Solution Of Integral Equation Research Articles

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.

Approximate solution of diffraction integral equation of resonator.

Utilizing the diffraction integral equation and the principle of slow amplitude approximation, we obtain a novel approximate solution of the transverse mode including the cavity parameters a (a is the section size of the resonator) and g = 1-L/R (L is the cavity length, R is the radius of curvature of the cavity). With this approximate solution, we can explore the influence of the resonator parameters a and g on the transverse mode. The theoretical analysis demonstrates that a and g have a certain influence on the shape and quality of the transverse mode, and selecting the appropriate a and g can effectively improve the quality of the transverse pattern. Moreover, laser experiments are conducted to validate analysis conclusion.

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.

Exploring Integral ϝ-Contractions with Applications to Integral Equations and Fractional BVPs

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ,Ω-Ϝ)-contraction has been introduced, which is called (ϰ,Γ1,2,Ω-Ϝ)-contraction. In the end, the applications of an extended integral Ϝ-contraction and (ϰ,Ω-Ϝ)-contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ,Γ1,2,Ω-Ϝ)-contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required.

On Data Dependency and Solutions of Nonlinear Fredholm Integral Equations with the Three-Step Iteration Method

In this study, the solution of the second type of homogeneous nonlinear Fredholm integral equations is investigated using a three-step iteration algorithm. In other words, it has been shown that the sequences obtained from this algorithm converge to the solution of the mentioned equations. Also, data dependency is obtained for the second type of homogeneous nonlinear Fredholm integral equations and this result is supported by an example.

Existence and stability of solution for a nonlinear Volterra integral equation with binary relation via fixed point results

The real-world problems that can be resolved by a dynamic mathematical model is an analogous form of a complex integral equation. In this article, we intend to exhibit the solution to a non-homogeneous, nonlinear Volterra integral equation endowed with an arbitrary binary relation utilizing the fixed point solution that is originally proved for a multivalued map and then reduced to a single valued self map. The results proved and the application demonstrated is an added asset to the present literature and can significantly contribute in determining the existence of a solution for a dynamic mathematical model in the form of a Volterra integral equation equipped with an arbitrary binary relation via fixed point findings. Furthermore, the stability of the solution of the Volterra integral equation is verified using the Hyers–Ulam stability concept.

Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay

In this paper, we are concerning with the existence of the solution \( \V \in L_1([0,\tau],L_2(\Omega))\) of the singular stochastic fractional-order integral equation with delay \(\varrho(.) \), \[ \V(t) = B(t) t^{\alpha - 1} + \lambda ~ \I^{\beta} \G(t,\V(\varrho (t))), ~~~t\in (0,\tau], \] where \(B(t)\) is a given second order mean square stochastic process, \( \lambda \) is a parameter, \(\varrho (t) \leq t\), and \(\G(t,\V) \) is a measurable function in \(t \in (0,\tau]\) and satisfies Lipschitz condition on the second argument. %and \(x\in C_{1-\beta}([0,T],L_2(\Omega)) $ will be proved The Hyers-Ulam and generalized Hyers-Ulam-Rassias stability will be proved. Moreover, the continuous dependence of the solution on the process \(B(t)\) and \(\lambda\) will be studied. As applications, some nonlocal, weighted and nonlocal-weighted integral problems of stochastic fractional-order differential equations will be studied.

On Ulam type of stability for stochastic integral equations with Volterra noise

Abstract This paper concerns the existence, uniqueness and stability of solutions of stochastic Volterra integral equations perturbed by some random processes. The obtained results extend, generalize and enrich the theory of stochastic Volterra integral equations in literature. Lastly, for illustration, we give an example that agrees with the theoretical analysis.

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γze−sz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients.