An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue

Accurate integral methods are applied to a one dimensional moving boundary problem describing the diffusion of oxygen in absorbing tissue. These methods have been well studied for classic Stefan problems but this situation is unusual because there is no condition which contains the velocity of the moving boundary explicitly. This paper begins by giving a short time solution and then discusses some of the previous integral methods found in the literature. The main drawbacks of these solutions are that they cannot be solved from $t=0$ and also cannot determine the end behaviour. This is due to the non-uniform initial profile which integral methods typically fail to capture. The use of a novel transformation removes this non-uniformity and, on applying optimal integral methods to the resulting system, leads to simple and yet very accurate approximate solutions that overcome the deficiencies of previous methods.     

第二积分中值定理“中间点”渐近性的完善

通过定义第二积分中值函数,用统一的方法继续探讨了第二积分中值定理“中间点”的一些渐近性质,得出一系列新结论,相信在积分学中有着很重要的作用.     

Estimation of the upper and lower bounds of solutions of the integral equations of viscoelasticity

The different methods of solving problems of viscoelasticity for hereditary media that employ Laplace transforms or exact solutions of integral equations for inaccurately approximated kernels inevitably introduce errors associated with the approximation and the inverse transformations. Accordingly, it is necessary to estimate the accuracy of these methods. It is shown that the kernels of the integral equations of viscoelasticity permit the estimation, with a certain accuracy, of upper and lower bounds directly for the solutions of these integral equations. In cases when the accuracy of the estimate is sufficient, there is no need to employ other methods of solution.Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 976–985, November–December, 1968.     

波动方程初值问题的解在两类积分中的应用

利用波动方程初值问题解的特点给出了圆域上一类反常二重积分和球面上第一类曲面积分的微分算子级数公式解和定积分公式解.通过举例说明了该方法相对于常规解法的简便实用性.     

关于亚音速绕流问题的强奇性基本解法

本文对求解亚音速流的偶极子基本解法作了新的处理,导出了一个关于求解偶极子强度的强奇性积分方程,给出了强奇性积分有效主值的定义及计算公式.由此可以导出多种整体连续分布的数值基本解法.适用于亚音速气动力计算.     

On the solution of a class of integral equations II

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

An algorithm based on the regularization and integral mean value methods for the Fredholm integral equations of the first kind

In this paper, an algorithm based on the regularization and integral mean value methods, to handle the ill-posed multi-dimensional Fredholm equations, is introduced. The application of this algorithm is based on the transforming the first kind equation to a second kind equation by the regularization method. Then, by converting the first kind to a second kind, the integral mean value method is employed to handle the resulting Fredholm integral equations of the second kind. The efficiency of the approach will be shown by applying the procedure on some examples.     

Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

Applying the combined integral method to one-dimensional ablation

In this paper the combined integral method is applied to a simple one-dimensional ablation problem. One of the drawbacks of heat balance integral methods is how to choose the approximating function. It is common to use a polynomial form but even then it is not clear what the power of the highest order term should be. Previous studies have determined exponents either from exact solutions or from expansions valid over short time scales; neither approach is satisfactory nor very accurate for larger times. We combine the heat balance and refined integral methods to determine this exponent as part of the solution process, and conclude that it is in fact time-dependent in the ablation stage. From comparing the approximate solutions with numerical and exact analytical solutions whenever possible, we show that this new method greatly improves the accuracy on standard methods, without overcomplicating the method.     

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.     

Convergence of high-order deterministic particle methods for the convection-diffusion equation

A proof of high-order convergence of three deterministic particle methods for the convection-diffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kernel that defines the integral operator include moment conditions and a condition on the kernel's Fourier transform. Explicit formulae for kernels that satisfy these conditions to arbitrary order are presented. © 1997 John Wiley & Sons, Inc.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.     

HIGH ORDER NUMERICAL METHODS TO TWO DIMENSIONAL HEAVISIDE FUNCTION INTEGRALS

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function integral is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the second- to fourth-order methods implemented in this paper achieve or exceed the expected accuracy.     

Numerical solution of the high thermal loss problem presented by a fractional differential equation

Three different numerical methods are used to solve singularly perturbed Able Volterra integral equation as presented by a fractional differential equation. Convergence and stability analysis together with the results of these methods are compared and contrasted when applied to the high thermal loss problem as an example of singularly perturbed Able Volterra integral equation.     

Linear functional equations of the first, second, and third kind in L 2

Under consideration are the functional equations of the first, second, and third kind with operators in wide classes of linear continuous operators in L ₂ containing all integral operators. We propose methods for reducing these equations by linear invertible changes either to linear integral equations of the first kind with nuclear operators or to equivalent linear integral equations of the second kind with quasidegenerate Carleman kernels. Some various approximate methods of solution are applicable to the so-obtained integral equations.     

L~1空间中直接离散法求解弱奇异积分方程特征值的探讨

在L~1空间中讨论弱奇异积分方程的特征值问题.利用弱奇异积分算子为紧算子,便可以对其直接进行离散化算法求出特征值.并将直接离散的方法与以往的迭代后离散的方法用实例通过Matlab作图进行对比,说明直接进行离散的方法更佳.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

A remedy for the failure of the numerical steepest descent method on a class of oscillatory integrals

In this paper we demonstrate that the numerical method of steepest descent fails when applied in a straight forward fashion to the most commonly occurring highly oscillatory integrals in scattering theory. Through a polar change of variables, however, the integral can be reformulated so that it can be solved efficiently using a combination of oscillatory integration techniques and classical quadrature. The approach is described in detail and demonstrated numerically with some oscillatory integral examples. The numerical examples demonstrate that our approach avoids the failure in some special cases, such as in an acoustic scattering model oscillatory integral with observation point located in the illuminated region. This paves the way for using the framework of numerical steepest descent methods on a wider class of problems, like the 3D high frequency scattering from convex obstacles, up to now only handled in a satisfactory way by methods due to Ganesh and Hawkins (J Comp Phys 230:104–125, 2011).