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The Ornstein-Zernike integral equation method has been employed for a single-component hard sphere fluid in terms of the Percus-Yevick (PY) and Martynov-Sarkisov (MS) approximations. The virial equation of state, excess chemical potential and free energy have been computed in both approximations. The excess entropy has been obtained with both thermodynamic relation and free energy expression for reduced densities of 0.1 to 0.9. It has been shown that the entropy values from thermodynamic relation in the MS approximation are better than those from the PY approximation, especially for high densities, and presents a reasonable comparison with available data in the literature, while the values from free energy expression in both approximations are not only close to each other, but also comparable to accurate ones. [ABSTRACT FROM AUTHOR]

The paper proposes a method for solving the problem of optimal performance for linear systems of ordinary differential equations in the presence of phase and integral restrictions, when the initial and final states of the system are elements of given convex closed sets, taking into account the control value restriction. The presented work refers to the mathematical theory of optimal processes from L.S. Pontryagin and his students and the theory of controllability of dynamic systems from R.E. Kalman. We study the problem of optimal speed for linear systems with boundary conditions from given sets close to the presence of phase and integral constraints, as well as constraints on the control value. A theory of the boundary value problem has been created and a method for solving it based on the study of solvability and the construction of a general solution to the Fredholm integral equation of the first kind has been developed. The main results are the distribution of all controls’ sets, each subject of which transfers the trajectory of the system from any initial state to any final state; reducing the initial boundary point to a special initial optimal control problem; constructing a system of algorithms for the gamma-algorithm study on the derivation of problems and rational execution with restrictions on the solution of the optimal speed’ problem with restrictions.

In a recent paper, Khojasteh et al. presented a new collection of simulation functions, said Z-contraction. This form of contraction generalizes the Banach contraction and makes different types of nonlinear contractions. In this article, we discuss a pair of nonlinear operators that applies to a nonlinear contraction including a simulation function in a partially ordered metric space. For this pair of operators with and without continuity, we derive some results about the coincidence and unique common fixed point. In the following, many known and dependent consequences in fixed point theory in a partially ordered metric space are deduced. As well, we furnish two interesting examples to explain our main consequences, so that one of them does not apply to the principle of Banach contraction. Finally, we use our consequences to create a solution for a particular type of nonlinear integral equation.

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.

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.

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.

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 Rn (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′(R3), 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.

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.

In present paper, an initial-boundary value problem is formulated in a rectangle for a higher even order partial differential equation with the Bessel operator. Applying the method of separation of variables to the considered problem a spectral problem is obtained for an ordinary differential equation of higher even order. The self-adjointness of the last problem is proved, which implies the existence of the system of its eigenfunctions, as well as the orthonormality and completeness of this system. The uniform convergence of some bilinear series and the order of the Fourier coefficients, depending on the found eigenfunctions, is investigated. The solution of the considered problem is found as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. The absolute and uniform convergence of this series, as well as the series obtained by its differentiating, have been proved. The uniqueness of the solution of the problem is proved by the method of spectral analysis. An estimate is obtained for the solution of the problem which implies the continuous dependence of the solution on the given functions.

In this paper, some generalized fixed point results of Banach and $ \acute{C} $iri$ \acute{c} $ type in the context of extended fuzzy $ b $-metric spaces are established. For authenticity of the aforesaid results a nontrivial supporting example is also provided. Eventually, an application for the existence of a solution for an integral equation is established which shows corporeality of the obtained results. The presented work generalizes some well known fixed point results from the existing literature.

In this article we have introduced a metric named complex valued controlled metric type space, more generalized form of controlled metric type spaces. This concept is a new extension of the concept complex valued b-metric type space and this one is different from complex valued extended b-metric space. Using the idea of this new metric, some fixed point theorems involving Banach, Kannan and Fisher contractions type are proved. Some examples togetheran application are described to sustain our primary results.

The Tricomi boundary value problem for a mixed-type equation with a delayed argument of the required function and its derivative is investigated. The problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

M.M. Lavrentiev's linear integral equation arises as a result of a special transformation of a nonlinear coefficient inverse wave sensing problem. The completeness of the set of products of regular harmonic functions and Newtonian potentials supported by a segment is proved. As a corollary, we establish the uniqueness of the solution to M.M. Lavrentiev's equation and a related inverse problem of wave sensing. We present results of an approximate solution of this equation by using parallel calculations. [ABSTRACT FROM AUTHOR]

