Type Integral Equations Research Articles

On existence of extremal integrable solutions and integral inequalities for nonlinear Volterra type integral equations

We prove the existence of maximal and minimal integrable solutions of nonlinear Volterra type integral equations. Two basic integral inequalities are obtained in the form of extremal integrable solutions which are further exploited for proving the boundedness and uniqueness of the integrable solutions of the considered integral equation.

Generalizations of Riemann–Liouville fractional integrals and applications

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs.

THE DEVELOPMENT OF A PIEZOELECTRIC DEFECT DETECTION DEVICE THROUGH MATHEMATICAL MODELING APPLIED TO POLYMER COMPOSITE MATERIALS

In this research article, the focus is on examining the strength characteristics of polymer composite materials under various stress states. The investigation into the material dissolution process is conducted in two distinct stages. Initially, the article employs a criterion approach to derive explicit formulas for the latent decay time (incubation period) of materials, considering singular, exponential and Abelian kernels of the damage operator. In the subsequent stage, the classical strength condition is applied to establish a Volterra type integral equation, which characterizes the damage process for hereditary materials. These resulting differential equations are then solved with appropriate initial conditions. The study utilizes the obtained results to create time-dependent graphs of the damage parameter and performs comparisons. A piezoelectric defect detection device for polymer composite materials has been proposed through the application of mathematical modeling.

3-Dimensional computational analysis of [formula omitted]-contraction in GV-fuzzy metric spaces with applications

It is a well known fact that contraction conditions play a major role in composing coincidence point results. In present work, a new ϕ-contraction is being corroborated to yield coupled coincidence points in partially ordered GV-fuzzy metric spaces. Due to the significant feature of H-type t-norm, in this communication, we endue GV-fuzzy metric spaces with this t-norm. We use the mixed monotone property of mappings with regard to partial ordering. Present work generalizes some already existing results. An applied example is also formulated to cast the 3-dimensional analysis of ϕ-contraction utilized in the main result. Computational analysis of the illustrative example has been done using the software MATLAB version R2022b which shows the experimental verification of our work. Furthermore, applications to the solution of the system of Fredholm type integral equations and solution of the system of equations in dynamic programming accords the validity of the present work. Endmost, the concluding remark projects the significance of current work.

An iterative method for solving one class of non-linear integral equations with Nemytskii operator on the positive semi-axis

A class of non-linear integral equations with monotone Nemytskii operator on the positive semi-axis is considered. This class of integral equations appears in many areas of modern natural science. In particular, such equations, with various restrictions on non-linearity and the kernel, arise in the dynamic theory of $p$-adic strings for the scalar field of tachyons, in the kinetic theory of gases and plasma within the framework of the usual and modified non-linear Bhatnagar-Gross-Crook models for the Boltzmann kinetic equation. Equations of similar nature appear also in the non-linear radiative transfer theory in inhomogeneous media and in the mathematical theory of the spread of epidemic diseases within the framework of the modified Diekmann-Kaper model. A constructive theorem for the existence of a bounded positive and continuous solution is proved. As a result, we get a uniform estimate of the difference between the previous and subsequent iterations, the corresponding successive approximations converge uniformly to a bounded continuous solution to the equation. The asymptotic behaviour of the constructed solution at infinity is studied. In particular, it is proved that the limit of this solution at infinity exists and is positive, and is uniquely determined from a certain characteristic equation. It is also proved that the difference between the limit and the solution is a summable function on the positive semi-axis. Using some geometric estimates for convex and concave functions, and employing the theorem on integral asymptotics obtained here, we prove the uniqueness of the solution in a certain subclass of non-negative non-trivial continuous and bounded functions. The results obtained are also applied to the study of a special class of non-linear Urysohn type integral equations on the positive semi-axis. In particular, the existence of a positive bounded solution to this class of equations is verified, and some qualitative properties of the constructed solution are studied. We also give specific applied examples of the corresponding kernels and non-linearities to illustrate the importance of the results obtained.

A Novel Approach with Comparative Computational Simulation of Analytical Solutions of Fractional Order Volterra Type Integral Equations

The main purpose of this article is to reconsider the theory of variational iteration method and embed it with Laplace transform to construct the conjoined technique variational iteration Laplace transform method. And then, by using the proposed method we solve a class of integral equations, especially fractional order Volterra type integro‐partial‐differential equations occurring in different fields of natural and social sciences. For this, we first turn to some prelude concepts and different types of equations especially integral equations of different kinds from the literature. Then, we will present step by step the embedded technique. Next, some fractional order model integral equations will be tackled using the proposed method along with some existing techniques, like the Adomian decomposition method and successive approximations method. The efficacy and authentication of the proposed technique will be analysed with the help of the obtained results and graphical exhibitions of the particular solved examples. Finally, use of the initiated method in the future for different types of equations occurring in different fields of science will be presented.

A system of Abel type integral equations

A system of Abel type integral equations

Coincidence and common fixed points for F-Contractive mappings

The purpose of this article is to establish the existence and uniqueness of coincidence and common fixed point of discontinuous non-compatible faintly compatible pair of self maps in non-complete metric space without using containment requirement of range space of involved maps satisfying Ciric type F-contraction and Hardy-Roger type F-contraction. Some illustrative examples associated with pictographic validations are provided to demonstrate the main results and to show the genuineness of our results. We consider the application of our results to the study of a two-point boundary value problem related to second order differential equation, solve the two-point boundary value problem of the second-order differential equation arising in electric circuit equation, and also apply our results to Volterra type integral equation using Ciric type F-contraction as well as Hardy Roger type F-contraction. Mathematics subject classification: 47H10; 54H25; 54E50

