System Of Fredholm Integral Equations Research Articles

Existence of common fuzzy fixed points via fuzzy F-contractions in b-metric spaces

The main goal of this study is to establish common fuzzy fixed points in the context of complete b-metric spaces for a pair of fuzzy mappings that satisfy F-contractions. To strengthen the validity of the derived results, non-trivial examples are provided to substantiate the conclusions. Moreover, prior discoveries have been drawn as logical extensions from pertinent literature. Our findings are further reinforced and integrated by the numerous implications that this technique has in the literature. Using fixed point techniques to approximate the solutions of differential and integral equations is very useful. Specifically, in order to enhance the validity of our findings, the existence result of the system of non-linear Fredholm integral equations of second-kind is incorporated as an application.

Effects of a Nonlinear Boundary Condition on the Steady Aerodynamics of Porous Airfoils

This paper considers the lift coefficient of an airfoil with nonzero prescribed thickness and camber, along with a prescribed porosity distribution in a steady incompressible flow. In similar prior work, considering a Darcy-type porosity condition on the airfoil surface provides a Fredholm integral equation that can be solved exactly for Hölder-continuous porosity distributions. However, the generated lift predicted by the model diverges from the experimental data for porous airfoils with large values of the porosity parameter. This indicates a fundamental characteristic is missing in the mathematical model. Consequently, in this paper, the (linear) Darcy porosity condition is replaced by the (nonlinear) Forchheimer porosity condition. The Forchheimer porosity condition is decomposed into linear sections and furnishes a modified system of Fredholm integral equations, enabling an approximate solution by superposition for Hölder-continuous porosity distributions. The solution is then verified against the SD7003 airfoil results, and the comparison shows better agreement than considering the Darcy porosity condition. The methodology and results presented here may be combined with previous work on the aeroacoustics of porous airfoils with a Forchheimer boundary condition to address the conflicting aims of improving aerodynamic performance while reducing unwanted aeroacoustic emissions.

Numerical Simulation of Initial Value Problem of Integro-differential Equation Models

In this article, the Volterra-Fredholm integral equation (V-FIE) is derived from an initial value problem of kind integro-differential equation (IVP). We discuss the existence and uniqueness of the solution to the problem in Hilbert space. A numerical method is used to reduce this type of equation to System of Fredholm integral equations of the second kind(SFIEs). In light of this, the clustering method and the Galerkin method to solve the system of second-order Fredholm integral equations(SFIEs) and calculate the error in each case. Finally, the approximate and exact solutions are plotted on the same coordinate plane Using MATLAB code (2022).

Об особенностях идентификации переменных термомеханических характеристик функционально-градиентного прямоугольника

The inverse thermoelastic problem of identification of the variable properties of a functionally graded rectangle is studied. Unsteady vibrations are excited by applying mechanical and thermal loads to the upper side of the rectangle. To solve the direct problem in Laplace transforms, the method of separation of variables and the shooting method for harmonics are used. Transformants are inverted by expanding the origin in terms of shifted Legendre polynomials. The method proposed for solving the direct problem is verified by comparison with a finite element solution. The influence of the laws of change of variable characteristics on the boundary physical fields is analyzed. The displacement components give additional information on the mechanical loading, and the temperature measured on the upper side of the rectangle over a certain time interval – on the thermal loading. Assuming that the additional information admits expansion in Fourier series, the two-dimensional inverse problem is reduced to one-dimensional problems for various harmonics. The solution of the obtained nonlinear inverse problems is carried out on the basis of an iterative process, at each stage of which, in order to find corrections for thermomechanical characteristics, systems of Fredholm integral equations of the 1st kind are solved. The possibility of simultaneous reconstruction of several characteristics is investigated. The results of computational experiments on the phased reconstruction of thermomechanical characteristics are presented. The influence of the thermomechanical coupling parameter on the results of the thermal stress coefficient reconstruction was clarified.

