Integral Equation Problems Research Articles

On global convergence for an efficient third-order iterative process

We establish a global convergence result for an efficient third-order iterative process which is constructed from Chebyshev’s method by approximating the second derivative of the operator involved by combinations of the operator. In particular, from the use of auxiliary points, we provide domains of restricted global convergence that allow obtaining balls of convergence and locate solutions. Finally, we use different numerical examples, including a Chandrashekar’s integral equation problem, to illustrate the study.

Bernstein basis functions based algorithm for solving system of third order initial value problems

For obtaining numerical solutions of the system of ordinary differential equations (ODEs) of third order, a new numerical technique is proposed by using operational matrices of Bernstein polynomials. These operational matrices can be utilized to solve different problems of integral and differential equations. The System of third-order ODEs occur in various physical and engineering models. In this paper, an iterative algorithm is constructed by using operational matrices of Bernstein polynomials for solving the system of third order ODEs. The proposed technique provides a numerical solution by discretizing the system to a system of algebraic equations which can be solved directly. The method will be verified by using appropriate examples which are arising in Physics and some Engineering problems. The comparison of approximate and exact solution of the given examples is demonstrated with the help of tables and graphs.

M‐Parameter Mittag–Leffler function, its various properties, and relation with fractional calculus operators

A number of Mittag–Leffler functions are defined in the literature which have many applications across various areas of physical, biological, and applied sciences and are used in solving problems of fractional order differential, integral, and difference equations. This paper aims to define the m‐parameter Mittag–Leffler function, which can be reduced to various already known extensions of the Mittag–Leffler function. Important properties like recurrence relations, differentiation formulae, and integral representations are discussed for the newly defined m‐parameter Mittag–Leffler function. Moreover, the new m‐parameter Mittag–Leffler function is expressed in terms of some well‐known special functions such as generalized hypergeometric function, Mellin‐Barnes integral, Wright generalized hypergeometric function, and Fox H‐function. The paper also deals with various integral transforms like Euler‐Beta, Whittaker, Laplace, and Mellin transforms of the m‐parameter Mittag–Leffler function. Fractional differential and integral operators are considered to highlight certain intriguing properties of the m‐parameter Mittag–Leffler function. We also use the above function to define a generalization of Prabhakar integral and discuss its properties. Further, the relation of the newly defined m‐parameter Mittag–Leffler function with various other functions such as exponential, trigonometric, hypergeometric, and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

Electric Field Integral Equation Fast Frequency Sweep for Scattering of Nonpenetrable Objects via the Reduced-Basis Method

A reduced-basis approximation to solve for the electromagnetic scattering from nonpenetrable targets in a frequency band of interest is studied in detail. Perfect electric scatterers in homogeneous media are considered. Even though the frequency behavior can be complicated, the electromagnetic scattering does not arbitrarily change in the frequency band of interest. Thus, instead of using computationally inefficient, large-dimension numerical approximation scenarios, such as the method of moments, to solve for integral equations in the band of interest, a low-dimension reduced-basis space is identified to approximate the smooth evolution of the electromagnetic scattering itself as frequency changes. As a result, the frequency-parameter scattering problem is solved within a reduced-order model approach. The electric field integral equation formulation is addressed and properly selected electric currents are considered to span the reduced-basis space in the frequency band of interest. The involved frequency dependence in the integral equation due to the wave propagation phenomena described in the free-space Green’s function has traditionally limited the applicability of reduced-basis approximations for fast frequency sweep in integral equation methods in electromagnetics. In this article, we focus on this difficulty and propose a physics-based approach where the frequency-parameter dependence shows an affine behavior in the integral equation, without relying on rather complicated interpolation schemes. As a result, a fast frequency sweep for integral equation problems in electromagnetics is proposed. Finally, real-life applications will illustrate the capabilities of this methodology.

Product Algebras for Galerkin Discretisations of Boundary Integral Operators and their Applications

Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems.