The integral equation method is one of the most successful computational models for microwave devices or integrated circuits in planar layered media. However, the efficient and accurate evaluation of the associated Green's function consisting of Sommerfeld integrals (SIs) is still a remaining challenge. To mitigate this difficulty, this work proposes a spatial domain rational function fitting technique (RFFT) for SIs so that the approximation accuracy is controllable. In conjunction with an adaptive sampling strategy, the proposed RFFT minimizes the orders of rational functions, and the resultant SI evaluation efficiency is optimized. In addition, we investigate the semi-analytical singularity treatment for the rational expression of SIs in method of moment (MoM) implementation. Extensive simulation of representative planar devices validates the correctness of the proposed method and demonstrates its superior performance over conventional SI approximation methods. [ABSTRACT FROM AUTHOR]

With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation φ (x) = ∫ D E (x , y) G (y , φ (y)) d y. This result is applied to branching diffusion processes with absorbing boundary. In particular, a question existing in the study by Pazy and Rabinowitz [A nonlinear integral equation with applications to neutron transport theory. Arch Ration Mech Anal. 1969;32:226-246] is solved. [ABSTRACT FROM AUTHOR]

This paper reviews a series of fast direct solution methods for electromagnetic scattering analysis, aiming to significantly alleviate the problems of slow or even non-convergence of iterative solvers and to provide a fast and robust numerical solution for integral equations. Then the advantages and applications of fast direct solution methods and the research trends are introduced in detail. Three different main methods are discussed, namely hierarchically off-diagonal low-rank matrices (HODLR) and skeletonization direct methods based on weak and strong admissibility condition. Numerical examples of computational complexity and electromagnetic scattering analysis of jet models are presented to demonstrate the efficiency and accuracy of each approach. Finally, a brief discussion is given on the main challenges and possible strategies of fast direct solution methods which still exist. [ABSTRACT FROM AUTHOR]

The problem of identifying the load acting on the elements of structures belongs to the class of inverse problems of the mechanics of a deformable solid, which are often incorrect. Solving such problems is associated with the instability of the calculation results, which requires the development of special methods for their research. This predetermines the relevance of this study. The object of the study is a single-pass cylindrical shell consisting of two rigidly fastened butt-fastened sections made of different materials. Each of the shells is assumed to be elastic isotropic, having a cross-section of medium thickness. The equations of axisymmetric deformation of shells are used within the framework of Timoshenko hypotheses. An approach to solving direct and inverse problems for such discretely heterogeneous objects is proposed, which implies the conditional separation of a discretely heterogeneous cylindrical shell along the length, followed by the addition of functions of fictitious loads. The main analytical relationships for building a system of integral Volterra equations are given, for which an analytic-numerical solution is derived. The final ratios have been obtained, which make it possible to calculate the kinematic and force parameters of the study object in the process of non-stationary deformation. The inverse problem of identifying arbitrary loads acting on a shell that is heterogeneous in length is solved in a general form. An algorithm for the restoration of pulse loads has been developed, which is robust to errors in the initial data (about 5 %). The material related to solving direct and inverse problems for shells that are discretely heterogeneous in length can significantly advance the methodology for identifying pulse loads acting on structural elements. [ABSTRACT FROM AUTHOR]

A controlled discrete-time stochastic process is proposed as a model for the state of a machine. In the proposed model, normal and random wear of the machine are considered. The random wear of the machine during a time unit can be either a discrete or a continuous random variable. The objective is to find a control policy that maximises the profits generated by the machine over its useful lifetime. In this problem, the optimiser must decide whether to do or not to do the maintenance work on the machine at each time unit. The significance of the paper is that the final time in the optimal control problem is a random variable denoting the first time the machine must be replaced. To obtain the optimal control, one can try to solve the dynamic programming equation, which reduces to a difference or an integral equation, satisfied by the value function. Finally, particular cases are solved exactly, or approximately, and explicitly to demonstrate and validate the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multidimensional inverse problem for a fractional diffusion equation. The inverse problem is reduced to the equivalent integral equation. For solving this equation the Schauder principle is applied. The local existence and uniqueness results are obtained. [ABSTRACT FROM AUTHOR]

