The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

Previous article Next article On Double Chebyshev Series ApproximationN. K. BasuN. K. Basuhttps://doi.org/10.1137/0710045PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractMany of the standard practical techniques for Chebyshev polynomial approximation in one-variable problems have been extended to two-variable problems.[1] Ferdinand Freudenstein, Bi-variate, rectangular, optimum-interval interpolation, Math. Comp., 15 (1961), 288–291 MR0129532 0168.14903 CrossrefGoogle Scholar[2] C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comput., 9 (1955), 118–120 MR0071856 0065.05403 CrossrefGoogle Scholar[3] David Elliott, Error analysis of an algorithm for summing certain finite series, J. Austral. Math. Soc., 8 (1968), 213–221 MR0230452 0159.45104 CrossrefGoogle Scholar[4] David Elliott, The numerical solution of integral equations using Chebyshev polynomials, J. Austral. Math. Soc., 1 (1959/1960), 344–356 MR0128087 0096.10201 CrossrefGoogle Scholar[5] L. Fox and , E. T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. Roy. Soc. London. Ser. A., 245 (1953), 501–534 MR0054355 0050.12902 CrossrefISIGoogle Scholar[6] Google Scholar[7] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205 MR0228149 0153.17502 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulusPhysics of Fluids, Vol. 34, No. 8 | 1 Aug 2022 Cross Ref Study of the free surface Green function at zero forward speed in finite water depth using Newman’s methodApplied Ocean Research, Vol. 120 | 1 Mar 2022 Cross Ref Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problemApplied Ocean Research, Vol. 81 | 1 Dec 2018 Cross Ref Nondimensionalized Univariate Equation Characterizing Optimal State Feedback Control for Collision AvoidanceIEEE Transactions on Intelligent Transportation Systems, Vol. 19, No. 10 | 1 Oct 2018 Cross Ref The automatic solution of partial differential equations using a global spectral methodJournal of Computational Physics, Vol. 299 | 1 Oct 2015 Cross Ref Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flowJournal of Differential Equations, Vol. 257, No. 8 | 1 Oct 2014 Cross Ref A new kind of double Chebyshev polynomial approximation on unbounded domainsBoundary Value Problems, Vol. 2013, No. 1 | 22 January 2013 Cross Ref On Approximate Solutions of Second-Order Linear Partial Differential EquationsApplied Mathematics, Vol. 03, No. 09 | 1 Jan 2012 Cross Ref A fitting formula for the falloff curves of unimolecular reactions, II: Tunneling effectsInternational Journal of Chemical Kinetics, Vol. 43, No. 1 | 19 October 2010 Cross Ref Direct numerical simulation of preferential particle concentration in decaying turbulence under the influence of magnetic fieldInternational Journal for Numerical Methods in Fluids, Vol. 419 | 1 Jan 2009 Cross Ref Large and small signal distortion analysis using modified Volterra seriesAnalog Integrated Circuits and Signal Processing, Vol. 54, No. 2 | 29 November 2007 Cross Ref Volterra Analysis Using Chebyshev Series2007 IEEE International Symposium on Circuits and Systems | 1 May 2007 Cross Ref Large and Small Signal Distortion Analysis Using Modified Volterra Series2006 NORCHIP | 1 Nov 2006 Cross Ref An efficient numerical approximation for the linear class of mixed integral equationsApplied Mathematics and Computation, Vol. 167, No. 2 | 1 Aug 2005 Cross Ref Chebyshev polynomial solutions of second-order linear partial differential equationsApplied Mathematics and Computation, Vol. 134, No. 1 | 1 Jan 2003 Cross Ref Chebyshev series solution of linear Fredholm integrodifferential equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 29, No. 4 | 1 Jul 1998 Cross Ref Fast polynomial multiplication and convolutions related to the discrete cosine transformLinear Algebra and its Applications, Vol. 252, No. 1-3 | 1 Feb 1997 Cross Ref Chebyshev series solutions of Fredholm integral equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 27, No. 5 | 9 July 2006 Cross Ref An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transformsJournal of Computational and Applied Mathematics, Vol. 8, No. 4 | 1 Dec 1982 Cross Ref Optimum acoustic design of free-running low speed propellersAircraft Systems and Technology Meeting | 16 August 1977 Cross Ref Clenshaw sums in several variablesConstructive Theory of Functions of Several Variables | 2 October 2006 Cross Ref Volume 10, Issue 3| 1973SIAM Journal on Numerical Analysis History Submitted:05 March 1971Published online:14 July 2006 InformationCopyright © 1973 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0710045Article page range:pp. 496-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

