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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 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 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]

This paper considers the inverse problem of determining the time‐dependent coefficient in the time‐fractional diffusion‐wave equation. In this case, an initial boundary value problem was set for the fractional diffusion‐wave equation, and an additional condition was given for the inverse problem of determining the coefficient from this equation. First of all, it was considered the initial boundary value problem. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag‐Leffler function and the generalized singular Gronwall inequality, we get a priori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. The stability estimate is also obtained. [ABSTRACT FROM AUTHOR]

In this research, two systems of nonlinear Volterra integral equation and Volterra integro‐differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. [ABSTRACT FROM AUTHOR]

The objective of this research is to propose some new fixed point theorems for fuzzy-dominated operators that satisfy a nonlinear contraction on a closed ball in a complete b-multiplicative metric space. Our strategy involves the use of a combination of two distinct kinds of mappings: one belongs to a weaker class of strictly increasing mappings, and the other is a class of dominated mappings. In order to demonstrate the validity of our new findings, we provide instances that are both illustrative and substantial. Finally, in order to illustrate the novelty of our findings, we provide applications that allow us to derive the common solution to integral and fractional differential equations. Our findings have a significant impact on the interpretation of a large number of previously published studies, both present and historical. [ABSTRACT FROM AUTHOR]

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 paper addresses a computational method to simulate more accurate models for population growth with immigration, focusing on integral equations (IEs) featuring a delay parameter in the time variable. The proposed method utilizes Legendre wavelets within the Galerkin scheme as a orthonormal basis. Legendre wavelets are known for their localized functions, offering suitable precision and stability in simulating time-delay biological models. This approach employs the composite Gauss-Legendre (CGL) quadrature rule to compute integrals appeared in the scheme. An error bound analysis demonstrates a convergence rate of order 2 − M k. Additionally, various numerical examples are presented to show the efficiency, accuracy and validate the theoretical error estimate of the novel technique. [ABSTRACT FROM AUTHOR]

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, (1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems ; (2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point (1); (3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, (4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem—which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

Given their high analytical speed, iterative physical optics (IPO) algorithms present an advantageous approximation method for analyzing the electromagnetic scattering of electrically large objects. IPO approaches for radar cross-section (RCS) prediction are of two types-one based on integral equations, and the other based on reflection coefficients. Integral equation (IE)-based IPO approaches have remarkably low computational complexity and facilitates the analyses of conducting and impedance media. In contrast, reflection (R)-based IPO approaches can consider the absorption effects of surface currents, yielding a higher accuracy in analyzing dielectric media. This study compared and analyzed the formulas of IE-based and R-based IPO and examined their respective advantages and limitations. Complex media targets composed of lossy metals and dielectrics, such as fighters, tailless airplanes, and missiles, above rough ocean surfaces were examined. The surface currents and RCS obtained using each approach were verified by comparing the results with those from FEKO or an in-house full-wave simulation (MLFMM) [ABSTRACT FROM AUTHOR]

The present work is devoted to finding numerical solutions of two types of nonlinear integral equations on half line with kernels depending on the sum and difference of arguments. These equations arise in various fields of mathematical physics: kinetic theory of gases, theoretical astrophysics, p-adic string theory, etc. The main result of the work is the derivation of an uniform estimate of the norm of difference between two successive approximations of solutions, which plays an important role for the control of the convergence of iterative schemes and number of iterations. The obtained results have been applied to determine numerical solutions of models from different areas of applications. [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.

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.

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]

The fundamental goal of our paper is to study θ-convex contractive mappings in metric spaces. We demonstrate some fixed point results for such mappings. Also, we give an application to integral equations of our results. Consequently, our results encompass numerous generalizations of the Banach contraction principle on metric space. [ABSTRACT FROM AUTHOR]

A fundamental solution matrix for elliptic systems of the second order with constant leading coefficients is constructed. It is used to obtain an integral representation of functions belonging to the Hölder class in a closed domain with a Lyapunov boundary. In the case of an infinite domain, these functions have power-law asymptotics at infinity. The representation is used to study a mixed-contact boundary value problem for a second-order elliptic system with piecewise constant leading coefficients. The problem is reduced to a system of integral equations that are Fredholm in the domain and singular at its boundary. [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]

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]

In the present work, an initial-boundary value problem for a high even order partial differential equation with a Bessel operator is considered and the existence, uniqueness and stability of the solution of the considered problem have been investigated. [ABSTRACT FROM AUTHOR]

This work investigates an initial-boundary value and an inverse coefficient problem of determining a space dependent coefficient in the fractional wave equation with the generalized Riemann–Liouville (Hilfer) time derivative. In the beginning, it is considered the initial boundary value problem (direct problem). By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get a priori estimate for solution via unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to the equivalent integral equation of Volterra type. To show existence unique solution to this equation the Schauder principle is applied. The local existence and uniqueness results are obtained. [ABSTRACT FROM AUTHOR]

