Approximated Solution of a Nonlinear Singular Equation of Prandtl's Type

A nonlinear strongly singular integral equation, which can be reduced to a nonlinear singular integro-differential equation of Prandtl's type, is considered. A collocation method for solution is treated and the convergence of the approximated solution to the unique solution of the nonlinear integral equation is proved.     

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.     

Uniform convergence of the rectangle method for singular integral equation with the Hölder density

We study an approximate method for solving singular integral equations. It implies an approximation of a singular operator by means of a compound quadrature formula similar to the rectangle one. The corresponding systems of linear algebraic equations are solvable if so is the integral equation, while its coefficients satisfy the strong ellipticity condition. Under these restrictions we obtain a bound for the rate of convergence of solutions of systems of linear equations to the solution of the considered integral equation in the uniform vector norm.     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

Constructive Method for Solving a Boundary Value Problem with Impedance Boundary Condition for the Helmholtz Equation

We substantiate the collocation method for the singular integral equation of a boundary value problem with impedance condition for the Helmholtz equation. We construct a sequence converging to the exact solution of the original problem and estimate the error.     

A probabilistic-informational algorithm for approximate solution of singular integral equations. The method of reduction to Riemann-Hilbert problems

We propose a numerical-analytic method of solving singular integral equations with a singular kernel of Cauchy type on an interval. The method relies on the construction of a regular operator of special form, whose action on the original singular equation leads to an integral equation that admits an explicit solution. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 36–40.     

闭光滑流形上的高阶线性微-积分方程

利用积分变换技巧,作者给出了C~n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。     

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.     

Problem with Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics for the gellerstedt equation with a singular coefficient

We study the well-posedness of a problem for the Gellerstedt equation with a singular coefficient and with the Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics.The uniqueness of the solution of the considered problem is proved with the use of the extremum principle, and the existence of the solution of the problem is justified with the use of the theories of singular integral equations, Wiener-Hopf equations, and Fredholm integral equations.     

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.     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Shape derivative of the volume integral operator in electromagnetic scattering by homogeneous bodies

We study the shape derivative of the strongly singular volume integral operator that describes time‐harmonic electromagnetic scattering from homogeneous medium. We show the existence and a representation of the derivative, and we deduce a characterization of the shape derivative of the solution to the diffraction problem as a solution to a volume integral equation of the second kind.     

A nonlocal problem for degenerate hyperbolic equation

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.     

Approximate solution of a complete singular integral equation of the first kind with a fixed hypersingularity by the overlapping method

We suggest a new method for the numerical solution of a singular integral equation of the first kind with a fixed hypersingularity, which arises in the problem on the flow past a profile with an ejector of the external flow. This method permits one to obtain a solution of the characteristic and complete integral equations with an interpolation degree of accuracy.     

On the solvability of a singular integral equation with a non-Carleman shift

We consider a singular integral equation with a non-Carleman shift on an interval. We prove the unique solvability of this equation in weighted Hölder classes under certain restrictions on the coefficients. We show that the solution of the equation can be written in quadratures.     

Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel

We construct an approximate solution of the singular integral equation with the Cauchy kernel with the use of Faber polynomials, prove its convergence to the exact solution, and indicate the order of the convergence rate.     

An analysis of a method for solving singular integral equations

We establish convergence rates for a method of approximate solution of certain singular integral equations. The method considered involves an expansion of the kernel of the equation in terms of Chebyshev polynomials.     

Algorithm for the classic dual cosine equation

An algorithm for numerically solving the classic dual cosine equation is obtained from an emended version of a singular integral solution due to Noble and Whiteman. A preliminary Fourier analysis is made of the data, and computational rules are derived by the systematic reduction of the singular integrals for each ordinary Fourier component of the data. Extensive numerical testing provides evidence for the correctness of the modified singular integral solution and the algorithm, whose implementation in ANSI FORTRAN is appended.     

两参数非线性反应扩散积分微分方程的内层激波渐近解

研究了一类两参数非线性反应扩散积分微分奇摄动问题.利用奇摄动方法,构造了问题的外部解、内部激波层、边界层及初始层校正项,由此得到了问题解的形式渐近展开式.最后利用积分微分方程的比较定理证明了该问题解的渐近展开式的一致有效性.