TOMIC FUNCTIONS AND LACUNARY INTERPOLATION SERIES IN BOUNDARY VALUE PROBLEMS FOR PARTIAL DERIVATIVES EQUATIONS AND IMAGE PROCESSING

In the paper we consider and solve the problem of construction of the so called tomic functions – the systems of infinitely differentiable functions which while retaining many important properties of the shifts of atomic function up(x) such as locality and representation of algebraic polynomials and being based on the atomic functions nevertheless have nonuniform character and therefore allow to take into account the inhomogeneous and changing character of the data encountered in real world problems in particular in boundary value problems for partial differential equations with variable coefficients and complex geometry of domains in which these boundary value problems must be solved. The same class of tomic functions can be applied to processing,denoising and sparse storage of signals and images by lacunary interpolation. The lacunary or Birkhoff interpolation of functions in which the function is being restored by the values of derivatives of orderin points in which values of function and derivatives of order k<r are unknown is of great importance in many real world problems such as remote sensing. The lacunary interpolation methods using the tomic functions possesss important advantages over currently widely applied lacunary spline interpolation in view of infinite smoothness of tomic functions.The tomic functions can also be applied to connect (to stitch) atomic expansions with different steps on different intervals preserving smoothness and optimal approximation properties. The equations for of construction oftomic functions tofuj(x) –analogues of the basic functions of the generalized atomic Taylor expansions are obtained – which are needed for lacunary (Birkhoff) interpolation. For the applications in variational and collocation methods for solving bondary value problems for partial derivative and integral equations the tomic functions ftupr,j(x) are obtained that are analogues of B-splines and atomic functions fupn(x). Using similar methods, the tomic functions based on other atomic functions such as Ξn(x) can be obtained.

Numerical solutions of 2D Fredholm integral equation of first kind by discretization technique

A novel numerical technique to solve 2D Fredholm integral equations (2DFIEs) of first kind is proposed in this study. This technique is based on the discretization of 2DFIEs by replacing the unknown function with two-dimensional Bernstein polynomial basis functions. We formulate the convergence analysis which shows the fast converges of this technique to the actual solution. Some problems of 2D linear Fredholm integral equations are illustrated to show the efficiency of the proposed scheme.

AGE Iterative Method with First-Order Quadrature Scheme for Fuzzy Fredholm Integral Equations of Second Kind

In this study, we propose Alternating Group Explicit (AGE) iterative method to solve the problem of linear fuzzy Fredholm integral equations of the second kind (FFIE-2) based on Trapezoidal quadrature rule. First, we generate a system of linear equations by using approximation equations. Next, we use AGE iterative method to solve the generated linear system of FFIE-2. Then, we illustrate the applicability of AGE iterative method on some numerical examples. Finally, we do the comparison to test the efficiency of AGE with Jacobi and Gauss-Seidel (GS) iterative method based on three parameters: number of iterations, execution time and Hausdorff distance. From our findings, we found that AGE is a better iterative method compared with Jacobi and GS to solve FFIE-2.

Some new weakly singular integral inequalities with discontinuous functions for two variables and their applications

This paper investigates some new retarded weakly singular integral inequalities with discontinuous functions for two independent variables. The inequalities given here can be used in the qualitative analysis of various problems for integral equations and differential equations. Some examples are also given to illustrate the application of the conclusion.

Discussion for Identifying H-matrices

H-matrices are special matrices and have widely applications in many fields such as numerically solving the problems of integral equations, partial differential equations, control theory, and also artificial neural networks, but it is not easy to judge that whether a matrix is an H-matrix or not. In this paper, some practically numerical methods to determine H-matrices are proposed.

Modifid Interpolatory Projection Method for Weakly Singular Integral Equation Eigenvalue Problems

This paper deals with eigenvalue problems for linear Fredholm integral equations of the second kind with weakly singular kernels. A new discrete method is proposed for the approximation of eigenvalues. Compactness of the integral operator in L1[0, 1] space is obtained. This method is based on the approximation of the integral operator by modified interpolatory projection. Different from traditional methods, norm convergence of operator approximation is proved theoretically. Further, convergence of eigenvalue approximation is obtained by analytical tools. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.

Solvability of some classes of singular integral equations of convolution type via Riemann\u2013Hilbert problem

In this paper, we study methods of solution for some kinds of convolution type singular integral equations with Cauchy kernel. By means of the classical boundary value problems for analytic functions and of the theory of complex analysis, we deal with the necessary and sufficient conditions of solvability and obtain the general solutions and the conditions of solvability for such equations. All cases as regards the index of the coefficients in the equations are considered in detail. Especially, we discuss some properties of the solutions at the nodes. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problems for analytic functions (that is, Riemann–Hilbert problems). Therefore, the classical theory of integral equations is extended.

