Solution Of Fredholm Integral Equations Research Articles

Approximate solution of Fredholm integral equations of the second kind through multinode Shepard operators

An efficient numerical method based on the Shepard method has been developed to deal with Fredholm integral equations of the second kind. The procedure involves expressing the unknown function as a linear combination of Shepard basis functions. Subsequently, a collocation method is used to get the matrix equation yielding the coefficients of such an approximate solution. The effectiveness of the proposed approach is illustrated by some examples.

Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by x(s)=1+λx(s)∫01st+sx(t)dt, can be very often encountered in many applications, where x is the function to be determined, λ is a parameter, and t,s∈[0,1]. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form χ(l)=ϱ(l)+χ(l)∫pqk(l,z)(Vχ)(z)dz are investigated in the space Cωp,q, where χ is the unknown function to be determined, V is a given operator, and ϱ,k are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.

Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces

For the numerical solution of Fredholm integral equations on [−1,1] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods.

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.

Polarized radiative transfer: Monte Carlo code Ventspils RTMC. Testing of it and RADMC3D

We have created Monte Carlo code Ventspils RTMC for polarized radiative transfer modelling in an isotropic, optically inactive medium of arbitrary configuration and morphology, and made it public. Its main advantages are simplicity of structure and usage, which can be a benefit for simple physical problems, together with use of sound and fully defined basic equations describing the scattering process. We compared the numeric results of Ventspils RTMC with those of RADMC3D created by Dullemond et al., as well as with the direct solution of Fredholm integral equations for source functions. A model of homogeneous sphere with Rayleigh scattering and concentric star inside it was used for comparison. The conclusion is that all three codes give essentially the same results if the radial optical thickness of the sphere does not exceed 5. If the optical thickness is between 5 and 10 then the statistical noise increases, and some small but systematic trends of differences appear between the results of three codes; however, the three codes are compatible. For optical thickness exceeding 10, the current version of Ventspils RTMC is too slow. There is a theoretical uncertainty about the adequacy of diffusion approximation of polarized radiative transfer used in RADMC3D for optical thickness exceeding 15. While RADMC3D is a very complex code usable for broad scope of applications, our Ventspils RTMC is a much simpler code usable for continuum transfer in case of modest optical thickness.

Karhunen-Loève expansion based on an analytical solution over a bounding box domain

This paper explores the accuracy and the efficiency of analytical solution of Fredholm integral equation to represent a random field on complex geometry. Because no analytical solution is available for arbitrary domains, it is proposed to use the analytical solution on simple bounding domains that enclose complex two- or three-dimensional geometries. It is a simple, accurate and robust approach for discretising a random field. The effect of the size of the bounding box on the resulting random field variance is investigated carefully and compared with the numerical solution given by the finite element method. The error variance is particularly localised near to the support domain boundary. Therefore, it is suggested to expand the bounding domain. This paper proposes a calibration of the correlation length to the ratio of the domain to maintain the convergence rate and the variance accuracy of the KLE without enlarging the stochastic dimension.

Gegenbauer Series for Numerical Solution of Fredholm Integral Equations of the Second Kind

Gegenbauer Series for Numerical Solution of Fredholm Integral Equations of the Second Kind

Solution of Fredholm Integral Equation via Common Fixed Point Theorem on Bicomplex Valued B-Metric Space

The notion of symmetry is the main property of a metric function. The area of fixed point theory has a suitable structure for symmetry in mathematics. The goal of this paper is to find fixed point and common fixed point results in a bicomplex valued b-metric space for mixed type rational contractions with control functions. Some well-known literature findings were generalized in our main findings. We provide an example to strengthen and validate our main results. As an example, in the context of bicomplex-valued b-metric space, we develop fixed point and common fixed point results for the rational contraction mapping.