This research is focused to introduce the new polyconvolution operator related to Hartley integral transforms H 1 , H 2 , H 2 study the algebraic fundamental properties, and at the same time, we apply this polyconvolution operator to solve an integral equation of polyconvolutions type, integro-differential equation of Barbashin type and a system of two integral equations of polyconvolution type. On the other hand, we study the Wastson-type integral for this polyconvolution operator. We establish necessary and sufficient conditions for these operators to be unitary in the space L 2 (ℝ) and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulate the Placherel-type theorem for the aforementioned operators, proved a sequence of function that convergence to the original function in the norm of L 2 (ℝ) and further show the boundedness of these operators. [ABSTRACT FROM AUTHOR]

This study investigates an effective method for solving one class of integral equations. It addresses various two dimensional problems related to do diffraction theory by metal screens, which are reduced to these integral equations. A novel approach for solving this class of integral equations is proposed. The study foceses on investigating diffraction on a strip using a combination of analytical and numerical methods and the results are obtained through simulation. [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]

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 this paper, we consider a two-point boundary-value problem with a Riemann–Liouville–Caputo fractional derivative of order α ∈ (1 , 2) . We solve this boundary-value problem in a sequence of processes, first using the shooting technique based on the secant iterative method, we convert the boundary-value problem into an initial-value problem, then the initial-value problem is transformed into an equivalent Volterra integral equation with weakly singular kernel. Finally, we find the approximate solution of the resultant equation by using a discretization scheme on a uniform mesh. The method's convergence analysis has been thoroughly established, and it demonstrates that the scheme is first-order convergent. To show the effectiveness of the suggested approach, numerical results are provided. [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 ($ \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.

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.

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.

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.

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]

In this paper, the inverse problem of determining the time‐dependent reaction–diffusion coefficient in the Cauchy problem for the time‐fractional diffusion equation with the Hilfer operator by a single observation at the point x=0$$ x&amp;#x0003D;0 $$ of the diffusion process is studied. To represent the solution of the direct problem, the fundamental solution of this equation is constructed and properties of this solution are investigated. The fundamental solution contains Fox's H$$ H $$‐functions widely used in fractional calculus. The fundamental solution for the fractional diffusion operator with the Hilfer derivative is also obtained in terms of the Yang Y$$ Y $$‐function, and it is shown that they coincide. Using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient, which will be used in investigation of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation, the contracted mapping principle is applied. The local existence and global uniqueness results and the conditional stability estimate of solution are proven. [ABSTRACT FROM AUTHOR]

Taking into account a wavelet transform associated with the quadratic‐phase Fourier transform, we obtain several types of uncertainty principles, as well as identify conditions that guarantee the unique solution for a class of integral equations (related with the previous mentioned transforms). Namely, we obtain a Heisenberg–Pauli–Weyl‐type uncertainty principle, a logarithmic‐type uncertainty principle, a local‐type uncertainty principle, an entropy‐based uncertainty principle, a Nazarov‐type uncertainty principle, an Amrein–Berthier–Benedicks‐type uncertainty principle, a Donoho–Stark‐type uncertainty principle, a Hardy‐type uncertainty principle, and a Beurling‐type uncertainty principle for such quadratic‐phase wavelet transform. For this, it is crucial to consider a convolution and its consequences in establishing an explicit relation with the quadratic‐phase Fourier transform. [ABSTRACT FROM AUTHOR]

The Convertible Bonds (CBs) market has witnessed an unprecedented level of activity over the last few years not only in developed countries such as the United States but also in BRICK countries such as China. Exploring new properties of CBs or CBs with clauses becomes important for academia communities in financial mathematics. In this paper, we build two coupled partial differential equations (PDEs) for pricing a callable CB, and find a newly identified inherent property of this bond. The new property is that the conversion ratio will not affect the critical recall time indicating the time beyond the callability. Besides this property, we also find that solving the critical recall time separately and superimposing later using a non-callable CB is the same as the method of a hybrid free boundary (the critical recall time) and a moving boundary (the optimal conversion price) though the callability and the American-style conversion are nonlinearly coupled. [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]

The main aim of this work is to establish a new polyconvolution operator with the weight function γ = sin ⁡ y for the Hartley integral transforms. After that, we will apply it to solve some classes of integral equations and a system of integral equations of polyconvolution type. On the other hand, we study the Wastson-type integral transform for this polyconvolution. We establish necessary and sufficient conditions for these operators to be unitary in the space L 2 (R) and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulate the Placherel-type theorem for the aforementioned operators, prove a sequence of functions that converges to the original function in the norm of L 2 (R) , and further show the boundedness of these operators. [ABSTRACT FROM AUTHOR]

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. [ABSTRACT FROM AUTHOR]

The problem of contact interaction of two identical stringers with an elastic wedge of arbitrary angle of solution, when a system of cracks is located on the bisector of the wedge, is considered under conditions of antiplane deformation. The stringers are symmetrically bisected to the faces of the wedge. Stringers and crack banks are loaded with tangential forces causing antiplane deformation (longitudinal shear) of the wedge. Using the Mellin integral transformation, the solution of the problem is reduced to a system of two singular integral equations (SIE), from which the tangential contact stresses under the stringers and the dislocation densities at the crack banks are determined. The determining system of the SIE is solved by the numerical-analytical method of mechanical quadratures. A numerical analysis was carried out in the case where there is only one crack on the bisector of the wedge. Graphs were constructed illustrating the dependence of the main required quantities on the wedge opening angle and the distance between the wedge tip and the crack. [ABSTRACT FROM AUTHOR]