TRANSFER OF LOADS FROM THREE HETEROGENEOUS ELASTIC STRINGERS TO AN INFINITE SHEET THROUGH ADHESIVE LAYERS

This paper considers the problem for an elastic infinite plate (sheet), which on parallel finite parts of its upper surface is strengthened by three finite stringers, two of which are located on the same line, having different elastic properties. The stringers are deformed under the action of horizontal forces. The interaction between infinite sheet and stringers takes place through thin elastic adhesive layers having other physical-mechanical properties and geometric configuration. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on three finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated. Further, for various values of changing characteristic parameters of the problem the multiple numerical results and its analysis are presented.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Solving Linear System of Fredholm Integral Equations with Homotopy Analysis and Genetic Algorithms

In this paper, we propose a technique called Homotopy Analysis Method (HAM) for solving linear systems of Fredholm integral equations to find relatively close solutions. The HAM approach involves an auxiliary parameter that offers a straightforward method for adjusting and managing the region where the series of solutions converge. We demonstrate the effectiveness of the HAM approach through our experimental results. Additionally, we improve the HAM approach by incorporating a genetic algorithm (HAM-GA) to further optimize the solutions. The performance of HAM-GA is evaluated by comparing its results to those obtained by HAM, using the residual error function as a fitness function for the genetic algorithm.

A Numerical Study for Solving the Systems of Fuzzy Fredholm Integral Equations of the Second Kind Using the Adomian Decomposition Method

In this paper, the Adomian decomposition method (ADM) is successfully applied to find the approximate solutions for the system of fuzzy Fredholm integral equations (SFFIEs) and we also study the convergence of the technique. A consistent way to reduce the size of the computation is given to reach the exact solution. One of the best methods adopted to determine the behavior of the approximate solutions. Finally, the problems that have been addressed confirm the validity of the method applied in this research using a comparison by combining numerical methods such as the Trapezoidal rule and Simpson rule with ADM.

Bernoulli wavelet method for numerical solution of linear system of Fredholm integral equation of the second kind

One of the key tools for many fields of applied mathematics is the integral equations. Integral equations are widely utilized in many models, atmosphere–ocean dynamics, fluid mechanics, mathematical physics, and many other physical and engineering disciplines. A new numerical strategy based on the Bernoulli wavelet is introduced to solve system of Fredholm integral equations of second kind. In this paper, the Bernoulli wavelets are first built. Second, the system of Fredholm integral equations has been reduced into an algebraic system. In order to demonstrate the viability, and accuracy of the suggested Bernoulli wavelet approach, some numerical examples are offered at the end. The derived numerical results are examined with those from other numerical techniques and with exact solutions, demonstrating the superiority of the proposed method over those techniques. The novelty of proposed technique is that it can be extended for the numerical solution of two dimensional integral equations and differential equations appearing in engineering models, however some modifications will be required.

Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space L2(−1,1)×C[0,T],(T<1). The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y.

C*-Algebra-Valued Partial Modular Metric Spaces and Some Fixed Point Results

In the present paper, we introduce the notion of C*-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space, C*-algebra-valued partial metric space, and C*-algebra-valued modular metric space and discuss it with examples. Some fixed point results using (ϕ,MF)-contraction mapping are discussed in such space. In addition, we study the stability of obtained results in the spirit of Ulam and Hyers. As an application, we also provide the existence and uniqueness of the solution for a system of Fredholm integral equations.

Mathematical models of spheroidal spiral-frame radiating structures

The article considers mathematical models of two spheroidal spiral-frame emitters, built on the basis of a general approach involving the use of an integral representation electromagnetic field. The internal problem of electrodynamics is reduced to a system of Fredholm integral equations of the 1st kind. The resulting system was solved by the method of moments with piecewise constant basis functions and delta functions as test functions. In this case, the local linearization of the generating conductors of the structures under consideration was carried out. The dependences of current distributions, input resistance, and radiation characteristics of structures on frequency have been studied. It is shown that standing, traveling, and mixed current waves can exist in the structures under consideration. The current regime is determined by the wave sizes and the geometry of the structures and determines the behavior of the wave resistance in the frequency range. Despite the similar geometry, the characteristics of the considered structures have certain differences.