SPECTRAL APPROXIMATION METHODS FOR FREDHOLM INTEGRAL EQUATIONS WITH NON-SMOOTH KERNELS

In this paper, polynomially based projection and modified projection methods for approximating the solution of Fredholm integral equations with a kernel of Green’s function type are studied. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and those of superconvergence of the iterated versions are analysed. A numerical example is given to illustrate the theoretical results.

Fixed point theorems for F- contraction mapping in complete rectangular M-metric space

In this paper, we prove a fixed point result for F- contraction principle in the framework of rectangular M-metric space. An example is also adopted to exhibit the utility of our result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equation.

The best approximate solution of Fredholm integral equations of the first kind via Gaussian process regression

In this paper, the best approximate solution of Fredholm integral equations of the first kind with some scattered observations is studied. An explicit approximate solution has been obtained by our proposed method and converges to the exact solution with minimum norm of the integral equations with probability 1, which is identical with the solution by means of the regularization method. In addition, based on the H-Hk formulation, the gap between the range space and its closure is fully characterized, in which the infinite reproducing kernel Hilbert spaces are strictly embedded. We prove that if the right-hand term of the integral equation lies in this gap, the H-norm of the best approximate solution will diverge to infinity, as the number of observations increases. Finally, an integral equation confirms our conclusions.

Legendre–Kantorovich method for Fredholm integral equations of the second kind

In the present paper, we consider polynomially based Kantorovich method for the numerical solution of Fredholm integral equation of the second kind with a smooth kernel. The used projection is either the orthogonal projection or an interpolatory projection using Legendre polynomial bases. The order of convergence of the proposed method and those of superconvergence of the iterated versions are established. We show that these orders of convergence are valid in the corresponding discrete methods obtained by replacing the integration by a quadrature rule. Numerical examples are given to illustrate the theoretical estimates.

Contradictions in the Plane Contact Problem of the Theory of Elasticity on the Compression of Cylinders in Contact with Parallel Generators

The inapplicability of the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic and physically linear material has been proved due to a well-known logarithmic feature in the plane classical problem of mechanics of elastic solids. In the special case of the elastic interaction of a cylinder with a half-plane, when one of the radii has an infinite length, it has been found that the convergence also becomes equal to infinity. This paradoxical result contradicts not only the physical and mechanical meaning of the process under study, but also confirms the inadequacy of Flamant model of a simple radial stress state in determining displacements. The authors have proposed an algorithm for eliminating the contradictions based on the solution of Fredholm integral equation of the first kind. In the future, it can be considered as a new fundamental and applied problem of the theory of elasticity, which is of a great importance for a refined assessment of the contact strength and stiffness of the cylindrical parts of load-bearing structures taking into account the general and local deformations (cylindrical rollers, gears, road surfaces, when they are compacted with steel rollers, etc.) on the basis of a flat Flamant calculation scheme considering three stress components and the width of the cylinder contact area previously developed and mathematically approximated by the authors.

Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given.

A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type Ω -contractions in the setting of extended b -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

On the Solution of Fredholm Integral Equations of the First Kind

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

Numerical solution of Fredholm integral equations of the first kind with singular logarithmic kernel and singular unknown function via monic Chebyshev polynomials

This paper provides a new method for the numerical solution of Fredholm integral equation of the first kind whose unknown function is singular at the end-points of the integration domain and has a ...

A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

Iterative Method for the Solution of Fredholm Integral Equations of the 2nd kinds via Matrices

Iterative Method for the Solution of Fredholm Integral Equations of the 2nd kinds via Matrices

A mixed collocation scheme for solving second kind Fredholm integral equations in [-1,1

In this paper we propose a suitable combination of two collocation methods based on the zeros of Jacobi polynomials in order to approximate the solution of Fredholm integral equations on [−1,1]. One of the main interesting aspects of this procedure is that our approach is cheaper than the usual collocation method based on standard Lagrange interpolation using Jacobi zeros. Moreover, we can successfully manage functions with algebraic singularities at the endpoints. The error of the method is comparable with the error of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method is proved, and some numerical tests, which confirm the theoretical estimates, are provided.