This article proposes and demonstrates a calibration-based integral formulation for resolving the forcing function in a mass–spring–damper system, given either displacement or acceleration data. The proposed method is novel in the context of vibrations, being thoroughly studied in the field of heat transfer. The approach can be expanded and generalized further to multi-variable systems associated with machine parts, vehicle suspensions, translational and rotational systems, gear systems, etc. when mathematically described by a system of constant property, linear, time-invariant ordinary differential equations. The analytic approach and subsequent numerical reconstruction of the forcing function is based on resolving a parameter-free inverse formulation for the equation(s) of motion. The calibration approach is formulated in the frequency domain and takes advantage of several observations produced by the dimensionality reduction leading to an algebratized system involving an input–output relationship and a transfer function possessing all the system parameters. The transfer function is eliminated in lieu of experimental data, from a calibration effort, thus leading to a reduction of systematic errors. These parameter-free, reduced systematic error aspects are the distinct and novel advantages of the proposed method. A first-kind Volterra integral equation is formed containing only the unknown forcing function and experimental data. As with all ill-posed problems, regularization must be introduced for system stabilization. A future-time technique is instituted for forming a family of predictions based on the chosen regularization parameter. The optimal regularization parameter is estimated using a combination of phase–plane analysis and cross-correlation principles. Finally, a numerical simulation is performed verifying the proposed approach. [ABSTRACT FROM AUTHOR]

The electromagnetic wave diffraction from perfectly conducting truncated wedges is considered on the rigorous level in cylindrical coordinates. An analytical regularization method is developed to obtain mathematically accurate problem solutions. The solution method is based on the unknown field representation through the principal value Kontorovich–Lebedev integral and the eigenfunctions series. We analyze the scattering from the semi-infinite truncated wedge, which consists of two non-parallel and non-intersecting perfectly conducting and infinitely thin half-planes, and develop this technique for analysis of more complicated problems of wave diffraction from the truncated wedge of finite length. The problems are reduced to the infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators and their inverse ones are used to reduce them to the ISLAE of the second kind applied to the analytical regularization procedure. Two versions of the procedure, such as left- and right-sides regularization, are considered. The developed technique is compared with the Wiener–Hopf method. The numerical examples of wave scattering from the truncated wedge, including its well-known geometries as the semi-plane and the slit in the infinite plane, are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of neutrosophic rectangular triple-controlled metric space, relaxing the symmetry requirement of neutrosophic metric spaces, by replacing triangular inequalities with rectangular inequalities, and prove fixed point theorems. We have derived several interesting results for contraction mappings supplemented with non-trivial examples. The derived results have been applied to prove the existence of a unique analytical solution as well as a closed form of the unique solution to the integral equation. [ABSTRACT FROM AUTHOR]

Many formulas of improper integrals are shown every day and need to be solved in different areas of science and engineering. Some of them can be solved, and others require approximate solutions or computer software. The main purpose of this research is to present new fundamental theorems of improper integrals that generate new formulas and tables of integrals. We present six main theorems with associated remarks that can be viewed as generalizations of Cauchy's results and I.S. Gradshteyn integral tables. Applications to difficult problems are presented that cannot be solved with the usual techniques of residue or contour theorems. The solutions of these applications can be obtained directly, depending on the proposed theorems with an appropriate choice of functions and parameters. [ABSTRACT FROM AUTHOR]

In this manuscript, a novel general family of contraction, called hybrid-interpolative Reich- Istrățescu-type (G-α-µ)-contraction is introduced and some fixed point results in generalised metric space that are not deducible from their akin in metric spaces are obtained. The preeminence of this class of contraction is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, a number of corollaries that reduce our result to other wellknown results in the literature are highlighted and analysed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution of a family of integral equations. [ABSTRACT FROM AUTHOR]

In this paper, the pricing problem of callable–puttable convertible bonds written on a single underlying asset is studied with an integral equation (IE) approach. The complication of the pricing exercise results from the tangled presence of callability, puttability, as well as conversion, which have led to possible coexistence of two moving boundaries at the same time, depending on the call price, the put price, and the conversion ratio. If a callable–puttable convertible bond needs to be priced at a time sufficiently far away from the expiry, only the moving boundary associated with the puttability needs to be dealt with. When the pricing time is closer to expiry beyond a critical point, it is then possible to have two distinct cases. While the two moving boundaries associated with conversion and puttability coexist in one case, they may both disappear in another case; callability remains to be the only issue that needs to be dealt with. Furthermore, there exists another critical value, beyond which a callable–puttable convertible bond can be treated as its vanilla counterpart. Mathematically, various different scenarios demand different systems of IEs to be formulated and solved numerically. [ABSTRACT FROM AUTHOR]

The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gausstype quadrature formula is used to approximate integrals during the discretization of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Stochastic arithmetics is also used based on the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. Applying this approach it is possible to find optimal parameters of the projective method. The numerical examples are included to illustrate the efficiency of proposed novel collocation method.

