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An integral equation is constructed relating the three-parameter Mittag-Leffler function with a perturbed one, from which a new property is derived. The latter helps in finding new sums of series of Mittag-Leffler functions. Also, a bound is obtained for the Mittag-Leffler function, allowing to perform a sensitivity analysis, being vital in the solution of fractional differential equations. The bound is also tested on a numerical example. [ABSTRACT FROM AUTHOR]

Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem. [ABSTRACT FROM AUTHOR]

This article presents the modified residual power series approach using Laplace transform, the method is used to solve partial integro differential equations. The basic definitions and theorems related to the method are presented and discussed. Moreover, the steps of the method are utilized and applied to solve various examples. [ABSTRACT FROM AUTHOR]

The probabilistic cone b-metric space is a novel concept that we describe in this study along with some of its fundamental topological properties and instances. We also established the fixed point theorem for the probabilistic nonlinear Banach contraction mapping on this kind of spaces. Many prior findings in the literature are generalized and unified by our findings. In order to illustrate the basic theorem in ordinary cone b-metric spaces, some related findings are also provided with an application to integral equation. [ABSTRACT FROM AUTHOR]

Previously, boundary control problems for a heat conduction equation with Dirichlet boundary condition were studied in a bounded domain. In this paper, we consider the boundary control problem for the heat conduction equation with Neumann boundary condition in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The studied initial boundary value problem is reduced to the Volterra integral equation of the first type using the method of separation of variables. It is known that the solution of Volterra’s integral equation of the first kind cannot always be shown to exist. In our work, the existence of a solution to the Volterra integral equation of the first kind is shown using the method of Laplace transform. For this, the necessary estimates for the kernel of the integral equation were found. Finally, the admissibility of the control function is proved.

In this publication, our objective was to introduce and establish the concepts of $ \kappa _{G_{m}} $-contraction and generalized $ (\alpha, \kappa _{G_{m}}) $-contraction in complete $ G_{m} $-metric spaces, which led to the discovery of novel fixed points, coincidence points, and common fixed points. Additionally, we demonstrated the usefulness of our main results by applying it to the investigation of the integral equation. Also, we presenting a noteworthy example demonstrating the practicality of our primary hypothesis.

In this manuscript, utilizing interpolative contractions with fractional forms, some unique fixed-point results were studied in the context of extended $ b $-metric spaces. For the validity of the presented results some examples are given. In the last section an existence theorem is provided to study the existence of a solution for the Fredholm integral equation.

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

This article investigates a time-fractional space-nonlocal diffusion equation in a bounded domain. The fractional operators are defined rigorously, using the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α <β ᾘ ≤-1. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated and the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series. [ABSTRACT FROM AUTHOR]

In this publication, our objective was to introduce and establish the concepts of KGm-contraction and generalized (a, KGm)-contraction in complete Gm-metric spaces, which led to the discovery of novel fixed points, coincidence points, and common fixed points. Additionally, we demonstrated the usefulness of our main results by applying it to the investigation of the integral equation. Also, we presenting a noteworthy example demonstrating the practicality of our primary hypothesis. [ABSTRACT FROM AUTHOR]

In this manuscript, utilizing interpolative contractions with fractional forms, some unique fixed-point results were studied in the context of extended b-metric spaces. For the validity of the presented results some examples are given. In the last section an existence theorem is provided to study the existence of a solution for the Fredholm integral equation. [ABSTRACT FROM AUTHOR]

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work. [ABSTRACT FROM AUTHOR]

It has been shown that an improved prediction of the thermodynamic quantity for the hard homonuclear diatomics can be performed with an interpolation scheme that relates the thermodynamic property of the hard sphere to that of tangent hard spheres at the same density. Using the analytic expressions based on the solution of an integral equation an excess Helmholtz free energy per particle and chemical potential for hard homonuclear diatomic fluid have been computed. Calculations are carried out for hard homonuclear diatomics with reduced internuclear separations of 0.1 to 1 at reduced densities of 0.2 to 0.9. Our findings for the excess Helmholtz free energy from the Percus–Yevickand Martynov–Sarkisov approximations presents good agreement with available accurate data, having maximum deviations of 15% from it over the separation and density ranges of the calculations. The excess chemical potential from the Martynov–Sarkisov approximation shows better agreement with accurate available data than those from the Percus–Yevick approximation. For the excess chemical potential, a maximum deviation of 9.5% over the range of the calculations has been shown up for the Percus–Yevick approximation.

