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Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay. [ABSTRACT FROM AUTHOR]

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder–Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [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]

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]

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]

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]

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 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]

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]

For homogeneous kernels, the memory requirements associated with H2 representations of integral equation matrices can be reduced by incorporating translational invariance. Starting with a nontranslationally invariant H2 representation, this can be accomplished using a left/right iterative algorithm. In this paper, it is shown that a similar algorithm can also be used to compress an existing fast multipole method (FMM). It is observed that the iterative compression converges faster when used to compress an FMM than when it is applied to an H2 representation. Resulting savings in floating-point operations are indicated, and extensions of the reported method are discussed. [ABSTRACT FROM AUTHOR]

The paper considers two weakly singular Fredholm boundary integral equations of the first kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on the spectra, where they are ill posed. For the first equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the transmission problem. The second equation in this case always has infinitely many solutions, with only one of them giving a solution of the transmission problem. The interpolation method for finding approximate solutions of the integral equations and the transmission problem in question is discussed. [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]

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 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]

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]

In the present paper, we consider the initial boundary value direct problem and zeroorder determination inverse problem for the time-fractional wave equation. First, we study the direct problem (using Robin-type boundary condition), and then using its properties, we investigate the inverse problem. We prove the local solvability of the inverse problem’s solution and will show its stability. [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]

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]

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]

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, 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$$ x=0 $$ 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]

Two recent analytical results concerning i) the direct initiation of gaseous detonations in free space and ii) the deflagration to detonation transition in tubes are revisited in this article. The first problem is treated by an asymptotic analysis in the limit of small heat release, enlightening the mechanisms controlling the direct initiation process of real detonations. The second one is limited to a one-dimensional mechanism of transition at work on the tip of elongated flames in tubes. Particular attention is paid to the physical mechanisms more than to the technical details of the analyses. However, the mathematical formulation as well as the different steps of the analysis are clearly presented in a synthetic manner.* [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]

Consider the integral equation$ \begin{eqnarray*} f^{m-1}(x)+f^{q-1}(x) = \int_{\Omega}^{}\frac{f(y)}{\left | x-y \right |^{n-\alpha } }dy, \ \ \ \end{eqnarray*} $where $ 1<\alphaq,m >\frac{2n}{n+\alpha} $, and the nonexistence of energy maximizing positive solutions in the critical and supercritical case $ 1

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]

Under the assumption of linear wave theory, this work investigates the interaction of surface gravity waves with two asymmetric horizontal flexible porous barriers. Three types of barriers are considered, namely, (i) step barriers, (ii) partially overlapped barriers and (iii) fully overlapped barriers. Green's integral theorem and the boundary condition on the barriers are utilised to produce four integral equations for functions related to potential difference and normal velocity of fluid across the barriers which are solved numerically by an expansion-collocation method. As a result, the role of asymmetric flexible porous barriers with an additional comparison between dual and single horizontal flexible porous barriers is studied by analysing numerical solutions of different hydrodynamic quantities. This study shows that the upper barrier plays a major role compared to the lower one in the propagation of surface waves. The current analysis is validated by reviving previous results and utilising energy balance relationship. [ABSTRACT FROM AUTHOR]

Let u ∈ L loc 2 n (p − 1) α + β (R n) be a non-negative solution of the integral equation u (x) = ∫ R n u p − 1 (y) | x − y | n − α ∫ R n u p (z) | y − z | n − β d z d y , x ∈ R n , where 0 < α , β < n and p ≥ 2. We prove that u ≡ 0 if 2 n 2 n − α − β < p < n + β n − α and u must assume an explicit form if p = n + β n − α . As an application, we obtain a similar result for non-negative distributional solutions of the corresponding static Choquard-type equation. The main tool we use is the method of moving planes in integral forms. [ABSTRACT FROM AUTHOR]

A modified exponentially weighted moving average (EWMA) scheme is a quality control chart which can quickly detect small shifts in autocorrelated data. In this article, an explicit formula is presented to calculate the average run length (ARL) for a modified EWMA chart with an autoregressive AR(p) model involving exponential white noise. Its performance is compared with a numerical integral equation method via a simulation study to determine whether the explicit formula can help decrease the processing time. Moreover, this scheme is tested against a classical EWMA control chart using the ARL and RMI results. Finally, the explicit formula for ARL is applied to a real numerical example from the health field to demonstrate the efficacy of our method. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for a nonhomogeneous nonlinear integral equation of the form bounded domain in Rn. We show that under suitable assumptions on f; the integral equation admits a positive solution in L 2n n+ (Our method combines the Ekeland variational principle, a blow-up argument and a rescaling argument which allows us to overcome the diffculties arising from the lack of Brezis-Lieb lemma in L 2n n+. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second 1-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. [ABSTRACT FROM AUTHOR]

