Asymptotic analysis of a volterra integral equation

A uniformly valid aymptotic solution is obtained for a class of perturbed Volterra integral equations, in which a naive expansion breaks down as t → ∞. The procedure used is an adaption of the formal methodology presented in [1] for the construction of a uniform asymptotic solution to Volterra equations which possess a boundary layer near t = 0.     

Singular perturbation analysis of an integrodifferential equation modelling filament stretching

A formal asymptotic scheme is used to determine the leading order behavior of a certain singularly perturbed integrodifferential equation which models the process of stretching a polymer filament. The inner solution describes the process during a boundary layer in time which occurs at the onset of elastic recovery. The outer solution describes the long term behavior of the elastic recovery, and a bound on the ultimate recovery is determined.

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Zusammenfassung Ein formales asymptotisches Schema wird benutzt, um das Verhalten der Leitordnung einer gewissen singular gestörten Integro-Differentialgleichung, die als Modell für den Dehnungsprozeß einer Polymerfaser dient, zu bestimmen. Die innere Lösung beschreibt den Prozeß in einem zeitlichen Randgebiet, das am Anfang der elastischen Rückstellung auftritt. Die äußere Lösung beschreibt das Langzeitverhalten der elastischen Rückstellung, und eine Grenze für die endgültige Rückstellung wird bestimmt.

     

Solution of a certain multiple integral equation

An explicit solution is derived formally for a certain multiple integral equation involving a multidimensional fractional integral of Riemann-Liouville type. The main inversion theorem proved here provides a generalization of a result due to W. L. Wainwright [in “Fractional Calculus and Its Applications” (B. Ross, Ed.), pp. 298–305, Springer-Verlag, Berlin/Heidelberg/New York, 1975]. A simple illustration of the theorem, involving the classical Laguerre function, is also presented.     

On a nonlinear volterra equation

Nonnegative solutions u of the nonlinear Volterra equation u = a * g(u) (g(0) = 0) in mathematical physics are considered. Under certain assumptions the nonhomogenuous equation u = a * g(u) + ? is studied. Some approximations of nonnegative solutions of the homogenuous equation are considered by the nonnegative solutions of the nonhomogenuous one.     

伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

Singular integral equation with Cauchy kernel on a complicated contour

We present a reasonably comprehensive exposition of the theory of a singular integral equation with Cauchy kernel for the case in which the integration contour is a set of disjoint smooth open arcs. We construct numerical schemes for this equation and give an order estimate for the accuracy of the approximate solutions.     

Singular perturbation analysis of bistable differential equation arising in the nerve pulse propagation

This paper deals with the numerical analysis of time dependent parabolic partial differential equation. The equation has bistable nonlinearity and models electrical activity in a neuron. A qualitative analysis of the model is performed by means of a singular perturbation theory. A small parameter is introduced in the highest order derivative term. This small parameter is known as singular perturbation parameter. Boundary layers occur in the solution of singularly perturbed problems when the singular perturbation parameter tend to zero. These boundary layers are located in neighbourhoods of the boundary of the domain, where the solution has a very steep gradient. Most of the conventional methods fails to capture this effect. A numerical scheme is constructed to overcome this discrepancy in literature. A rigorous analysis is carried out to obtain a-priori estimates on the solution of the problem and its derivatives. It is then proven that the numerical method is unconditionally stable. Convergence and stability analysis is carried out. A set of numerical experiment is carried out and it is observed that the scheme faithfully mimics the dynamics of the model.     

A certain integral equation of first kind

Kulyab, Tazik. SSR. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 190–193, January–February, 1989.     

Solution methods for the generalized integral volterra equation of the first kind

This paper is devoted to exact and approximate methods (first of all, direct ones) for the solution of integro-operational equations. Themost attention is paid to the theoretical substantiation of the collocation method for the solution of the mentioned equations within the general theory of approximate methods developed by L. V. Kantorovich.     

Infinite-horizon optimal controls for problems governed by a volterra integral equation with state-and-control-dependent discount factor

In this work, we concern ourselves with the existence of optimal solutions to optimal control problems defined on an unbounded time interval with states governed by a nonlinear Volterra integral equation. These results extend both the work of Baum and others in infinite-horizon control of ordinary differential equations as well as the work of Angell concerning integral equations. In addition, we incorporate into the objective functional (described by an improper integral) a discount factor which reflects a hereditary dependence on both state and control. In this manner, we are able to generalize the recent results of Becker, Boyd, and Sung in which they establish an existence theorem in the calculus of variations with objective functionals of the so-called recursive type. Our results are obtained through the use of appropriate lower-closure theorems and compactness conditions. Examples are presented in which the applicability of our results is demonstrated.This research was supported by the National Science Foundation, Grant No. DMS-87-00706.     

On a semilinear volterra integrodifferential equation

The Volterra integrodifferential equation $$\begin{array}{*{20}c} {u_t (t,x) + \smallint '_0 a(t - s)( - \Delta u(s,x) + f(x,u(s,x)))ds = h(t,x),,} \\ {t > 0,x \in \Omega \subset R^N ,} \\ \end{array} $$ together with boundary and initial conditions is considered. The existence of global solutions (in time) is established under weak assumptions onf. An application in heat flow is also indicated.     

Numerical solution of a certain hypersingular integral equation of the first kind

In this paper, we first discuss the midpoint rule for evaluating hypersingular integrals with the kernel sin ⁻²[(x−s)/2] defined on a circle, and the key point is placed on its pointwise superconvergence phenomenon. We show that this phenomenon occurs when the singular point s is located at the midpoint of each subinterval and obtain the corresponding supercovergence analysis. Then we apply the rule to construct a collocation scheme for solving the relevant hypersingular integral equation, by choosing the midpoints as the collocation points. It’s interesting that the inverse of coefficient matrix for the resulting linear system has an explicit expression, by which an optimal error estimate is established. At last, some numerical experiments are presented to confirm the theoretical analysis.     

Singular integral equation with Cauchy kernel on the real axis

We find closed-form formulas for the solution of the simplest singular integral equation with Cauchy kernel on the real axis and use them to reduce the full singular integral equation considered in the paper to a Fredholm equation. We construct numerical schemes for the above-mentioned equations and estimate the accuracy order of the approximate solution.     

Backward perturbation analysis of certain characteristic subspaces

Summary This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=x, the generalized eigenvalue problem Ax=Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.This subject was supported by the Swedish Natural Science Research Council and the Institute of Information Processing of the University of Umeå.     

Asymptotic analysis of a certain random differential equation

We prove a limit theorem for the mathematical expectation of the solution of an initial value problem in a Hilbert space. The random differential equations considered here satisfy a strong mixing condition which is weaker than the one imposed in analogue results (Cogburn and Hersh, 1973; Papanicolaou and Varadhan, 1973). Our motivation to develop this analysis comes from a system (Nogueira, preprint) formed by coupling an external source to the Martin-Emch model (Martin and Emch, 1975).