Solvability of Chandrasekhar’s Quadratic Integral Equations in Banach Algebra

In this paper, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contain various integral and functional equations that are considered in nonlinear analysis. Our considerations will be discussed in Banach algebra using a fixed point theorem instead of using the technique of measure of noncompactness. An important special case of that functional equation is Chandrasekhar’s integral equation which appears in radiative transfer, neutron transport and the kinetic theory of gases [1].

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].

Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

Existence of Solutions to Nonlinear Impulsive Volterra Integral Equations in Banach Spaces

In this paper, the existence of solutions is studied for nonlinear impulsive Volterra integral equations with infinite moments of impulse effect on the half line R^+ in Banach spaces.By the use of a new comparison result and recurrence method, the new existence theorems are achieved under a weaker compactness-type condition, which generalize and improve the related results for this class of equations with finite moments of impulse effect on finite interval and infinite moments of impulse effect on infinite interval.

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

The moment-method form of Pocklington's integral equation above ground

Pocklington’s integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expressed as the elementary functions using the Fresnel plane wave reflection coefficients method; and the Pocklington’s integral equation will be rearranged into a linear equation with solution easily obtained by using the method of moments, when the sinusoidal sub domain expansion is chosen to express the current distributions.

L_（loc）~p-solutions of Nonlinear Impulsive Volterra Integral Equation in Abstract Space

In this paper,we discuss Llocp-solutions of a kind of nonlinear impulsive Volterra integral equation and present an existence theorem of solutions in Banach space.

Numerical Solution of Generalized Abel’s Integral Equation by Variational Iteration Method

In this paper, a user friendly algorithm based on the variational iteration method (VIM) is proposed to solve singular integral equations with generalized Abel’s kernel. It is observed that an approximate solutions yn(x) converges to the exact solution irrespective of the initial choice y0 (x). Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the method in solving these types of singular integral equations.

S-asymptotically ω-periodic Solutions of R-L Fractional Derivative-Integral Equation

The aim of this paper is to study the S-asymptotically ω-periodic solutions of R-L fractional derivative-integral equation:v′(t)=∫t0(t-s)α-2/Γ(α-1)Av(s)ds+∫+∞-∞e-|τ|f(u(t-τ))dτ,(1)v(0)=u0∈X,(2)where 1 <α <2, A:D(A)X→X is a linear densely defined operator of sectorial type on a completed Banach space X, f is a continuous function satisfying a suitable Lipschitz type condition. We will use the contraction mapping theory to prove problem(1) and(2) has a unique S-asymptoticallyω-periodic solution if the function f satisfies Lipshcitz condition.

Controllability of Semilinear Integrodifferential Degenerate Sobolev Equations

In this paper,the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is investigated in the sense of integral solution in Hilbert spaces.Some sufficient and necessary conditions are obtained.Firstly,the existence and uniqueness of integral solutions of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions are considered by GE-evolution operator theory and Sadovskii’s fixed point theorem,the existence and uniqueness theorem of solutions is given.Secondly,the approximate controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions is studied in the sense of integral solution.The criterion for approximate controllability is provided.The obtained results have important theoretical and practical value for the study of controllability of semilinear integrodifferential degenerate Sobolev equations with nonlocal conditions.

The Mixture of New Integral Transform and Homotopy Perturbation Method for Solving Discontinued Problems Arising in Nanotechnology

In this paper, a reliable algorithm based on mixture of new integral transform and homotopy perturbation method is proposed to solve a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. The technique finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. Comparison of the approximate solution with the exact one reveals that the method is very effective. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems.

A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation

In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

Elastoplastic Large Deformation Using Meshless Integral Method

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust.

一类具扰动Chandrasekhar H-方程的可积正解

本文讨论了迁移理论和气体动力学中一类具扰动的 Ｃｈａｎｄｒａｓｅｋｈａｒ Ｈ－方程：这里Ｇ是Ｒ～ｎ中的有界闭集，为已知函数，λ为常数。本文研究并得到了方程（０．６）可积正解的存在性、个数、逼近性、稳定性，改进和发展了〔４〕［５］［７］的结果。

一类具扰动的Chandrasekhar H-方程的可积极小正解

讨论了迁移理论和气体动力学中一类具扰动的ＣｈａｎｄｒａｓｅｋｈａｒＨ方程：Ｈ（ｘ）＝φ（ｘ）＋λＨ（ｘ）∫ＧＫ（ｘ）Ｋ（ｘ）＋Ｋ（ｔ）Ｈ（ｔ）ｄｔ＋∫ＧＰ（ｘ，ｔ，Ｈ（ｘ），Ｈ（ｔ））ｄｔλ＞０式中，Ｇ是Ｒｎ中的紧集；φ（ｘ），Ｋ（ｘ），Ｐ（ｘ，ｔ，ｕ，ｖ）为已知函数；λ为常数．并得到了上述方程可积极小正解的存在性、逼近性、稳定性．

LESSER KNOWN MIRACLES OF BURGERS EQUATION

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.

Noether's and Poisson's methods for solving differential equation x_s^((m))=F_s(t,x_k^((m-2)) ,x_k^((m-1)))

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.

基于N-S方程的尾迹面法翼型气动阻力计算

研究了计算翼型阻力改进的方法——尾迹面法及尾迹面积分的位置和相应的积分技术．在亚跨声速时分别采用了表面积分和尾迹积分求得给定翼型RAE2822的阻力．结果显示，2种方法具有相同的结果，表明尾迹面积分方法是有效的，与实验值吻合较好．尾迹面法中，尾迹面位置应处于离后缘相对弦长距离0．6-1．0之间．尾迹面积分方法中积分结果不依赖于物体的详细几何外形，可以预计对曲率变化大的三维复杂外形，该方法有更大的优势．