Product Integration Method Research Articles

A Comparative Survey of Numerical Methods for the Linear Generalized Abel Integral Equation

AbstractA numerical comparison is made between a number of important representatives of the following classes of methods: (i) collocation, (ii) product integration and (iii) global methods. Special attention is paid to the performance of these methods for problems with a non‐smooth solution. It turned out that, when only relatively low accuracy is required, a good choice would be a second or third order collocation method of Branca. — A new collocation method which accounts for possible non‐smoothness of the solution near the origin, turned out to be advantageous when high accuracy is required, both for problems with a non‐smooth solution, and for problems with a smooth solution.

A family of high order product integration methods for an integral equation of lighthill

Lighthill has derived a nonlinear singular Volterra integral equation to describe the temperature distribution of the surface of a projectile moving through a laminar boundary layer at high Mach numbers. This paper presents high order product integration methods for its numerical solution and analyses their convergence. Numerical results are given.

The Analysis of Product Integration Methods for Abel's Equation using Discrete Fractional Differentiation

On analyse la convergence de methodes d'integration produit pour l'equation integrale d'Abel generalisee de premiere espece. On introduit une analogue discrete a la differentiation fractionnaire

Product integration and solution of ordinary differential equations

by the method of product integration. We will prove an existence and uniqueness theorem and related results for solutions of (0.2) when A is a (time dependent, possibly non-linear) vector field on a Banach space, satisfying a Lipschitz condition in the “space” variable and an integrability condition in the “time” variable. These results are, of course, not new, and can be proved in the usual way by iteration of Eq. (0.2); however, the use of product integration in the generality presented here will, it is hoped, be of sufficient novelty to justify the present work. To give an idea of our method, we first recall (see [ 1 ] for a discussion of the finite-dimensional case) that if the vector field A satisfies appropriate continuity conditions in the time variable, then the solution x(t) of (0.1) can be obtained by associating with partitions

Product integration methods for second-kind Abel integral equations

The construction and convergence of high-order product integration methods for the second-kind Abel equation are discussed and the results of De Hoog and Weiss are generalised. Backward difference methods are introduced, and numerical results are presented which verify the theoretical rates of convergence.

High‐accuracy product integration methods for the Abel equation, and their application to a problem in scattering theory

AbstractIn this paper we present orsder three, four and five backward product integration methods for the generalized Abel equation equation image We illustrate their use by silving the small angle deflection problem, and show computationally that these methods are convergent for all ϵ ϵ(0,1). The coefficients of order three and four methods are given in Applenidx I.

Product Integration Methods for the Nonlinear Basset Equation

Product integration methods are derived through backward difference interpolation for the solution of a nonlinear generalisation of the Basset equation; this equation models the motion of a particle in a turbulent fluid at reasonably high Reynolds numbers. It is proved that the product integration algorithm using $n + 1$ points with ackward difference interpolation is convergent of order $n + 1$ if $n \leqq 6$. Simple numerical results are given. Numerical stability is discussed and a diagram of stability is exhibited.

Two species diffusion in a sphere with inter-species linkage at the boundary

The concentration of carbon in stainless steel has been shown to satisfy a singular Volterra integral equation by Skyrme & Norbury[1]. This note is concerned with a simple product integration method for solving this equation. The method is described and numerical results are presented in graphical form. A concise convergence analysis making use of a generalisation of the usual discrete Gronwall's lemma is presented. Finally it is pointed out how a continuous form of the lemma can be used to demonstrate uniqueness.

Numerical evaluation of integrals containing a spherical Bessel function by product integration

A method is developed for numerical evaluation of integrals with k-integration range from 0 to ∞ that contain a spherical Bessel function jl(kr) explicitly. The required quadrature weights are easily calculated and the rate of convergence is rapid—only a relatively small number of quadrature points is needed—for an accurate evaluation even when r is large. The quadrature rule is obtained by the method of product integration. With the abscissas chosen to be those of Clenshaw–Curtis and the Chebyshev polynomials as the interpolating polynomials, quadrature weights are obtained that depend on the spherical Bessel function. An inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error. The procedure is summarized as an easily implementable algorithm. Questions of convergence are discussed and the rate of convergence demonstrated for several test integrals. Alternative procedures are given for generating the integration weights and an error analysis of the method is presented.

Analysis of general quadrature methods for integral equations of the second kind

This paper is concerned with a class of approximation methods for integral equations of the form $$y(t) = f(t) + \int\limits_a^b {k(t,s)y(s)ds} $$ , wherea andb are finite,f andy are continuous and the kernelk may be weakly singular. The methods are characterized by approximate equations of the form $$y_n (t) = f(t) + \sum\limits_{i = 1}^n {W_{ni} (t)y_n (s_{ni} )} $$ ; such methods include the Nystrom method and a variety of product-integration methods. A general convergence theory is developed for methods of this type. In suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule. The theory is used to deduce convergence results, some of them new, for a number of specific methods.

Product integration for weakly singular integral equations

The product integration method is used for the numerical solution of weakly singular integral equations of the second kind. These equations often have solutions which have derivative singularities at the endpoints of the range of integration. Therefore, the order of convergence results of de Hoog and Weiss for smooth solutions do not hold in general. In this paper it is shown that their results may be regained for the general case by using an appropriate nonuniform mesh. The spacing of the knot points is defined by the behavior of the solution at the endpoints. If the solution is smooth enough the mesh becomes uniform. Numerical examples are given.