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length (MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. Doi: 10.28991/HIJ-2024-05-01-02 Full Text: PDF

The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended b-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended b-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.

This paper introduces an original approach to the joint inversion of airborne electromagnetic (EM) data for three-dimensional (3D) conductivity and chargeability models using hybrid finite difference (FD) and integral equation (IE) methods. The inversion produces a 3D model of physical parameters, which includes conductivity, chargeability, time constant, and relaxation coefficients. We present the underlying principles of this approach and an example of a high-resolution inversion of the data acquired by a new active time domain airborne EM system, TargetEM, in Ontario, Canada. The new TargetEM system collects high-quality multicomponent data with low noise, high power, and a small transmitter–receiver offset. This airborne system and the developed advanced inversion methodology represent a new effective method for mineral resource exploration. [ABSTRACT FROM AUTHOR]

The problem of frictional thermomechanical contact of a rigid cylindrical flat-based punch with an elastic half-space has been solved numerically. The partial slipping with friction between the coupled surfaces, as a result of a compressive load, at the first stage, and a uniform heating of the bodies, at the second stage, has been investigated. To take into account the friction, we have used Coulomb's law. This problem is reduced to systems of nonlinear boundary integral equations that correspond to the mechanical and the thermal stress stages. The discretization of integral equations and iterative process have been used to obtain a numerical solution of these systems. The dependences of the normal and tangential contact stress distributions and the configuration of the stick and slip zones on the temperature have been analyzed for two possible cases of the relationship between the coefficients of thermal expansion of the bodies. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study some coupled fixed point theorems in C*-algebra-valued Sb-metric spaces. As applications, existence and uniqueness results for one type of integral equation x(t)= ∫E (K1(t,s) + K2(t,s)))f(s,x(s)) + g(s, x(s)))ds +h(t), t∈E where E is the Lebesque measurable set and m(E) < +∞, and under some other conditions are given. [ABSTRACT FROM AUTHOR]

The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended b-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended b-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature. [ABSTRACT FROM AUTHOR]

Background. The purpose of the work is the use of neural network technologies in the problems of restoring the parameters of inhomogeneities within the body. This problem occurs in acoustics, electrodynamics, flaw detection, as well as in medicine. Materials and methods. The process of propagating an acoustic wave inside various objects is described using the Helmholtz equation. After setting the boundary problem, the transition to the Lippmann- Schwinger integral equation is performed. To solve the reverse problem, a two-step method was used. Results. The problem is solved numerically. The order of the matrix obtained in the calculation is about 10,000 elements. Graphic illustrations of the recovery of the function of inhomogeneities within the body are presented. An experiment was conducted demonstrating the peculiarities of restoring body parameters when using neural networks. Conclusions. Proposed and implemented is software complex for determination of parameters of inhomogeneities inside body. The advantages of using neural networks as a method are shown.

Background. The problem of solvability of the electric field integral equation for non-absorbing media is considered. Materials and methods. The method of quadratic forms is applied to investigation of the operators of the problem. Results and conclusions. The study proves the continuous reversibility of the operator of the electric field equation in the case of plane screens and nonabsorbing media.

The probabilistic cone b-metric space is a novel concept that we describe in this study along with some of its fundamental topological properties and instances. We also established the fixed point theorem for the probabilistic nonlinear Banach contraction mapping on this kind of spaces. Many prior findings in the literature are generalized and unified by our findings. In order to illustrate the basic theorem in ordinary cone b-metric spaces, some related findings are also provided with an application to integral equation.

Как хорошо известно, наличие характеристик является очень существенным при исследовании задачи Коши для дифференциальных уравнений с частными производными независимо от его порядка. В случае, если дифференциальное уравнение с частными производными является нагруженным, то для однозначной разрешимости задачи Коши возникают дополнительные условия разрешимости, зависящие от вида следа нагрузки. Эти условия возникают даже для простейших линейных нагруженных дифференциальных уравнений с частными производными, начиная с первого порядка и выше. Основная цель данной работы – проиллюстрировать возникающие эффекты на примере исследования задачи Коши для линейного нагруженного уравнения в частных производных первого порядка. Так как корректность поставленной задачи Коши эквивалентным образом редуцируется к интегральному уравнению второго рода, то основной метод, применяемый для доказательства его разрешимости – метод последовательных подстановок. Основной вывод заключается в том, что разрешимость задачи Коши для нагруженного уравнения в частных производных существенным образом зависит от выбора следа нагрузки. В случае, когда разрешимость задачи Коши доказана, оказывается, что область влияния данных Коши не ограничивается только характеристиками, а появляются новые не характеристические линии, за которые данные Коши однозначно продолжаться не могут.

