Logarithmic Kernels Research Articles

Superconvergent spectral projection and multi-projection methods for nonlinear Fredholm integral equations

In this article, we apply Galerkin and multi-Galerkin methods based on Kumar Sloan technique using Legendre polynomial basis functions for approximating the nonlinear Fredholm Hammerstein integral equations with the smooth kernels as well as the weakly singular algebraic and logarithmic kernels. We show that we are getting the improved superconvergence rates for Galerkin and multi-Galerkin methods based on Kumar Sloan technique without the need for the iterated Galerkin and the iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates equal to the convergence rates of iterated Galerkin and iterated multi-Galerkin methods. Theoretical results have been illustrated with numerical experiments.

Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo–Hadamard Differential Equations

In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L∞. In addition, we provide numerical results to support our theoretical analysis.

Extremal functions for the critical Trudinger–Moser inequality with logarithmic Kernels

In this paper we study Moser–Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we sharpen the results in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935] under critical growth assumptions and give answers to some questions left open in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935].

Singular asymptotic expansion and Legendre collocation method for two-term weakly singular Volterra integral equation of the second kind

This paper develops a singularity-separation Legendre collocation method for solving two-term linear Volterra integral equation with algebraic and logarithmic singular kernels. The psi-series solution at the origin is obtained by an efficient Picard iteration, which is very accurate near the origin. By splitting the interval into a singular subinterval and a regular one, the equation is converted to one with a sufficiently smooth solution on the regular interval, which is discretized by the Legendre collocation method. The efficient computational process is discussed in detail, and the convergence under the square norm is proved. Finally, three numerical examples, including a Caputo fractional differential equation and a Basset equation with initial conditions, are provided to illustrate the efficiency and accuracy of the proposed singularity-separation method.

On the Theory of Capacities on Locally Compact Spaces and its Interaction with the Theory of Balayage

The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede’s theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely new even for the logarithmic, Newtonian, Green, α-Riesz, and α-Green kernels.

Error analysis of reiterated projection methods for Hammerstein integral equations

In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing as space of piecewise polynomials subspace of degree , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders and , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders , for algebraic kernels and for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders for algebraic kernels and for logarithmic kernels. Numerical results are given for the justification of our theoretical results.

Pentagon functions for one-mass planar scattering amplitudes

We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive-particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.

Balayage of Measures with Respect to Polynomials and Logarithmic Kernels on the Complex Plane

Balayage of Measures with Respect to Polynomials and Logarithmic Kernels on the Complex Plane

On the Negativity of the Walsh–Kaczmarz–Riesz Logarithmic Kernels

The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

Approximation methods for system of linear Fredholm integral equations of second kind

In this paper, Galerkin, multi-Galerkin methods and their iterated versions are developed for solving the system of linear Fredholm integral equations of the second kind for both smooth and weakly singular algebraic and logarithmic type kernels. Here first we develop the theoretical framework for Galerkin and iterated Galerkin methods to solve the system of linear second kind Fredholm integral equations using piecewise polynomials as basis functions and then obtain the superconvergence results similar to that of single linear Fredholm integral equation of the second kind. We show that iterated Galerkin approximation yields better convergence rates than Galerkin approximate solution. Further we enhance these results by using multi-Galerkin and iterated-multi-Galerkin methods and show that the iterated multi-Galerkin approximation yields improved superconvergence rates over iterated Galerkin and multi-Galerkin approximations. The theoretical results are justified by the numerical results.

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

In this article, we apply projection methods and their iterated versions to approximate the solution of system of Fredholm–Hammerstein integral equations with both smooth and weakly singular kernels of algebraic and logarithmic type using the piecewise polynomial basis functions. We show that the iterated Galerkin approximate solution converges to the exact solution faster than the Galerkin approximate solution for both smooth and weakly singular algebraic and logarithmic type kernels. We improve these results further in iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.

A Numerical Method for Solving Fredholm Integral Equations of the First Kind with Logarithmic Kernels and Singular Unknown Functions

In this paper, we present a numerical method for solving singular Fredholm integral equations of the first kind. The method is based on the application of the shifted Chebyshev polynomials of the second kind using two techniques. By using the first technique, we solve singular Fredholm integral equations of the first kind with singular logarithmic kernels and singular unknown functions, while the second technique determines the associated data function for a specified exact solution. The unknown function is factorized into two functions; the first is badly-behaved and the second an unknown regular function. By expanding these two functions into shifted Chebyshev polynomials of the second kind through monomial basis functions, the unknown function’s singularity is completely removed. We develop an algebraic formula for evaluating the Chebyshev coefficients instead of using the Chebyshev integral formula. This minimizes the steps of the solution’s procedure and reduces the roundoff errors. The singularity of the kernel is treated by numerical integration. Two boundary integral equations related to the radar, electromagnetism, and scattering engineering problems are solved. The obtained results ensure the efficiency and the high accuracy of the presented method compared with other methods.