An Analytical Approximate Solution of Linear, System of Linear and Non Linear Volterra Integral Equations Using Variational Iteration Method

In this paper we have solved Linear Volterra Integral Equations (LVIE), System of Linear Volterra Integral Equation (SLVIE) and non Linear Volterra Integral Equation (NLVIE) using variational Iteration Method (VIM). VIM is a powerful tool to solve the differential equations as well as integral equations. VIM gives consecutive approximations which converge rapidly to the exact solution in the given domain for the given initial condition, if it exists, without any transformation or any restrictive assumptions to non linear terms, which may change the physical behaviour of the problem, otherwise few approximations are enough only for numerical purposes. To exhibit the ability, accuracy and reliability of the method we have provided two problems of Linear Volterra Integral Equations, one problem of system of Linear Volterra Integral Equation and one problem of Non Linear Volterra Integral Equation. The study highlights the heaviness of the method.

Yang-Laplace transform method Volterra and Abel's integro-differential equations of fractional order

This study outlines the local fractional integro-differential equations carried out by the local fractional calculus. The analytical solutions within local fractional Volterra and Abel’s integral equations via the Yang-Laplace transform are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the local fractional integral equations.

Singular integral equations of convolution type with Cauchy kernel in the class of exponentially increasing functions

In this paper we study some classes of generalized singular integral equations of convolution type with Cauchy kernel in the class of exponentially increasing functions. Such equations can be transformed into Riemann boundary value problems with two unknown functions on two parallel straight lines via Fourier transformation. The general solutions and the conditions of solvability are obtained by means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation. This paper will be of great significance for the study of improving and developing complex analysis, integral equation and boundary value problem. Therefore, the classic Riemann boundary value problem is extended further.

Linear quadratic control problems of stochastic Volterra integral equations

This paper is concerned with linear quadratic control problems of stochastic differential equations (SDEs, in short) and stochastic Volterra integral equations (SVIEs, in short). Notice that for stochastic systems, the control weight in the cost functional is allowed to be indefinite. This feature is demonstrated here only by open-loop optimal controls but not limited to closed-loop optimal controls in the literature. As to linear quadratic problem of SDEs, some examples are given to point out the issues left by existing papers, and new characterizations of optimal controls are obtained in different manners. For the study of SVIEs with deterministic coefficients, a class of stochastic Fredholm−Volterra integral equations is introduced to replace conventional forward-backward SVIEs. Eventually, instead of using convex variation, we use spike variation to obtain some additional optimality conditions of linear quadratic problems for SVIEs.

Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations

We extended the moving least squares (MLS) and modified moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations. This modification is proposed on the quadratic base functions $$(m=2)$$ by imposing additional terms based on the coefficients of the polynomial base functions. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Additionally, finding the optimum value for the radius of the domain influence is an open problem for MLS-based methods. So an efficient algorithm is introduced for computing a suitable value of dilatation parameter to determine the radius of the support domain. This algorithm able to prevent the singular matrix which is an outcome of adverse selection of the radius of influence domain. In numerical examples are provided to enable us to compare MMLS method and standard MLS method by the new proposed algorithm. Comparing the errors of MMLS and MLS method determines the capability and accuracy of applied techniques to solve systems of integral equation problems. This indicates the advantage of the proposed method respect to MLS method.

Coincidence Point Theorems for (α,β,γ)-Contraction Mappings in Generalized Metric Spaces

The result of our study is that a coincidence point of two mappings P and Q can be achieved when the ordered pair (P,Q) is an (α,β,γ)-contraction with respect to a generalized metric space. Moreover, with some additional condition, a common fixed point can be obtained as a consequence of our main theorems. Further, we apply our findings to some examples and integral equation problems.

Some applications of fixed point results for generalized two classes of Boyd\u2013Wong\u2019s F-contraction in partial b-metric spaces

In this paper, we will present some fixed point results for two classes of generalized contractions of Boyd–Wong type in partial b-metric spaces. More precisely, the structure of the paper is the following. In section one, we present some useful notions and results. The aim of section two is to introduce the concepts of Boyd–Wong F-contractions of type A and of type B and establish some new common fixed point results in partial b-metric spaces. We show the validity and superiority of our main results by suitable examples which are visualized by corresponding surfaces and related graphs. In section three, we correct some slip-ups in some recent papers. Finally, in section four, two applications to integral equation and periodic boundary value problem are included which make effective the new concepts and results.

The analytical solutions for conformable integral equations and integro-differential equations by conformable Laplace transform

In this article, existence theorem for conformable Laplace transform is expressed. Then by using basic properties of conformable Laplace transform such as convolution theorem, conformable Laplace transform of fractional derivative and fractional integral, authors obtained the exact solution of initial value problems for integral equations and integro-differential equations where the derivatives and integrals are in conformable sense. In the literature it is the first time that obtaining the solutions of integro differential equations, integral equations by means of conformable fractional derivative.