On the solution of a certain class of nonlinear singular integro-differential equations with Carleman shift

The solution of a class of nonlinear singular integro-differential equations with Carleman shift is obtained in the space H_μ⁽ⁿ⁾(Γ). The approach followed depends essentially on solution of linear singular integro-differential equation with shift. A criterion for the Noetherity of a correspondence singular integral functional operator of second order with Carleman shift preserving orientation is obtained and the index formula is given.     

THE DIFFERENTIAL INTEGRAL EQUATIONS ON SMOOTH CLOSED ORIENTABLE MANIFOLDS

1 IntroductionSillce tl1e limit value fOrlnula, viz. tl1e Plemelj fOrn1ula, of the Cauthe type integraJ withBochner-Martinelli kernel was proved in 1957[1], it has beell successfully used to the study Ofsingular i1ltegral equatious, solvi11g the 0b--equation, holomorphic extension, 0--closed exten-sion and C-R 111al1ifolds[2-51. Evideutly, the researcl1 of higher order singular integrals withBochuer-Martinelli kerllel itself also l1as important significallce. In 1952, J. Hadanmrd firstde…     

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。     

Approximate solution of one singular integro-differential equation

In this paper we construct and theoretically justify a computational scheme for solving the Cauchy problem for a singular integro-differential equation of the first-order, where the integral over a segment of the real axis is understood in the sense of the Cauchy principal value. In addition, we study and solve approximately the integral equation with a special logarithmic kernel. We obtain uniform estimates for errors of approximate formulas. Orders of errors of approximate solutions are proved to be proportional to the order of the approximation error for the derivative of the density of the singular integral in the integro-differential equation.     

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.      

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

一类带弱奇异核偏积分微分方程二阶差分空间半离散格式的全局稳定性

本文给出了数值求解一类带弱奇异核偏积分微分方程的二阶差分空间半离散格式;借助于Laplace变换及Parseval等式,得到了全局稳定性的证明.     

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.     

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.     

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

Singularly perturbed higher order periodic boundary value problems

The existence of positive periodic solutions is proved for the higher order singular differential equation

(0.1)     

Banach空间非线性二阶脉冲奇异积分-微分方程积分边值问题的正解

The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory.Moreover,an application is also given to illustrate the main result.     

Banach空间混合型一阶奇异脉冲微分-积分方程初值问题的正解

通过构造一个特殊的闭凸集,利用著名的Mnch不动点定理,在Banach空间中获得了一类奇异脉冲微分-积分方程正解的存在性,所使用的方法本质上不同于已有文献.     

抽象空间中一类带奇异性的混合型积分-微分方程边值问题的正解

利用不动点指数理论,研究了Banach空间中一类带奇性的混合型积分——微分方程边值问题正解的存在性,得到了多个正解存在的充分条件,并给出了相应的例子以说明所得结果的合理性和应用性.     

Exponential stability of nonlinear impulsive neutral integro-differential equations

In this paper, a nonlinear impulsive neutral integro-differential equation with time-varying delays is considered. By establishing a singular impulsive delay integro-differential inequality and transforming the n$n$-dimensional impulsive neutral integro-differential equation to a 2n$2n$-dimensional singular impulsive delay integro-differential equation, some sufficient conditions ensuring the global exponential stability in PC¹$P{C}^1$ of the zero solution of an impulsive neutral integro-differential equation are obtained. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.     

THE NUMERICAL SOLUTION FOR A PARTIAL INTEGRO-DIFFERENTIAL EQUATION WITH A WEAKLY SINGULAR KERNEL

In this paper, a first order semi-discrete method of a partial integro-differential equation with a weakly singular kernel is considered. We apply Galerkin spectral method in one direction, and the inversion technique for the Laplace transform in another direction, the result of the numerical experiment proves the accuracy of this method.     

A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel

In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L₁ discrete formula and the Riemann–Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis.     

Multiple Positive Solutions for First Order Impulsive Singular Integro-Differential Equations on the Half Line

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.