Isospin symmetry and analyticity in D→K¯ππ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ D\ o \\overline{K}\\pi \\pi $$\\end{document} decays

We perform a detailed study of the consequences of isospin symmetry in the Cabibbo favoured D→K¯ππ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ D\ o \\overline{K}\\pi \\pi $$\\end{document} decays. These processes are important for precision testing of the Standard Model and for hadronic physics. Combining isospin symmetry with a dispersive reconstruction theorem we derive a representation in terms of one-variable functions which allows one to predict all the D→K¯ππ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ D\ o \\overline{K}\\pi \\pi $$\\end{document} amplitudes given inputs from one D+ mode and one D0 mode. From this, using dispersion relations and unitarity, we derive a set of 6+6 Khuri-Treiman type integral equations which enable to take three-body rescattering effects into account. A first test of this approach is presented using experimental results on the D+→ KSπ0π+ and the D0→ KSπ−π+ modes.

Bazı İntegral Denklemlerin Nokta Kollokasyon Yöntemiyle Çözümü

In several engineering or physics problems, particularly those involving electromagnetic theory, thermal and radiation effects, acoustics, elasticity, and some fluid mechanics, it is not always easy or possible to find the analytical solution of integral equations that describe them. For this reason, numerical techniques are used. In this study, Point-collocation method was applied to linear and nonlinear, Volterra and Fredholm type integral equations and the performance and accuracy of the method was compared with several other methods that seem to be popular choices. As the base functions, a suitably chosen family of polynomials were employed. The convergence of the method was verified by increasing the number of polynomial base functions. The results demonstrate that the collocation method performs well even with a relatively low number of base functions and is a good candidate for solving a wide variety of integral equations. Nonlinear problems take longer to calculate approximate solution coefficients than linear problems. Furthermore, it is necessary to use the real and smallest coefficients found in order to obtain a suitable approximate solution to these problems.

THREE OVER DETERMINED SYSTEM VOLTERRA TYPE INTEGRAL EQUATION WITH THREE SINGULAR AND SUPER-SINGULAR DOMAINS

In this paper, for the first time, an overridden system of Volterra-type integral equations with singular or super-singular kernels is considered. It is proved that, when certain conditions are met, the problem of finding a solution to an over determined Volterra type system of integral equations reduces to the problem of finding a solution to one-dimensional Volterra-type integral equations with singular or super-singular kernels, the theory of which is developed in the works of the author of this article. Depending on the sign of the values of functions in the kernels of the system of Volterra type integral equations, found manifold solution consideration system integral equation.

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

Solution of a two-dimensional parabolic model problem in a degenerate angular domain

In this paper, the boundary value problem of heat conduction in a domain was considered, boundary of which changes with time, as well as there is no the problem solution domain at the initial time, that is, it degenerates into a point. To solve the problem, the method of heat potentials was used, which makes it possible to reduce it to a singular Volterra type integral equations of the second kind. The peculiarity of the obtained integral equation is that it fundamentally differs from the classical Volterra integral equations, since the Picard method is not applicable to it and the corresponding homogeneous integral equation has a nonzero solution.

Common fixed point theorems in intuitionistic fuzzy metric spaces with an application for Volterra integral equations

This work aims to demonstrate some common fixed point results in light of certain common limit range properties. Such results are associated with certain weakly compatible mappings in intuitionistic fuzzy metric spaces that meet a specific implicit relation. In particular, the accomplished results in this paper generalize several significant theorems reported in the literature. As an application of our work, we utilize a certain novel result to establish the existence and uniqueness of a common solution for a system of Volterra type integral equations.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

We have four kinds of solutions for Cauchy type integral equations: by expanding on known functions and using the Maclaurin series, we can convert these four kinds of solutions into linear combinations of some elements that are basis of these solutions. Using these bases gives exact solutions for polynomials and, for some other functions, a high-accuracy approximate solution.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations

In this paper, orthogonal fuzzy versions are reported for some celebrated iterative mappings. We provide various concrete conditions on the real valued functions J,S:(0,1]→(−∞,∞) for the existence of fixed-points of (J,S)-fuzzy interpolative contractions. This way, many fixed point theorems are developed in orthogonal fuzzy metric spaces. We apply the (J,S)-fuzzy version of Banach fixed point theorem to demonstrate the existence and uniqueness of the solution. These results are supported with several non-trivial examples and applications to Volterra-type integral equations and fractional differential equations.

Iterative schemes for linear equations of the second kind and related inverse problems

This paper consists of two parts. The first one deals with the generation of an iterative algorithm to obtain an approximate solution of a linear equation of the second kind in a Banach space. This generation is based on a perturbed version of the geometric series theorem which, in particular, allows us to find a family of unisolvent linear Fredholm integral equations of the second kind, even when the associated linear operator has norm greater than or equal to 1. When we consider Fredholm equations of this type and linear Volterra integral equations of second kind, the numerical schemes obtained when appropriate Schauder bases are also introduced in the spaces where the equations operate, enable us to approximate their respective solutions iteratively. The second part of this work focuses on the design of a numerical method for solving an inverse problem associated with a linear equation of the second kind in a Banach space, a method which we apply to problems of parameter estimation related to the two classes of integral equations mentioned above.

A new integral equation for Brownian stopping problems with finite time horizon

For classical finite time horizon stopping problems driven by a Brownian motion V(t,x)=supt≤τ≤0E(t,x)[g(τ,Wτ)],we derive a new class of Fredholm type integral equations for the stopping set. For a large class of discounted problems, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes.

Applications in Integral Equations through Common Results in C*-Algebra-Valued Sb-Metric Spaces

We study some common results in C*-algebra-valued Sb-metric spaces. We also present an interesting application of an existing and unique result for one type of integral equation.