Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations

We consider a linear two-point boundary value problem for systems of integro-differential equations. By using the parametrization method and an approximation of the integro-differential equation by a loaded differential equation, we establish coefficient tests for the well-posedness of the considered problem and suggest an algorithm for finding the solution.     

带非局部积分项常微分方程的讨论及其应用

研究带非局部积分项的二阶线性常微分方程及其在金融保险上的应用.首先讨论带非局部积分项的二阶常微分方程解的存在唯一性,通过变量代换和累次积分交换积分顺序将非局部项简化,将方程化为方程组,然后完成了对方程组解的存在唯一性的证明.接着分析了带非局部项的二阶常微分方程解的结构,给出了方程解的形式.最后通过推导,指出带非局部项的线性常微分方程在保险公司的破产概率研究中的应用,重点放在二阶方程的应用上,并且在某一特定情况下,举出了一个可以给出解析解的例子.     

Unbounded perturbation of the controllability for evolution equations

In this paper we prove that the controllability for evolution equations in Banach spaces is not destroyed, if we perturb the equation by “small” unbounded linear operator. This is done by employing a perturbation principle from linear operator theory and a characterization of surjective operators in Banach spaces. Finally, we apply these to a control system governed by partial integro-differential equations.     

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.     

Constant Dividend Barrier in a Risk Model with a Generalized Farlie-Gumbel-Morgenstern Copula

In this paper, we consider the classical surplus process with a constant dividend barrier and a dependence structure between the claim amounts and the inter-claim times. We derive an integro-differential equation with boundary conditions. Its solution is expressed as the Gerber-Shiu discounted penalty function in the absence of a dividend barrier plus a linear combination of a finite number of linearly independent particular solutions to the associated homogeneous integro-differential equation. Finally, we obtain an explicit solution when the claim amounts are exponentially distributed and we investigate the effects of dependence on ruin quantities.     

An application of the Hurwitz theorem to the root analysis of the characteristic equation

In this work we show that a classical result of A. Hurwitz is still very effective in studying the root analysis of the characteristic equation for a linear functional differential equation. A conjecture was made by Funakubo et al. (2006) [3] regarding the asymptotic stability condition of the zero solution of a linear integro-differential equation of Volterra type. We applied the Hurwitz theorem to the characteristic equation in question and showed the existence of a root with positive real part and solved the conjecture. The Hurwitz theorem is expected to work well for the root analysis in critical cases.     

Hölder continuous solutions for integro-differential equations and maximal regularity

We characterize existence and uniqueness of solutions for a linear integro-differential equation in Hölder spaces. Our method is based on operator-valued Fourier multipliers. The solutions we consider may be unbounded. Concrete equations of the type we study arise in the modeling of heat conduction in materials with memory.     

Generalized solvability of parabolic integro-differential equations

For a linear integro-differential equation of parabolic type, we obtain theorems of the generalized solvability by the method of a priori inequalities.     

Generalized Collocation Method for Integro-Differential Equations in an Exceptional Case

We study a linear integro-differential equation with a coefficient that has finite-order zeros. To solve the equation approximately in a distribution space, we suggest and substantiate a generalized collocation method based on special interpolation polynomials.     

Divided powers and composition in integro-differential algebras

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.     

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.     

Local existence of solutions for the Neumann problem of the nonlinear elastodynamic system outside a domain

In this paper, we study the mixed initial-boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.     

An approximate method for solving a class of weakly-singular Volterra integro-differential equations

In this paper, we present a new approach to resolve linear and nonlinear weakly-singular Volterra integro-differential equations of first- or second-order by first removing the singularity using Taylor’s approximation and then transforming the given first- or second-order integro-differential equations into an ordinary differential equation such as the well-known Legendre, degenerate hypergeometric, Euler or Abel equations in such a manner that Adomian’s asymptotic decomposition method can be applied, which permits convenient resolution of these equations. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained demonstrate this approach is indeed practical and efficient.     

Numerical methods for singular nonlinear integro-differential equations

For the numerical integration of singular nonlinear integro-differential equations we consider fractional linear multistep methods. We prove convergence of these methods and discuss their stability (as an extension of A-stability for stiff differential equations). Numerical experiments with the Basset equation are included.     

A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.     

On the uniform asymptotic stability for a linear integro-differential equation of Volterra type

In this paper, we give a necessary and sufficient condition on the uniform asymptotic stability of the zero solution of a linear integro-differential equation of Volterra type where the ordinary part is ax(t). We put emphasis on the case a>0. The proofs of our results are carried out by using the root analysis of the characteristic equation. In Section 5 we give some conjectures.     

New version of the collocation method for integro-differential equations in a special case

We consider a linear integro-differential equation whose coefficient has polynomial zeros. For its approximate solution in the space of distributions, we suggest and justify a new version of the collocation method based on the use of special polynomials.     

Necessary and sufficient condition for the unique existence of periodic solution of linear volterra equations

In this paper, we obtained the sufficient and necessary condition for the unique existence of periodic solution of the linear Volterra integro-differential equations of the form $$x'(t) = \int_0^\infty {(dE(s))x(t - s) + f(t)} $$ . We also proved that the mentioned equation has unique periodic solution is a generic property.     

线性边界下两步保费率风险模型的Gerber-Shiu罚金函数

在经典复合泊松模型的基础上,研究线性红利边界下两步保费率风险模型的Gerber-Shiu贴现罚金函数.根本目的是推导出它的微积分方程和偏微积分方程.同时给出了线性红利边界下Lundberg基本方程;利用Laplace变换求出了最终破产概率.     

分数阶积分微分方程的近似解

本文给出了分数阶积分微分方程的一种新的解法.利用未知函数的泰功多项式展开将分数阶积分微分方程近拟转化为一个涉及未知函数及其n阶导数的线性方程组.数值例子表明该方法的有效性.