Fredholm Type Integral Equations Research Articles

Wave interaction with multiple thin flexible porous barriers in water of uniform finite depth

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock’s inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin’s approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters.

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.

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.

Common Fixed Point Results on a Double-Controlled Metric Space for Generalized Rational-Type Contractions with Application

In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results.

Water wave scattering by pair of porous barriers in the presence of a bottom-standing rectangular obstacle

ABSTRACT Oblique wave scattering by a pair of partially immersed porous barriers, placed (i) on one side of (Position I), (ii) just above (Position II) and (iii) on the two sides (Position III) of a thick rectangular bottom-standing rigid obstacle, is studied here. The mathematical formulation and solution procedure of the corresponding boundary value problem are discussed using Havelock's expansion and Galerkin's approximation technique. During the solution procedure, a set of first-kind Fredholm-type integral equations is solved approximately by choosing suitable basis functions (Chebyshev and Gegenbauer polynomials). To address the singularities of order half and one-third that appear at the submerged edges of the thin barriers and the corners of the thick obstacle, appropriate basis polynomials with suitable weight functions are chosen in Galerkin's approximation. The reflection and transmission coefficients, dissipating wave energy, and dimensionless wave force are computed for various values of different parameters and depicted graphically in quite a number of figures. The numerical results for some special cases are compared with those available in the literature and excellent agreement has seen to have been achieved thus validating the correctness of the numerical method presented here. Based on numerical results it is observed that Position III of the barriers is more effective than Position I and Position II in resisting wave loads on marine areas and attenuating wave energy.

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.

Analysis of the family of integral equation involving incomplete types of I and Ī-functions

ABSTRACT The present article introduces and studies the Fredholm-type integral equation with an incomplete I-function (IIF) and an incomplete -function (I F) in its kernel. First, using fractional calculus and the Mellin transform principle, we solve an integral problem involving IIF. The idea of the Mellin transform and fractional calculus is then used to analyse an integral equation using the incomplete -function. This is followed by the discovery and investigation of several important exceptional cases. This article's general discoveries may yield new integral equations and solutions. The desired outcomes seem to be very helpful in resolving many real-world problems whose solutions represent different physical phenomena. And also, findings help solve introdifferential, fractional differential, and extended integral equation problems.

Complex-valued double controlled metric like spaces with applications to fixed point theorems and Fredholm type integral equations

<abstract><p>In this paper we introduce the concept of complex-valued double controlled metric like spaces. These new results generalize and extend the corresponding results about complex-valued double controlled metric type spaces. We prove some complex-valued fixed point theorems in this new complex-valued metric like spaces and, as application, we give an existence and uniqueness of the solution of a Fredholm type integral equation result. Moreover, some examples are also presented in favor of our given results.</p></abstract>

Fixed point results for generalized contractions in controlled metric spaces with applications

<abstract><p>The purpose of this article is to establish some common fixed point results for generalized contractions including some precise control functions of two variables in the setting of controlled metric spaces. As consequences of our leading results, we derive common fixed point and fixed point results for contractions with control functions of one variable and constants. We also discuss controlled metric spaces endowed with a graph and obtain some common fixed point results in this newly introduced space. As an application of our leading result, we examine the solution of a Fredholm type integral equation.</p></abstract>

Biermann Battery and Magnetic Fields of Accretion Discs

Magnetic fields should be studied to understand different processes in accretion discs, such as transition of angular momentum which is closely connected with them. There are different ways to describe the growth of magnetic fields of cosmic objects. One of them is based on Biermann battery mechanism which previously has been used to study the origin of magnetic fields in galaxies. It is based on different masses and same absolute values of charges of protons and electrons, which move across the disc. They interact with rotating medium, get azimuthal velocity and produce circular currents that are associated with the magnetic fields in the disc. Such magnetic field influences the motion of the charged particles that created it, and we should take into account the feedback. Mathematical description of this mechanism leads us to the Fredholm-type integral equation of the 2nd kind. In the presented work it is solved numerically using Galerkin methods. Here we give the solutions of different cases.

Symmetrical Hybrid Coupled Fuzzy Fixed-Point Results on Closed Ball in Fuzzy Metric Space with Applications

In this research, we establish some new fixed-point results for a symmetric coupled dominated fuzzy mapping satisfying a new advanced contraction on a closed ball in the setting of complete fuzzy metric spaces. In addition, the new notion of hybrid fuzzy-graph-dominated mappings introduced in fuzzy metric spaces achieves some new advanced fuzzy fixed-point problems. Some new definitions and illustrative examples are given to validate our new findings. Lastly, to demonstrate the originality of our new results, we present an application to the Fredholm-type integral equation.