The major objectives of this research article are to introduce the notion of ($ \alpha, \psi $)-contraction in the context of $ \mathfrak{F} $-bipolar metric space and establish fixed point theorems. In this way, coupled fixed point results are obtained by applying the leading theorems. Some non-trivial examples are also furnished to show the validity of established results. As applications of the main result, we investigate the solution of an integral equation and a homotopy problem.

Integral transforms are a versatile mathematical technique that can be applied in a wide range of science and engineering fields. We consider the general integral transform with the Caputo derivative and Constant Proportional Caputo derivative in this work. We present some applications to show the effect of the general integral transform with different fractional derivatives.

Statistical process control (SPC) is used for monitoring and detecting anomalies in processes in the areas of manufacturing, environmental studies, economics, and healthcare, among others. Herein, we introduce an innovative SPC approach via mathematical modeling and report on its application via simulation studies to examine its suitability for monitoring processes involving correlated data running on advanced control charts. Specifically, an approach for detecting small to moderate shifts in the mean of a process running on a homogenously weighted moving average (HWMA) control chart, which is symmetric about the center line with upper and lower control limits, is of particular interest. A mathematical model for the average run length (ARL) of a moving average process with exogenous variables (MAX) focused only on the zero-state performance of the HWMA control chart is derived based on explicit formulas. The performance of our approach was investigated in terms of the ARL, the standard deviation of the run length (SDRL), and the median run length (MRL). Numerical examples are given to illustrate the efficacy of the proposed method. A detailed comparative analysis of our method for processes on HWMA and cumulative sum (CUSUM) control charts was conducted for process mean shifts in many situations. For several values of the design parameters, the performances of these two control charts are also compared in terms of the expected ARL (EARL), expected SDRL (ESDRL), and expected MRL (EMRL). It was found that the performance of the HWMA control chart was superior to that of the CUSUM control chart for several process mean shift sizes. Finally, the applicability of our method on a HWMA control chart is provided based on a real-world economic process. [ABSTRACT FROM AUTHOR]

This article studies new classes of contractions called the p-cyclic Reich contraction and p-cyclic Reich contraction pair and develops certain best proximity point results for such contractions in the setting of partial metric spaces. Furthermore, the best proximity point results for p-proximal cyclic Reich contractions of the first and second types are also discussed. [ABSTRACT FROM AUTHOR]

This article presents a study on surface gravity wave scattering by a rectangular (box-type) breakwater with thin side plates in the situation of oblique incident waves in deep water. Applying the continuity of fluid pressure and velocity to Havelock’s expansion of velocity potentials, the problem is converted to an integral equation of the Fredholm type, whose solution is the horizontal component of fluid velocity. The integral equation is solved by employing Galerkin’s approximation with polynomials as basis functions multiplied by suitable weight functions. The wave reflection and transmission coefficients are calculated numerically to find the breakwater’s performance on wave scattering. The accuracy of the results is verified through numerical convergence and checking of the energy balance equation. The rectangular breakwater reflects long waves to some extent in water of infinite depth, in contrast to a thin breakwater. The thin plates attached to the rectangular breakwater show a reduction in wave transmission. Furthermore, the attachment of thin plates leads to an increment in horizontal force and a reduction in vertical force. [ABSTRACT FROM AUTHOR]

The major objectives of this research article are to introduce the notion of (α, ψ)-contraction in the context of F-bipolar metric space and establish fixed point theorems. In this way, coupled fixed point results are obtained by applying the leading theorems. Some non-trivial examples are also furnished to show the validity of established results. As applications of the main result, we investigate the solution of an integral equation and a homotopy problem. [ABSTRACT FROM AUTHOR]

In Electromagnetics, the field scattered by an ensemble of particles—of arbitrary size, shape, and material—can be obtained by solving the Lippmann–Schwinger equation. This singular vectorial integral equation is generally formulated in the direct space R n (typically n = 2 or n = 3 ). In the article, we rigorously computed the Fourier transform of the vectorial Lippmann–Schwinger equation in the space of tempered distributions, S ′ (R 3) , splitting it in a singular and a regular contribution. One eventually obtains a simple equation for the scattered field in the Fourier space. This permits to draw an explicit link between the shape of the scatterer and the field through the Fourier Transform of the body indicator function. We compare our results with accurate calculations based on the T-matrix method and find a good agreement. [ABSTRACT FROM AUTHOR]