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., D(ψ1,ψ2, s0). 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 FROM AUTHOR]

In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an integral system u(x)=Wβ,γup−1v(x),u>0inRn,v(x)=Iαup(x),v>0inRn,$$ \left\{\begin{array}{l}u(x)={W}_{\beta, \gamma}\left({u}^{p-1}v\right)(x),u>0\kern0.5em \mathrm{in}\kern0.5em {R}^n,\\ {}v(x)={I}_{\alpha}\left({u}^p\right)(x),v>0\kern0.5em \mathrm{in}\kern0.5em {R}^n,\end{array}\right. $$where p≥1,0<α,βγ1$$ p\ge 1,0<\alpha, \beta \gamma 1 $$. Base on the integrability of positive solutions, we obtain some Liouville theorems and the decay rates of positive solutions at infinity. In addition, we use the properties of the contraction map and the shrinking map to prove that u$$ u $$ is Lipschitz continuous. In particular, the Serrin type condition is established, which plays an important role to classify the positive solutions. [ABSTRACT FROM AUTHOR]

In this work, we investigate a generalization of the Tricomi problem for a loaded mixed‐type equation with the Riemann–Liouville fractional differential operator. By using the method of integral equations, a unique solvability of the formulated problem given in piecewise non‐parallel characteristics is proven. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, an a-complete extended Branciari b-distance space and rational α -- λ -- JS contractive conditions in the underlying spaces are introduced. Further, we establish the coincidence point, common fixed points, and uniqueness of fixed points for two pairs of mappings in new spaces under the aforementioned contractive conditions. We illustrate the work with an example and apply the results to determine the existence of solutions for a system of Urysohn integral equations. [ABSTRACT FROM AUTHOR]

Regime-switching models have been heavily studied recently, as they have some clear advantages of over other non-constant volatility model to resolve the so-called smirk effect displayed when constant volatility models are used to price financial derivatives such as options. However, due to the increased model complexity, the associated computational effort usually increases as well, particularly when they are used to price American-style options. In this paper, a novel computational approach based on integral equations is presented. A distinctive feature of our approach, in comparison with other numerical approaches, is that the coupled partial differential equations (PDEs) in a PDE system have been decoupled in the Fourier space, resulting in a completely decoupled integral equation for each economical states, and thus has greatly reduced computational effort. Some examples with preliminary results for a two-state regime-switching model are used to demonstrate our approach. [ABSTRACT FROM AUTHOR]

The paper presents formulations of inverse problems for the Helmholtz equation on finding its right-hand side with a Samarskii–Ionkin-type additional integral condition and the justification of their well-posedness in the Hadamard sense in the class of regular solutions. The uniqueness of solutions of the formulated problems is proved on the basis of integral identities. Solutions of the problem are found in explicit form by the methods of separation of variables and integral equations. [ABSTRACT FROM AUTHOR]

The control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed ball of the space Lq (Ω;Rm) q > 1 with radius r and centered at the origin. The trajectory of the system is defined as p-integrable multivariable function from the space Lp Ω;Rn), 1/q, + 1/p = 1, satisfying the system's equation almost everywhere. It is shown that the system's trajectories are robust with respect to the fast consumption of the remaining control resource. Applying this result it is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the total control resource. [ABSTRACT FROM AUTHOR]

In this article, a numerical method is presented to solve Abel's equations. In the given method, the solution of the equation is found as a finite expansion of the shifted Legendre polynomials. To this end, the integral and differential parts of the equation are converted to vector-matrix representations. Therefore, the equation is converted to an algebraic system of the equations and by solving it, the solution of the equation is obtained. Further, the numerical example is given to illustrate the method's efficiency. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a new integral transform named the W transform. We have been obtained some important results about the W transform. At the same time, the relation between the W transform and other transforms has been established. In order to prove the efficiency of this transform, we have solved the differential equations and integral equations. [ABSTRACT FROM AUTHOR]

The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations. [ABSTRACT FROM AUTHOR]

Our concern is to consider linear integral equations with two delays. A new result on necessary and sufficient conditions for the exponential stability is presented by careful study of the characteristic equation. Explicit expressions of limits of solutions in the critical case where integral equations lose exponential stability are also given by use of an asymptotic formula for solutions. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of a solution of Integral equations of fractional order via Hadamard's fractional integral by reducing the research to the search of the existence of fixed points of appropriate operators. The nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Frechet spaces due to Avraniescu and a fractional version of Pachpatte's inequality are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example. [ABSTRACT FROM AUTHOR]