Solution Of Integral Equation Research Articles

A Qualitative Study of Solutions of Nonlinear Volterra Integral Equation in the Plane, from the Perspective of Shiba Bounded Variation

In this article a nonlinear Volterra integral equation is studied in the space of functions of the bounded variation in the Shiba sense on the plane. A series of conditions is established for the functions and the kernel involved in this equation which guarantee the global existence and uniqueness of the solution in this space.

A THEORY OF MULTIDIMENSIONAL FREDHOLM INTEGRAL EQUATIONS HAVING SEPARABLE KERNELS: SOLVABLE IN A REGION SURROUNDING BY THE HYPERPLANES

In this article, we present a theory of multidimensional Fredholm integral equations, having separable kernels, are solvable in a region surrounded by hyperplanes. In derivation of their solutions, we employ the generalized Hilbert-Schmidt theory involving eigenvalues and corresponding normalized eigen functions obtained by separable kernels in a region surrounded by the hyperplanes. Finally, we apply two variables Gegenbauer polynomials and derive a result on the inequality of the solution for the double symmetric Fredholm integral equation.

Some New Rational Contractions Approach to the Solution of Integral Equations via Unique Common Fixed Point Theorems in Complex Valued -Metric Spaces

Some New Rational Contractions Approach to the Solution of Integral Equations via Unique Common Fixed Point Theorems in Complex Valued -Metric Spaces

Oscillatory Solution of a Convolutional Volterra Integral Equation

Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.

Modeling of induced currents in a type-II superconducting cylinder of finite length in a transverse magnetic field

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation.

The Existence of a Unique Solution of Nonlinear Integral Equation in Fréchet Space

In this article, the existence of a unique solution on a semi-infinite interval for an integral equation in Fréchet space has been studied by applying a nonlinear alternative of the Leray-Schauder type for contraction maps.

Application of Rishi transform to the solution of nonlinear Volterra integral equation of the second kind ∗

This paper aims to investigate the solution of non-linear Volterra integral equation of the second kind by using the Rishi transform. The solutions of two numerical problems in compact form are determined by applying the Rishi transform, which sug gests that the Rishi transform can be used as a tool for solving these and the other types of related real world problems across various disciplines.

Location, separation and approximation of solutions of nonlinear Hammerstein-type integral equations

From Newton's method, we construct a Newton-type iterative method that allows studying a class of nonlinear Hammerstein-type integral equations. This method is reduced to Newton's method if the kernel of the integral equation is separable and, unlike Newton's method, can be applied to approximate a solution if the kernel is nonseparable. In addition, from an analysis of the global convergence of the method, we can locate and separate solutions of the nonlinear Hammerstein-type integral equations involved. For this study of the global convergence, we use auxiliary functions and obtain restricted global convergence domains that are usually balls.

“Spectral Method” for Determining a Kernel of the Fredholm Integral Equation of the First Kind of Convolution Type and Suppressing the Gibbs Effect

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.

Motion of a disk embedded in a nearly inviscid Langmuir film. Part 1. Translation

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number, ${\textit {Bq}}\gg 1$ . A semianalytic solution of the dual integral equations governing the flow at arbitrary ${\textit {Bq}}$ was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit ${\textit {Bq}}\to 0$ , it produces the value $8$ for the dimensionless translational drag, which is $50\,\%$ larger than the classical $16/3$ -value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit ${\textit {Bq}}\to 0$ from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction $(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$ , $\gamma _E$ being the Euler–Mascheroni constant.

Regularization of the Solution of a Nonlinear Integral Equation of the First Kind

Integral equations, the main branch of mathematics, are widely used in physics, engineering, mechanics, control theory and other fields. Related to the application of integral equations, new fields are developing, such as economics, some sections of biology, etc. The theory of integral equations mainly developed in the late nineteenth-early twentieth century, starting with Vito Volterra (1982, 1986), Eric Ivar Fredholm (2010), David Hilbert, Erhard Schmidt, etc. scientists began to study it. Nevertheless, within the framework of mathematical concepts that existed before the first half of the twentieth century, such problems were considered incorrect because a small change in the given functions led to a greater change in the desired functions. The Volterra equation of the first kind is an integral equation that has an exact solution only in some cases. The limit of integration has been carried out in very small quantities on non-classical linear and nonlinear integral equations with variable limits and the construction of solutions in these works is based on numerical methods. Therefore, for the so-called non-classical Volterra integral equations, it is relevant to determine the conditions that ensure the uniqueness and regularization of their solutions. In this paper, the uniqueness of the solution of the non-classical nonlinear integral Volterra equation of the first kind is resolved. The aim of the study is to solve the non-classical Volterra integral equation of the first kind, that is, to determine the conditions that ensure the uniqueness of the solution of the nonlinear non-classical Volterra integral equation of the first kind. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems.

Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant

For the telegraph equation with a nonlinear potential in a curvilinear quadrant, we consider a mixed problem with the Cauchy conditions on a spatial half-line and the Dirichlet condition on a noncharacteristic curve. The solution of the problem is constructed by the method of characteristics in an implicit analytical form as a solution of integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. A mild solution is constructed in the case of insufficiently smooth data of the problem.

Integral Equations of Volterra Typewith Two Boundary and One Interior Singular Point

Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.

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.

A highly accurate indirect boundary integral equation solution for three dimensional elastic scattering problem

In this paper, we propose a numerical formula for three dimensional elastic wave scattering problem. Different from the classical boundary integral equation method, the three dimensional elastic scattering problem is transformed into a coupled boundary value problem including a scalar Helmholtz equation and a vector Helmholtz equation by the decomposition of the displacement. The coupled boundary value problem can be solved by the boundary integral equations based on the Helmholtz equation. A denseness result is given to support this algorithm. A comparative study with the classical method of fundamental solutions getting from the boundary integral equation method is given to check the accuracy with different types of boundary conditions. From the numerical results, the present method can give more accurate results than the classical method of fundamental solutions method.

Gauss-Legendre type quadrature iterative method for fuzzy Fredholm integral equations

In this paper we obtain an iterative method for the numerical solution of second kind fuzzy Fredholm integral equations by applying appropriate composite fuzzy quadrature formula. We approach a general fuzzy quadrature formula and show that the three-point and four-point Gauss-Legendre type fuzzy quadrature formulas are the best choice from computational cost and accuracy point of view. The convergence of the method is proved by providing the error estimates expressed in terms of the Lipschitz constant of the Picard iterations. Some numerical results confirm the obtained theoretical result and illustrate the performances of the three point Gauss-Legendre method in comparison with the techniques generated by the fuzzy Newton-Cotes and Gauss-Lobatto quadrature formulas, the results being tested on the Simpson and on the four point Gauss-Legendre and Gauss-Lobatto quadrature formulas in particular. The numerical experiment suggests that five point or more point fuzzy quadrature formulas are useless due to the accumulation of errors and the grown computational cost.

Application of Fixed Point Result in Complex Valued Extended b-Metric Space

The aim of the present research work is to investigate the solution of Urysohn integral equation by common fixed point result in the setting of complex valued b-metric space. To obtain the objective, we used a generalized rational contraction involving control functions and a pair of self-mappings. In this way, we generalize some well-known results of literature. Some non-trivial examples are also flourished to demonstrate the innovation of our principal result.

Averaged Nyström interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nyström methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose and introduces new stability properties of them.

Existence of Solutions of Linear Set-Valued Integral Equations and Their Properties

Existence of Solutions of Linear Set-Valued Integral Equations and Their Properties

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.