The aim of the present research article is to investigate the existence and uniqueness of a solution to the integral equation and homotopy result. To achieve our objective, we introduce the notion of ( α , η , ψ )-contraction in the framework of F -bipolar metric space and prove some fixed point results for covariant and contravariant mappings. Some coupled fixed point results in F -bipolar metric space are derived as outcomes of our principal theorems. A non-trivial example is also provided to validate the authenticity of the established results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the b-triangular property in fuzzy b-metric space. Furthermore, we give some new fixed point results in fuzzy b-metric space for non-continuous mappings. Our results generalize and expand some results from the related literature. Two applications of our results, to solving Fredholm integral equation and in dynamic programming, are also given. [ABSTRACT FROM AUTHOR]

Integral transforms are a versatile mathematical technique that can be applied in a wide range of science and engineering fields. We consider the general integral transform with the Caputo derivative and Constant Proportional Caputo derivative in this work. We present some applications to show the effect of the general integral transform with different fractional derivatives. [ABSTRACT FROM AUTHOR]

An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green's 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method's 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K + ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results. [ABSTRACT FROM AUTHOR]

We investigate the radiation of the THz natural waves of the dielectric waveguide with graphene plane scattered by finite number of graphene strips. Our mathematically accurate analysis uses the singular integral equations method. The discretization scheme employs the Nystrom-type algorithm. The complex-valued propagation constants of the natural waves and corresponding fields are determined numerically from the equation, which also involves the kernel-function of the singular integral equation. The method we use is meshless and full-wave. The convergence is provided by the mathematical theorems. By varying the chemical potential of graphene and structural geometrical parameters, we examine the elevation angle of the main lobe of the radiation pattern and the radiated power. [ABSTRACT FROM AUTHOR]

In this study, the development, design, and software implementation of the methods for solving the nonlinear diffraction problem were performed. The influence of nonlinear medium defined by the Kerr law on the propagation of a wave passing through an object was examined. The differential and integral formulations of the problem and the nonlinear integral equation were considered. The problem was solved for different bodies with the use of various computational grids. Convergence graphs of the iterative processes were generated. The obtained graphical results were presented. The explicit and implicit methods for solving the integral equation were compared.

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form $ \begin{equation} u^{p-1}(x) = \int_{\Omega} \frac{u(y)}{|x-y|^{n-\alpha}} d y+\int_{\Omega} \frac{f(y)}{|x-y|^{n-\alpha}} d y, \ x \in \bar{\Omega}\nonumber \end{equation} $ where $ \frac{2n}{n+\alpha}\leq p < 2, $ $ 1 < \alpha < n $, $ n > 2, $ $ \Omega $ is a bounded domain in $ \mathbb R^{n} $. We show that under suitable assumptions on $ f, $ the integral equation admits a positive solution in $ L^{\frac{2n}{n+\alpha}}\left(\Omega\right) $. Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the difficulties arising from the lack of Brezis-Lieb lemma in $ L^{\frac{2n}{n+\alpha}}(\Omega) $.

In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second $ \kappa $-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space.

Previously, boundary control problems for parabolic type equations were considered. A portion of the thin rod boundary has a temperature-controlled heater. Its mode of operation should be found so that the average temperature in some region reaches a certain value. In this article, we consider the boundary control problem for the pseudo-parabolic equation. The value of the solution with the control parameter is given in the boundary of the interval. Control constraints are given such that the average value of the solution in considered domain takes a given value. The auxiliary problem is solved by the method of separation of variables, and the problem under consideration is reduced to the Volterra integral equation. The existence theorem of admissible control is proved by the Laplace transform method.

This article investigates Wardowski's contraction in the setting of extended distance spaces known as $ M $-metric spaces using hybrid operators based an $ M $ -dynamic iterative process. The main purpose is to observe new set-valued Chatterjea type common fixed point theorems for hybrid operators with respect to an $ M $-dynamic iterative process, i.e., $ \check{D}(\Psi _{1}, \Psi _{2}, s_{0}) $. We realize an application: the existence of a solution for a multistage system and integral equation. Also, we give a graphical interpretation of our obtained theorems. The main results are explicated with the help of a relevant example. Some important corollaries are extracted from the main theorems as well.

