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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 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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Surface tension plays a central role in the mechanical behavior of soft materials such as gels. Elastocapillary deformation of elastic graded substrates is ubiquitous in soft materials. In this work, the effect of a partially wetting sessile liquid droplet on the elastocapillary deformation of a soft elastic graded substrate is studied. The modulus is assumed to have an exponential form along the thickness direction. By applying the Fourier transformation, a mixed boundary-value problem is reduced into a dual integral equation. The numerical results show that the surface displacement is strongly affected by the inhomogeneity of the material. The study of the wetting properties of gel substrates is essential for both understanding the wetting phenomena of gels and developing gels for applications as soft actuators and sensors that can be used in wearable electronics and soft robotics. [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 manuscript, our objective is to introduce the notion of neutrosophic extended metric-like spaces. We establish some fixed point theorems in this setting. Neutrosophic extended metric-like spaces metric space uses the idea of continuous triangular norms and continuous triangular conorms in an extended intuitionistic fuzzy metric-like space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. Triangular norms and triangular conorm are very significant for fuzzy operation. The obtained results boost the approaches of existing ones in the literature and are supported by some examples and an application. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the fractional (p , q) -Mellin transform of order α , denoted by M p , q α , an appropriate convolution and the inversion formula for M p , q α . We provide several properties of M p , q α including those involving fractional integral and fractional derivative of Riemann–Liouville type. Towards applications, using M p , q α , we solve some integral equations, a fractional (p, q)-difference equation and a Titchmarsh type theorem is proved. [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]

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]

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]

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]

We consider a perforated domain Ω(ε) of R² with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. [ABSTRACT FROM AUTHOR]

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L¹ (a, b) and C[a, b] when n ≥ 2. We establish the equivalence in C[a, b] of the IVP of the nth order FDE subject to the initial condition u(i) (a) = ui for i ∈ {0, 1, . . ., n − 1} when n ≥ 2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n ≥ 2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n ≥ 2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of bicomplex-valued controlled metric spaces and prove fixed point theorems. Our results mainly focus on generalizing and expanding some recently established results. Finally, we explain an application of our main result to a certain type of fractional differential equation. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this study is to analyze the features of a moving Griffith crack in an initially stressed infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and singularities of the Cauchy-type are used to formulate the present model, Dirac delta function is employed to analyze point load located at the moving crack edge, and Hilbert transformation properties are used for obtaining stress intensity factor (SIF) with constant point loading. Numerical simulations and graphical illustrations are performed to analyze the influences of the prevalent parameters, viz. initial stresses, punch pressure, distinct positions of point load, length and speed of the crack on the SIF for the considered isotropic material strip. [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]

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]

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]

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]

It is well known that any classical Lotka–Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved. [ABSTRACT FROM AUTHOR]

The aim of this research article is to obtain fixed point results in the context of F -bipolar metric spaces. The obtained results extend some fixed point theorems in the existing literature. We also provide a non-trivial example to validate our claims. The existence and uniqueness of the solution of the integral equation are proved as applications of our leading results. Furthermore, the existence of the unique solution in homotopy theory is also investigated. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]