The purpose of this paper is to present a new concept of a Banach algebra in a fuzzy metric space (FM-space). We define an open ball, an open set and prove that every open ball in an FM-space over a Banach algebra A is an open set. We present some more topological properties and a Hausdorff metric on FM-spaces over A. Moreover, we state and prove a fuzzy Banach contraction theorem on FM-spaces over a Banach algebra A. Furthermore, we present an application of an integral equation and will prove a result dealing with the integral operators in FM-spaces over a Banach algebra.

Our aim is to prove some new fixed point theorems for a hybrid pair of multivalued α∗-dominated mappings involving a generalized Q-contraction in a complete modular-like metric space. Further results involving graphic contractions for a pair of multi-graph dominated mappings have been considered. Applying our obtained results, we resolve a system of nonlinear integral equations.

In this paper, we determine in a closed form the current distribution in a thin two-layer tubular conductor with a sinusoidal current in the presence of a filament with an equal but opposite current. The method of treating the problem consists in solving an integral equation for the current density in one layer which enables to find the current density in the other layer by imposing equality of the tangential components of the electric fields at the layer interface. An important factor, the AC to DC resistance ratio for the conductor is also found in a closed form. Some numerical examples, for current distribution and resistance ratio, versus frequency or geometry parameters, are presented.

Abstract We introduce a variable annuity (VA) contract with a surrender option and lookback benefit, that is, the benefit of the VA contract is linked to the maximum process of the policyholder’s account value. In contrast to the constant guarantee model provided in Bernard et al. (Insur. Math. Econ. 55:116–128, 2014), it is optimal for the policyholder of the VA contract with lookback benefit to surrender the VA contract when the policyholder’s account value is below or equal to the optimal surrender boundary. Thus, from the perspective of the insurer to construct a portfolio of VA contracts, utilizing the VA contracts with lookback benefit along with VA contracts with constant guarantee provides the diversification of early surrenders. The valuation of this contract can be described as a two-dimensional parabolic variational inequality. By converting this into the one-dimensional problem, we obtain the integral equations for the value function and the free boundary. The recursive integration method is applied to obtain the numerical solutions. We also provide comparative statics of the optimal surrender boundaries with respect to various parameters.

Motivated by applications in population models, we consider S -asymptotically periodic solution of fractional differential equations with periodic environment forces or asymptotically periodic ones. The system is quasi-monotone, and the existence of positive S -asymptotically periodic solution is established by using upper and lower solutions. The sufficient conditions that ensure the uniqueness of positive S -asymptotically periodic solution are also established on the basis of theory of sublinear operator. The applications of the general conclusions to classical population models yield the global convergence of positive S -asymptotically periodic solution in logistic equation with or without weak Allee effect, and the model of two cooperative populations. • Existence of asymptotically periodic solution. • Uniqueness of asymptotically periodic solution. • Upper and lower solutions. • Monotonic operator. • Ecological applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose a sharp and quantitative criterion, which focuses solely on Q curvature, to demonstrate the Chern-Gauss-Bonnet integral. In contrast to the previous results [6,7,11] , we use a new approach that involves estimating the singular integral. Furthermore, we derive the asymptotic formula for the solution to the general Q curvature equation. [ABSTRACT FROM AUTHOR]

• A series solution for an interface crack under remote mixed-moded loading is given. • Fracture mode mixity based on the non-oscillatory cohesive-spring interface solution is presented. • Machine-learning model was developed for fast evaluation of phase angle. • Can be applied to interface fracture characterization for IC interconnect structure. The oscillatory crack-tip field in the classical interface fracture mechanics solution often leads to complications in identifying the mode mixity in bimaterial interface fracture problems. In this paper, a non-singular series solution for the continuum problem of a crack on the cohesive-spring interface under mixed-mode loading is presented and compared to the classical oscillatory solution. A Gaussian process regression model was also developed to enable quick evaluation of the phase angle for interface delamination characterization by using mixed-mode bending fracture test. [ABSTRACT FROM AUTHOR]