Numerical Solution of Linear System of Fredholm Integral Equations Using Haar Wavelet Method

The aim of this paper is to present the numerical method for solving linear system of Fredholm integral equations, based on the Haar wavelet approach. Many test problems, for which the exact solution is known, are considered.Compare the results of suggested method with the results of another method (Trapezoidal method). Algorithm and program is written by Matlab vergion 7.Keywords: linear system of Fredholm integral equations, Haar wavelet metho

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

RECONSTRUCTION OF THE VARIABLE PROPERTIES OF A PIEZOELECTRIC ROD

The problem of reconstruction of variable characteristics (piezoelectric modulus and elastic compliance) of a functionally graded electroelastic rod with steady oscillations is considered when some additional information is given. To formulate the operator relations connecting the desired and measured characteristics, two types of influence are considered: a) by applying a potential difference to the electrodes, b) by applying a force to the end of the rod. Moreover, in the first case, the ends of the rod are free from stress and the current is measured, and in the second case, one of the ends of the rod is rigidly clamped, there are no electrodes, and the oscillation amplitude of the free end is measured. Statements of the corresponding boundary value problems are given in dimensionless form. Asymptotic formulas (quadratic in the frequency parameter) are constructed for the amplitude-frequency characteristics of the current and displacements in the low-frequency range. The inverse problem is solved on the basis of data on the amplitude-frequency characteristics of the current and displacements in a certain frequency range. The solution of the inverse problem is began with the procedure for choosing the initial approximation, and then it is constructed in an iterative manner. At each iteration, the direct problem with known characteristics is solved and corrections are found based on the solution of the system of Fredholm integral equations of the first kind with smooth kernels using the regularization method of A.N. Tikhonov. To find the initial approximation, the constructed asymptotic formulas are used, as well as the method of minimizing the residual functional. The conditions are described under which the non-uniqueness of the solution of the inverse problem is possible. The results of computational experiments on the simultaneous restoration of two functions are presented. The choice of the most informative frequency ranges is discussed. The various methods for choosing the initial approximation are considered. To control the iterative process, the plots of the residual in the frequency response and plots of the reconstruction error depending on the iteration number are given.

New algorithms for solving nonlinear mixed integral equations

<abstract><p>In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.</p></abstract>

Linearization-Discretization process to solve systems of nonlinear Fredholm integral equations in an infinite-dimensional context

In this paper, we propose a different way for solving systems of nonlinear Fredholm integral equations of the second kind. We construct our new strategy in two steps, through beginning with the linearization phase of the system of Fredholm integral equations by applying Newton method, then we pass to the discretization phase for some involved integral operator using Nystr\"{o}m method. The convergence analysis of our new method is proved under some necessary conditions. At last, a numerical application to approach a nonlinear Fredholm integro-differential equation by using this new process is taken to confirm its advantage.

C*-algebra valued quasi metric spaces and fixed point results with an application

In this paper, we introduce the notion of C*-algebra valued quasi metric space to generalize the notion of C*-algebra valued metric space and investigate the topological properties besides proving some core fixed point results. Finally, we employ our one of the main results to examine the existence and uniqueness of the solution for a system of Fredholm integral equations.

Reduction an Inverse Problem to a System of Second Kind Fredholm Integral Equations with No Singularities

In this article, we deal with an inverse problem concerning the two-dimensional Laplace equation with local boundary conditions on a bounded region. In this problem, the goal is to reduce it into a system of Fredholm integral equations of the second kind involving kernels with weakly/or no singularities (Fredholm property) by considering some additional conditions on the parameters of the problem. Finally, the method is carried out on an example, to show the simplicity and efficiency of the method.

Iterative Kernel Technique to Solve System Fredholm Integral Equation First Kind for Degenerate Kernel

Many problems associated with the engineering technology field can be transformed into Fredholm integral equations of the first kind to achieve problem-solving strategies. In this paper, the iterative kernel technique was reformulated to treat the numerical solution for the system of Fredholm integral equations of the first kind for the degenerate kernel. Three new theorems have been proposed and proved. This technique was programmed via Matlab and achieved a good result.