Numerical Solution of a Certain Class of Singular Fredholm Integral Equations of the First Kind via the Vandermonde Matrix

In this paper, a computational method is presented to solve potential-type Fredholm integral equations of the first kind, equations in which the unknown functions are singular at the endpoints of the integration domain, in addition to the weakly singular logarithmic kernels. This method provides a numerical solution based on the Newton interpolation technique via the Vandermonde matrix, which can accommodate an approximation of the unknown function, in such a manner that its singularity is easily removed, as well as the removal of kernel singularity. In addition, the Gauss–Legendre formula is adapted and applied for the computations of the obtained convergent integrals. Thus, the obtained numerical solution is equivalent to the solution of an algebraic equation in matrix form without applying the collocation method. The numerical solutions of the illustrated example are strongly converging to the exact solution for all values of 1 x  including the end-points 1 whereas the exact solution fails to find the functional values at these end-points; which ensures the powerful and high accuracy of the presented computational technique

Интегро-дифференциальные уравнения на замкнутом контуре с функцией Гаусса в ядре

Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.

Approximation methods for second kind weakly singular Volterra integral equations

Projection methods are applied to obtain the convergence rates for Volterra integral equations with weakly singular kernels. We consider Galerkin and multi Galerkin methods and their iterated versions to solve Volterra integral equations with weakly singular kernels, in the space of piecewise polynomials subspaces based on graded mesh. We will show that the iterated multi-Galerkin method improves over iterated Galerkin method. In fact, we show that iterated multi-Galerkin solution converges with the convergence rates O(n−3m) and O(n−3m(logn)2), for algebraic and logarithmic type kernels, respectively. We prove that iterated Galerkin method, for algebraic kernel, converges with the convergence rate O(n−2m) and for logarithmic type kernel converges with the convergence rate O(n−2mlogn), where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations. Theoretical results are justified by the numerical results.

Integral Equations of Curvilinear Convolution Type with Hypergeometric Function in a Kernel

We study the class of integral equations of first kind over the circumference in the complex plane. The kernels of the equations contain a Gaussian hypergeometric function and depend on the arguments ratio. This class includes such specific cases as the equations with power and logarithmic kernels. To set the Noetherian property of equations correctly the method of operator normalization with a non-closed image is applied. The space of the right-hand sides of equations is described as the space of fractional integrals of curvilinear convolution type. The solutions of equations in explicit form are obtained as a result of consequent solving characteristic singular equations with a Cauchy kernel and inversion of a curvilinear convolution operator by means of Laurent transform of the functions defined on the circumference.

Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm–Hammerstein integral equations

In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O ( n − r ) , whereas the iterated Legendre Galerkin method converges with the order O ( n − r − α + 1 2 ) for the algebraic kernel, and order O ( log n n − r − 1 2 ) for logarithmic kernel in both L 2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O ( ( 1 + c l o g n ) n − r − 2 α + 1 2 ) for the algebraic kernel, and order of convergence O ( ( log n ) 2 ( 1 + c l o g n ) n − r − 3 2 ) for logarithmic kernel in both L 2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results.

Application of dual-Chebyshev wavelets for the numerical solution of boundary integral equations with logarithmic singular kernels

In this paper, the discrete Galerkin method based on dual-Chebyshev wavelets has been presented to approximate the solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of a boundary value problem of Laplace's equation with linear Robin boundary conditions. The discrete Galerkin methods for solving logarithmic boundary integral equations with Chebyshev wavelets as a basis encounter difficulties for computing their singular integrals. To overcome this problem, we establish the dual-Chebyshev wavelets, such that they are orthonormal without any weight functions. This property adapts Chebyshev wavelets to discrete Galerkin method for solving logarithmic boundary integral equations. We obtain the error bound for the scheme and find that the convergence rate of the proposed method is of $$O(2^{-Mk})$$O(2-Mk). Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique and confirm the theoretical error analysis.

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.

A meshless local discrete collocation (MLDC) scheme for solving 2‐dimensional singular integral equations with logarithmic kernels

AbstractIn the current investigation, we study a numerical technique to solve 2‐dimensional singular integral equations of the second kind with logarithmic kernels. These integral equations are a general form of the problem of obtaining the cross‐sectional separation of current in an infinitely long thin filament of conduct bar which arises in electrical engineering. The moving least squares scheme estimates a function without mesh generation on the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation method in addition to the shape functions of moving least squares established on scattered points is used to approximate the solution of integral equations. To compute logarithm‐like singular integrals appeared in the process of the scheme, we use a particular dual nonuniform composite Gauss‐Legendre integration rule on normal domains. The proposed scheme does not require any meshes, so it is meshless and does not depend to the domain form. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.