Oblique Wave Diffraction by a Bottom-Standing Thick Barrier and a Pair of Partially Immersed Barriers

Abstract On the basis of the linearized theory of water waves, the present study has demonstrated a semi-analytical method to assess the hydrodynamic performance of a pair of partially immersed barriers just above a thick bottom-standing barrier. By means of the eigenfunction expansion method, a system of the first kind Fredholm-type integral equation involving a horizontal component of velocity as unknown functions is developed for the interaction of water waves with both types of barriers. The multiterm Galerkin approximation is adopted to determine these unknown functions having square root singularities at the submerged edge of the thin barriers and one-third singularities at the corners of the thick barrier. In order to overcome such types of singularities, Chebychev polynomials for half-singularities and ultra-spherical Gegenbauer polynomials for one-third singularities with suitable weight functions have been taken into consideration. The numerical examples of both reflection and transmission coefficients are presented to examine the hydrodynamic performance of breakwater. Some fascinating results like resonant frequencies are obtained for practical engineering. At the same time, reflection coefficients for the present breakwater agree reasonably for the limiting cases with the previously available results.

Numerical modeling of the performance of a well with an arbitrary piecewise-smooth external boundary in an anisotropic heterogeneous reservoir

The extraction of oil and gas from productive reservoirs of complex geological structures is constantly increasing. Thus, it is obvious that reliable mathematical models able to manage such porous media are of particular interest. In this paper, a model of filtration flow to a well located in an anisotropic heterogeneous soil formation, with a piecewise-smooth external contour, is numerically studied. The permeability of soil, as a porous medium, is characterized by a second-rank tensor. Soil heterogeneity is modeled by a power function of one of the coordinates. We consider a particular case with a positive exponent. The solution of the stated boundary problem of well operation in a porous reservoir causes significant mathematical difficulties associated with the complicated form of the basic equation. The formulation of the problem is reduced to a canonical form, which greatly simplifies the solution. To do this, we turn on the auxiliary plane using homeomorphic (affine) transformation. The problem is reduced to a system that includes a Fredholm type integral equation and an integral relation. The results were obtained using the discrete singularity method. It was found that the algorithm convergence is slightly better under weak anisotropy. The influence of anisotropy and heterogeneity of the soil on the production in the case of the productive reservoir with a rectangular boundary was assessed. The anisotropy can greatly change the production rate compared to the rate of a well in an isotropic soil. Permeability tensor components are located on the main diagonal and have a major impact on the rate of the flow in the anisotropic and heterogeneous soil. This method can be used to solve different problems of fluid filtration in an anisotropic porous medium.

Kannan-Type Contractions on New Extended b -Metric Spaces

This article is focused on the generalization of some fixed point theorems with Kannan-type contractions in the setting of new extended b -metric spaces. An idea of asymptotic regularity has been incorporated to achieve the new results. As an application, the existence of a solution of the Fredholm-type integral equation is presented.

Fredholm-type integral equation in controlled metric-like spaces

In this article we make an improvement in the Banach contraction using a controlled function in controlled metric like spaces, which generalizes many results in the literature. Moreover, we present an application on Fredholm type integral equation.

Solutions of Fredholm type integral equations via the classical Schauder fixed point theorem

We examine the existence of solutions for more general integral equations with respect to other work in Caballero Mena, Nalepa and Sadarangani (2014). Our techniques are based on the classical Schauder fixed point theorem and a new result about the relative compactness in Hölder spaces. Moreover, an example is presented to illustrate the obtained results.

An analytical approach to rotor blade modulation

Scattering of electromagnetic plane waves normally incident on a flat perfect electric conductor strip in arbitrary rotational motion in its own plane is investigated in the context of Hertzian Electrodynamics. This canonical problem is an analytical-asymptotic approach that reveals the parametric dependence of interference of electromagnetic waves with rotating blades, a subject which is known as rotor blade modulation. Scattering by a stationary perfect electric conductor strip under TE and TM incidences stand as two sub-problems that are incorporated in the main formulation. These sub-problems are formulated by Wiener–Hopf Technique and solved using a spectral iteration method that converges rapidly in the solution of corresponding coupled dual Fredholm type integral equations in case of electrically wide strips. Numerical implementations that simulate electromagnetic scattering by helicopters blades and wind turbines are provided and discussed.

Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel

Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

On Complex-Valued Triple Controlled Metric Spaces and Applications

In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

Water wave propagation over multiple porous barriers with variable porosity in the presence of an ice cover

A semi-analytical method is applied to investigate the propagation of flexural wave over multiple bottom-standing porous barriers with variable porosity beneath an ice cover under the assumption of linearised theory of water waves. Eigenfunction expansion method is used to express the velocity potential explicitly in terms of non-orthogonal eigenfunctions. Utilizing mode coupling relations satisfied by aforesaid eigenfunctions, the boundary value problem is reduced to a set of coupled Fredholm-type integral equations. These integral equations are solved by multi-term Galerkin’s method involving the Chebychev polynomials (multiplied by proper weights) as basis functions . The reflection and transmission coefficients, hydrodynamic forces and dissipated wave energy at the barriers are presented both analytically and graphically. A notable effect of the porosity of barriers and flexural rigidity of the ice cover on wave propagation is recorded. Bragg resonance of the flexural gravity waves due to the presence of four vertical porous barriers is observed and shown graphically. Efficiency of the present study is confirmed through a good agreement of the present results and the existing results available in the literature.