concluding remarks and future research ideas are discussed. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
23 pages, 341 KiB  
Article
Multistep Methods of the Hybrid Type and Their Application to Solve the Second Kind Volterra Integral Equation
by Vagif Ibrahimov and Mehriban Imanova
Symmetry 2021, 13(6), 1087; https://doi.org/10.3390/sym13061087 - 18 Jun 2021
Cited by 8 | Viewed by 2003
Abstract
There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from [...] Read more.
There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the construction methods, with the best properties of the intersection of them. We also show some connection of the methods constructed here with the methods which are using solving of the initial-value problem for ODEs of the first order. Some of the constructed methods have been applied to solve model problems. A formula is proposed to determine the maximal values of the order of accuracy for the stable and unstable methods, constructed here. Note that to construct the new methods, here we propose to use the system of algebraic equations which allows us to construct methods with the best properties by using the minimal volume of the computational works at each step. For the construction of more exact methods, here we have proposed to use the multistep second derivative method, which has comparisons with the known methods. We have constructed some formulas to determine the maximal order of accuracy, and also determined the necessary and sufficient conditions for the convergence of the methods constructed here. One can proved by multistep methods, which are usually applied to solve the initial-value problem for ODE, demonstrating the applications of these methods to solve Volterra integro-differential equations. For the illustration of the results, we have constructed some concrete methods, and one of them has been applied to solve a model equation. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
17 pages, 2249 KiB  
Article
Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions
by Alexander A. Minakov and Christoph Schick
Symmetry 2021, 13(2), 256; https://doi.org/10.3390/sym13020256 - 03 Feb 2021
Cited by 9 | Viewed by 2611
Abstract
An integro-differential equation describes the non-equilibrium thermal response of glass-forming substances with a dynamic (time-dependent) heat capacity to fast thermal perturbations. We found that this heat transfer problem could be solved analytically for a heat source with an arbitrary time dependence and different [...] Read more.
An integro-differential equation describes the non-equilibrium thermal response of glass-forming substances with a dynamic (time-dependent) heat capacity to fast thermal perturbations. We found that this heat transfer problem could be solved analytically for a heat source with an arbitrary time dependence and different geometries. The method can be used to analyze the response to local thermal perturbations in glass-forming materials, as well as temperature fluctuations during subcritical crystal nucleation and decay. The results obtained can be useful for applications and a better understanding of the thermal properties of glass-forming materials, polymers, and nanocomposites. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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19 pages, 352 KiB  
Article
A Type of Time-Symmetric Stochastic System and Related Games
by Qingfeng Zhu, Yufeng Shi, Jiaqiang Wen and Hui Zhang
Symmetry 2021, 13(1), 118; https://doi.org/10.3390/sym13010118 - 12 Jan 2021
Cited by 2 | Viewed by 1567
Abstract
This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some [...] Read more.
This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
15 pages, 1124 KiB  
Article
Effects of Second-Order Velocity Slip and the Different Spherical Nanoparticles on Nanofluid Flow
by Jing Zhu, Ye Liu and Jiahui Cao
Symmetry 2021, 13(1), 64; https://doi.org/10.3390/sym13010064 - 31 Dec 2020
Cited by 9 | Viewed by 1854
Abstract
The paper theoretically investigates the heat transfer of nanofluids with different nanoparticles inside a parallel-plate channel. Second-order slip condition is adopted due to the microscopic roughness in the microchannels. After proper transformation, nonlinear partial differential systems are converted to ordinary differential equations with [...] Read more.
The paper theoretically investigates the heat transfer of nanofluids with different nanoparticles inside a parallel-plate channel. Second-order slip condition is adopted due to the microscopic roughness in the microchannels. After proper transformation, nonlinear partial differential systems are converted to ordinary differential equations with unknown constants, and then solved by homotopy analysis method. The residual plot is drawn to verify the convergence of the solution. The semi-analytical expressions between NuB and NBT are acquired. The results show that both first-order slip parameter and second-order slip parameter have positive effects on NuB of the MHD flow. The effect of second-order velocity slip on NuB is obvious, and NuB in the alumina–water nanofluid is higher than that in the titania–water nanofluid. The positive correlation between slip parameters and Ndp is significant for the titania–water nanofluid. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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16 pages, 2283 KiB  
Article
Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels
by Elham Hashemizadeh, Mohammad Ali Ebadi and Samad Noeiaghdam
Symmetry 2020, 12(12), 2105; https://doi.org/10.3390/sym12122105 - 17 Dec 2020
Cited by 10 | Viewed by 2144
Abstract
In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this [...] Read more.
In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority of the proposed method. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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15 pages, 295 KiB  
Article
A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind
by Sanda Micula
Symmetry 2020, 12(11), 1862; https://doi.org/10.3390/sym12111862 - 12 Nov 2020
Cited by 15 | Viewed by 2343
Abstract
This paper presents a numerical iterative method for the approximate solutions of nonlinear Volterra integral equations of the second kind, with weakly singular kernels. We derive conditions so that a unique solution of such equations exists, as the unique fixed point of an [...] Read more.