Abstract In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

В последнее время интенсивно изучаются начально – граничные задачи в прямоугольной области для дифференциальных уравнений в частных производных как четного, так и нечетного порядка. При этом в качестве объекта исследования, в основном, берется не вырождающееся уравнение или уравнение, вырождающееся на одной стороне четырехугольника. Начально – граничные задачи (как локальные, так и нелокальные) для уравнений с двумя или тремя линиями вырождения остаются неизученными. В данной работе в прямоугольной области рассмотрено уравнение четвёртого порядка, вырождающееся на трех сторонах четырехугольника и содержащее оператор дробного дифференцирования Герасимова –Капуто. Для этого уравнения сформулирована и исследована одна начально – граничная задача с нелокальными условиями, связывающими значения искомой функции и её производных до третьего порядка (включительно), принимаемых на боковых сторонах прямоугольника. Сначала методом интегралов энергии доказана единственность решения поставленной задачи. Затем, исследована спектральная задача, возникающая при применении метода Фурье, основанном на разделении переменных, к поставленной начально – граничной задаче. Построена функция Грина спектральной задачи, с помощью чего она эквивалентно сведена к интегральному уравнению Фредгольма второго рода с симметричным ядром, откуда следует существование счетного числа собственных значений и собственных функций спектральной задачи. Доказана теорема разложения заданной функции в равномерно сходящийся ряд по системе собственных функций. С помощью найденного интегрального уравнения и теоремы Мерсера доказана равномерная сходимость некоторых билинейных рядов, зависящих от найденных собственных функций. Установлен порядок коэффициентов Фурье. Решение изучаемой задачи выписано в виде суммы ряда Фурье по системе собственных функций спектральной задачи. Исследована равномерная сходимость этого ряда и рядов, полученных из него почленным дифференцированием. Получена оценка для решения задачи, откуда следует его непрерывная зависимость от заданных функций.

Background. The purpose of this work is to study the 3-D scalar problem of scattering a plane wave from an inhomogeneous solid covered with an infinitely thin layer of graphene. Material and methods. The transmission problem for the Helmholtz equation with special boundary conditions is considered; this problem, which has a unique solution, is reduced to a weakly singular integral equation; the operator of the equation is investigated in a Sobolev space. Results. The diffraction problem is reduced to an integral equation; the equivalence of the integral equation and the transmission is proved; Fredholm property and continuous invertibility of the operator of the integral equation are proved. Conclusions. Important results on existence and uniqueness of the solution to the diffraction problem are obtained and can be used for validation of projection methods for numerical solving of the diffraction problem.

Background. The problem of electromagnetic waves’ propagation in a dielectric rod of arbitrary cross-section covered with a layer of graphene, which is considered infinitely thin, is considered. The main problem in describing the process of wave propagation in the waveguiding structure is to obtain and analyze the system of integral equations to determine propagation constants. Materials and methods. Maxwell’s equations are solved in the frequency domain. The coupling conditions contain the conductivity of graphene. The method of Green’s functions is applied. Results and conclusions. The system of integral equations for determining the propagation constants is solved numerically. Numerical results are presented.

The focus of this study is to establish the existence and uniqueness of solutions for differential and integral equations within specific metric spaces. Our investigation begins by introducing the concept of the so-called cone Eb-metric space and presenting crucial findings in this particular space. We have presented fixed point results for specific contractions, particularly in the context of non-solid cones that possess semi-interior points. Not only do the results enhance specific previous fixed points outcomes, but they also encompass and extend previous findings documented in the literature. Furthermore, we apply our findings in the cone Eb-metric space to various examples and applications. The ultimate outcome is the rigorous validation of the existence and uniqueness of solutions for certain differential and integral equations.

Transitional and turbulent boundary layer (BL) flows possess dramatically larger wall-shear stresses and surface heat fluxes than their laminar counterparts. Locally, this can be explained by appealing to the presence of spatiotemporally coherent velocity and temperature regions that enhance the momentum and thermal transport across the BL. However, these coherent structures are seldom present alone, but usually influenced by other physical phenomena, such as the streamwise growth of BLs or freestream pressure gradients. A moment of enthalpy integral equation is introduced to quantify how turbulence and other flow phenomena enhance surface heat flux. Results from a direct numerical simulation of a transitional and turbulent BL are used to demonstrate the proposed analysis method and to form a quantitative assessment of the BL physics. The integral analysis demonstrated in this paper for canonical turbulent flows provides an interpretable analysis tool for unlocking key BL physics, including future extensions to compressible BLs, freestream pressure gradients, and the evaluation of flow control schemes. [ABSTRACT FROM AUTHOR]