Product Integration for Weakly Singular Integral Equations

The product integration method is used for the numerical solution of weakly singular integral equations of the second kind. These equations often have solutions which have derivative singularities at the endpoints of the range of integration. Therefore, the order of convergence results of de Hoog and Weiss for smooth solutions do not hold in general. In this paper it is shown that their results may be regained for the general case by using an appropriate nonuniform mesh. The spacing of the knot points is defined by the behavior of the solution at the endpoints. If the solution is smooth enough the mesh becomes uniform. Numerical examples are given.

Produktintegration mit nicht-�quidistanten St�tzstellen

For the numerical evaluation of $$\int\limits_a^b {(t - a)^{\alpha - 1} x(t)dt}$$ , 0<?<1 andx `smooth', product integration rules are applied. It is known that high-order rules, e.g. Gauss-Legendre quadrature, become `normal'-order rules in this case. In this paper it is shown that the high order is preserved by a nonequidistant spacing. Furthermore, the leading error terms of this product integration method and numerical examples are given.

Product-integration with the Clenshaw-Curtis and related points

This paper is concerned with the theoretical properties of a productintegration method for the integral $$\int\limits_{ - 1}^1 {k(x)f(x)dx}$$ , wherek is absolutely integrable andf is continuous. The integral is approximated by $$\sum\limits_{i = 0}^n {w_{ni} f(x_{ni} )}$$ , where the points are given byx ni =cos(i?/n, 0?i?n, and where the weightsw ni are chosen to make the rule exact iff is any polynomial of degree ?n. The principal result is that ifk?L p [?1, 1] for somep>1, then the rule converges to the exact result asn?? for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely $$\int\limits_{ - 1}^1 {|k(x)|dx}$$ . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.

Product integration in reflexive Banach spaces

LetX denote a reflexive Banach space and {A(t)|t∈[0,T]} a time dependent family of accretive operators defined onX. Conditions are placed on {A(t)|t∈[0,T]} which are sufficient to guarantee the existence of solutions to the Cauchy initial value problem:u′(t,x)+A(t)u(t,x)=0; u(0,x)=x. These solutions are obtained via the method of product integration; however limits for the infinite product are taken with respect to the weak topology.

Abel type integral equations in stereology: I. General discussion

SUMMARYIn developing numerical methods for the solution of computational problems of an involved and pragmatic nature, as occur for example in stereology, the approach should be first to classify the distinct types of numerical problems which arise. If available, appropriate results from numerical analysis can then be used to develop stable numerical processes for their solution. In fact, many of the results relating to the construction of basic numerical methods, such as spectral methods for the stable differentiation of experimental data and product integration methods for the evaluation of integrals with oscillatory and singular integrands, have been resolved. Thus, we are now in a position to apply the above approach to the more complex computational problems which arise in stereology.Many of these problems belong to one of the following two classes: (i) the solution of integral equations of Abel type; and (ii) the solution of some numerical differentiation problem.In Part I of this paper, we illustrate the generality in stereology of the Abel type integral equation and numerical differentiation formulations. As well, we derive the specific properties required in Part II which is mainly concerned with the construction of stable computational methods for (i). Included among these properties are the explicit inversion formulae which are known for general Abel type integral equations. It is these inversion formulae which we use to construct stable computational methods.Often, when estimates for linear properties of the solutions of (i) and (ii) are required, the numerical solution of (i) and (ii) can be circumvented by estimating the linear properties directly from the given observational data. In deriving such estimates, use of the properties of the Abel type integral equation and differentiation formulations plays an essential role. Because of the close connection between such estimation problems and (i) and (ii), the estimation of linear properties from truncated observational data is also examined in Part I.

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.

Time Dependent Nonlinear Cauchy Problems in Banach Spaces

The method of product integration is used to obtain solutions to the nonlinear evolution equation $u’(t) + A(t)u(t) = 0$ where $\{ A(t):t \in [0,T]\}$ is a family of nonlinear accretive operators mapping a Banach space X to itself. The main requirements are that $R(I + \lambda A(t)) \supseteq {\text {cl}}(D(A(t))),D(A(t))$ is time independent, the resolvent ${(I + \lambda A(t))^{ - 1}}x$ satisfies a local Lipschitz condition, and that each A(t) satisfies Condition M.

Product integration methods for Volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be accomplished if the integral is replaced by certain simple quadrature rules, such as the midpoint or the trapezoidal methods. When the kernel of the integral equation oscillates more rapidly than the solution one can use product integration techniques to increase the accuracy. Such an approach is investigated in this paper.

A product integration method for a class of singular first kind Volterra equations

Convergence of a midpoint product integration method for singular first kind Volterra equations with kernels of the formk(t, s)(t?s) ?? , 0<?<1, wherek(t, s) is continuous, is examined. It is shown that convergence of order one holds if the solution of the Volterra equation has a Lipschitz continuous first derivative andk(t, s) is suitably smooth. In addition, convergence is shown to hold when the solution has only Lipschitz continuity and the same conditions onk(t, s) apply. An existence theorem of Kowalewski is used to relate these conditions on the solution to conditions on the data andk(t, s).