This paper presents a numerical iterative method for the approximate solutions of nonlinear Volterra integral equations of the second kind, with weakly singular kernels. We derive conditions so that a unique solution of such equations exists, as the unique fixed point of an integral operator. Iterative application of that operator to an initial function yields a sequence of functions converging to the true solution. Finally, an appropriate numerical integration scheme (a certain type of product integration) is used to produce the approximations of the solution at given nodes. The resulting procedure is a numerical method that is more practical and accessible than the classical approximation techniques. We prove the convergence of the method and give error estimates. The proposed method is applied to some numerical examples, which are discussed in detail. The numerical approximations thus obtained confirm the theoretical results and the predicted error estimates. In the end, we discuss the method, drawing conclusions about its applicability and outlining future possible research ideas in the same area. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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13 pages, 248 KiB  
Article
Matrix Expression of Convolution and Its Generalized Continuous Form
by Young Hee Geum, Arjun Kumar Rathie and Hwajoon Kim
Symmetry 2020, 12(11), 1791; https://doi.org/10.3390/sym12111791 - 29 Oct 2020
Cited by 10 | Viewed by 2654
Abstract
In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional [...] Read more.
In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
22 pages, 490 KiB  
Article
Detecting Optimal Leak Locations Using Homotopy Analysis Method for Isothermal Hydrogen-Natural Gas Mixture in an Inclined Pipeline
by Sarkhosh S. Chaharborj, Zuhaila Ismail and Norsarahaida Amin
Symmetry 2020, 12(11), 1769; https://doi.org/10.3390/sym12111769 - 26 Oct 2020
Cited by 6 | Viewed by 1856
Abstract
The aim of this article is to use the Homotopy Analysis Method (HAM) to pinpoint the optimal location of leakage in an inclined pipeline containing hydrogen-natural gas mixture by obtaining quick and accurate analytical solutions for nonlinear transportation equations. The homotopy analysis method [...] Read more.
The aim of this article is to use the Homotopy Analysis Method (HAM) to pinpoint the optimal location of leakage in an inclined pipeline containing hydrogen-natural gas mixture by obtaining quick and accurate analytical solutions for nonlinear transportation equations. The homotopy analysis method utilizes a simple and powerful technique to adjust and control the convergence region of the infinite series solution using auxiliary parameters. The auxiliary parameters provide a convenient way of controlling the convergent region of series solutions. Numerical solutions obtained by HAM indicate that the approach is highly accurate, computationally very attractive and easy to implement. The solutions obtained with HAM have been shown to be in good agreement with those obtained using the method of characteristics (MOC) and the reduced order modelling (ROM) technique. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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16 pages, 376 KiB  
Article
Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library
by Samad Noeiaghdam, Aliona Dreglea, Jihuan He, Zakieh Avazzadeh, Muhammad Suleman, Mohammad Ali Fariborzi Araghi, Denis N. Sidorov and Nikolai Sidorov
Symmetry 2020, 12(10), 1730; https://doi.org/10.3390/sym12101730 - 20 Oct 2020
Cited by 35 | Viewed by 2674
Abstract
This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the [...] Read more.
This paper studies the second kind linear Volterra integral equations (IEs) with a discontinuous kernel obtained from the load leveling and energy system problems. For solving this problem, we propose the homotopy perturbation method (HPM). We then discuss the convergence theorem and the error analysis of the formulation to validate the accuracy of the obtained solutions. In this study, the Controle et Estimation Stochastique des Arrondis de Calculs method (CESTAC) and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are used to control the rounding error estimation. We also take advantage of the discrete stochastic arithmetic (DSA) to find the optimal iteration, optimal error and optimal approximation of the HPM. The comparative graphs between exact and approximate solutions show the accuracy and efficiency of the method. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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18 pages, 831 KiB  
Article
On (ϕ, ψ)-Metric Spaces with Applications
by Eskandar Ameer, Hassen Aydi, Hasanen A. Hammad, Wasfi Shatanawi and Nabil Mlaiki
Symmetry 2020, 12(9), 1459; https://doi.org/10.3390/sym12091459 - 05 Sep 2020
Cited by 5 | Viewed by 2132
Abstract
The aim of this article is to introduce the notion of a ϕ,ψ-metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology τϕ,ψ in such spaces [...] Read more.
The aim of this article is to introduce the notion of a ϕ,ψ-metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology τϕ,ψ in such spaces and discuss their topological properties. We also establish the Banach contraction principle in the context of ϕ,ψ-metric spaces and we illustrate the significance of our main theorem by examples. Ultimately, as applications, the existence of a unique solution of Fredholm type integral equations in one and two dimensions is ensured and an example in support is given. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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