This book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

Purpose. Testing of numerical solution algorithm for integral equation for calculation of plane meridian magnetostatic field source distribution at interfaces of piecewise homogeneous magnetized medium by means of electrostatic analogy. Methodology. The piecewise homogeneous medium consists of three regions with different magnetic permeabilities: the shell of arbitrary meridian section, external unlimited medium outside the shell, and the medium inside the shell. For testing external homogeneous magnetic field effect on spherical shell is considered. The analytical solution of this problem on the basis of electrostatic analogy from the solution of the problem uniform electrostatic field effect on dielectric shell is obtained. We have compared results of numerical solution of integral equation with the data obtained by means of analytical solution at the variation of magnetic permeabilities of regions of medium. Results. Integral equation and the algorithm of its numerical solution for calculation of source field distribution at the boundaries of piecewise homogeneous medium is validated. Testing of integral equations correctness for calculation of fictitious magnetic charges distribution on axisymmetric boundaries of piecewise homogeneous magnetized medium and algorithms of their numerical solutions can be carried out by means of analytical solutions of problems of homogeneous electrostatic field effect analysis on piecewise homogeneous dielectric medium with central symmetry of boundaries – single-layer and multilayer spherical shells. In the case of spherical shell in wide range of values of the parameter λk, including close to ± 1, numerical solution of integral equation is stable, and relative error in calculating of fictitious magnetic charges surface density and magnetic field intensity inside the shell is from tenths of a percent up to several percent except for the cases of very small values of these quantities. Originality. The use analytical solutions for problems of calculation of external electrostatic field effect on piecewise homogeneous dielectric bodies for testing integral equations of magnetostatics and algorithms for their numerical solutions. Practical value. The described method of testing integral equations of magnetostatics and their numerical solutions can be used for calculation of magnetic fields of spacecraft control system electromagnets.

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

This study presents an extension of the piecewise quadratic polynomial technique to solve singular integral equations with logarithmic- and Hadamard-type singularities. For completeness and continuity, the evaluation of the weights for logarithmic-, Cauchy- and Hadamard-type singularities are given explicitly. Numerical results concerning logarithmic- and Hadamard-type singular integral equations are compared with the known solutions. The solution of Cauchy-type singular integral equations was discussed in detail in a previous study by Miller and Keer. © 1998 John Wiley & Sons, Ltd.

Qualitative Problems in Matrix Theory

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

In a recent paper (L. Greengard and V. Rokhlin, On the Numerical Solution of Two‐Point Boundary Value Problems. in Communications on Pure and Applied Mathematics, Volume XLIV, 1991, pages 419‐452). L. Greengard and V. Rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two‐point boundary value problems for second‐order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high‐order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N.p2.n3) operations with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to endpoint singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.We in addition present a Newton method for solving boundary value problems for nonlinear first‐order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. © 1994 John Wiley & Sons. Inc.

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

The work is devoted to the study and solution of one class of nonlinear integral equations with monotone and noncompact Hammerstein-type operators on whole line. Equations of this kind arise in many branches of natural science. In particular, such equations are applied in the dynamic theory of [Formula: see text]-adic open–closed strings, in the kinetic theory of gases, in the mathematical theory of the spatial-temporal spread of pandemic. A constructive theorem for existence of non-negative nontrivial and essentially bounded solution is proved. The asymptotic behavior of constructed solution at [Formula: see text] is studied. It is also proved that the difference between the limit in [Formula: see text] and the solution represents summable function on [Formula: see text] Using results obtained a non-negative integrable and bounded solution for new class of integral equations on whole line with monotone and sign-alternating nonlinearity is constructed. Under additional restrictions on nonlinearity, a uniqueness theorem for a solution in a certain class of continuous and bounded functions is also proved. At the end of the work, the specific examples of above equations, satisfying all conditions of the formulated theorem are given.

The purpose of this paper is to present numerical methods for the computation of samples of a Gaussian random process x(t) with prescribed autocorrelation or power spectral density.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.