This paper is concerned with water wave scattering of an obliquely incident surface wave train on a new type of breakwater model, which is called inverse T-type bottom-standing breakwater. Based on linear water wave theory, the physical problem reduces to a BVP (boundary value problem) which is solved by using the methodology of Havelock's expansion and Galerkin's approximation technique. In the solution process, a set of first kind Fredholm-type integral equations is solved approximately by choosing Gegenbauer polynomial as basis function in the multi-term Galerkin's technique. To address the singularities of order one-third that appears at the corners of the inverse T-type breakwater, appropriate basis polynomial with suitable weight functions is considered for solving the system of integral equations which consist cube root singularity. This study validates its findings by comparing the results with prior research in the field. The efficiency of the breakwater system is analyzed by evaluating the reflection coefficient and horizontal drift force for a broad variety of structural and wave parameters which are found to feature significant effects on the structures. The identification of Bragg resonance and the analogy to the Doppler effect in this study provide a remarkable benchmark for analyzing the design of water wave breakwaters. • The physical problem involving inverse T-type breakwater reduces to a BVP. • Havelock's expansion and Galerkin technique are used to solve the physical problem. • Cube root singularities are tackled using Gegenbauer polynomial as basis function. • Drift force, reflection and transmission coefficients are presented graphically. • Bragg resonance and Doppler effect have been observed in different graphs. [ABSTRACT FROM AUTHOR]

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

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

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.

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.

In this paper, we introduce the notion of a modular $ p $-metric space (an extended modular $ b $-metric space) and establish some fixed point results for $ \alpha $-$ \widehat{\nu} $-Meir-Keeler contractions in this new space. Using these results, we deduce some new fixed point theorems in extended modular metric spaces endowed with a graph and in partially ordered extended modular metric spaces. Also, we develop an important relation between fuzzy-Meir-Keeler and extended fuzzy $ p $-metric with modular $ p $-metric and get certain new fixed point results in triangular fuzzy $ p $-metric spaces. We provide an example and an application to support our results which generalize several well known results in the literature.

Statistical process control (SPC) plays a necessary role in manufacturing industry processes. An essential tool for SPC used for monitoring, measuring, controlling, and improving quality in various fields is the control chart. The modified exponentially weighted moving average (modified EWMA) control chart is widely used in various fields, and a measure commonly used to elucidate its efficiency is average run length (ARL). The main purpose of this study is to derive explicit formulas for the ARL to detect changes in the process mean of modified EWMA control chart for an autoregressive processes with explanatory variables (ARX(p,r)) with exponential white noise. In addition, the performances of the modified EWMA are compared with EWMA control charts based on the relative mean index (RMI). It was found that the explicit formulas for the ARL of the modified EWMA control chart performed better than on the EWMA control chart for monitoring process mean.

A new type of non-polynomial fractional spline function for approximating solutions of Fredholm-integral equations has been presented. For this purpose, we used a new idea of fractional continuity conditions by using the Caputo fractional derivative and the Riemann Liouville fractional integration to generate fractional spline derivatives. Moreover, the convergence analysis is studied with proven theorems. The approach is also well-explained and supported by four computational numerical findings, which show that it is both accurate and simple to apply.

In this paper, the direct-current (DC) resistivity inversion problem, based on the integral equation (IE), is expanded to data sets, including uneven topography, which has never been done before. For the inversion procedure, a weighted damped minimum length solution is implemented and depth weighting is used as the model weighting matrix. The regularisation parameter is estimated utilising an initial guess equal to the maximum value of the kernel matrix (forward operator). Then, the optimal value is adopted by a few trials and errors. To investigate the productivity of the presented method, the inversion results are compared with the inversion models derived from Res2dinv software, which works based on nonlinear equations. At first, a model of a conductive dyke under a valley is considered for which the retrieved model by the extended IE technique is even superior to the Res2dinv result. Because of the promising consequence derived from the synthetic case, the efficiency of the presented method is tested on the data set of the Robat Sang mine in Iran. The inversion model obtained from the extended IE method shows a high consistency with Res2dinv. [ABSTRACT FROM AUTHOR]

In this study, we obtain certain coupled fixed-point results for generalized contractions involving two control functions in a controlled metric space. Additionally, we establish some coupled fixed-point results in graph-enabled controlled metric spaces. Many well-known results from the literature will be expanded upon and modified by our results. In order to demonstrate the validity of the stated results, we also offer some examples. Finally, we apply the theoretical results to obtain the solution to a system of integral equations. [ABSTRACT FROM AUTHOR]

New mathematical results are given for the radiative transfer equations alone and coupled with the temperature equation of a fluid: existence, uniqueness, a maximum principle, and a convergent monotone iterative scheme. Numerical tests for Earth's atmosphere and the heating of a pool by the Sun are included. [ABSTRACT FROM AUTHOR]

This paper develops a finite approximation approach to find the solution F(x) of integral equations in the form F(x) = ξ(x) + R R κ(x, u)dF(u) ∀x ∈ R, where the unknown distributional solution F(x) defines the measure associated with the integration, and F(x) may not be continuous. To the best of our knowledge, the equation solutions in this framework have never been studied before. However, such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available. [ABSTRACT FROM AUTHOR]