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.

MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SUPERLINEAR 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 superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Mixed Monotone Iterative Technique and Boundary Value Problem for a Class of Impulsive Integro-Differential Equations

In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.

Numerical Methods for a Class of Hybrid Weakly Singular Integro-Differential Equations

This paper proposes numerical methods for solving hybrid weakly singular integro-differential equations of the second kind. The terms in these equations are in the following order: derivative term of a state, integro-differential term of a state with a weakly singular kernel, a state, integral term of a state with a smooth kernel, and force. The original class of weakly singular integro-differential equations of the first kind is derived from aeroelasticity mathematical models. Among the proposed methods, the method for solving linear cases is fully based on previously reported approximation scheme for equations of the first kind. For nonlinear cases, a revised method is proposed. Examples are presented to demonstrate the effectiveness of the proposed methods, and the results indicate that the proposed methods facilitate achieving satisfactory and accurate approximations.

A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel

Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical methods.In this paper,a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel.The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative.The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation(IDE).The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time integrator.The scheme is validated using a homogeneous and two nonhomogeneous test problems.Conditioning of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization parameters.Comparison of results of the present approach with Sinc collocation method and quasi-wavelet method are also made.

Numerical Algorithms for Solving One Type of Singular Integro-Differential Equation Containing Derivatives of the Time Delay States

This study presents numerical algorithms for solving a class of equations that partly consists of derivatives of the unknown state at previous certain times, as well as an integro-differential term containing a weakly singular kernel. These equations are types of integro-differential equation of the second kind and were originally obtained from an aeroelasticity problem. One of the main contributions of this study is to propose numerical algorithms that do not involve transforming the original equation into the corresponding Volterra equation, but still enable the numerical solution of the original equation to be determined. The feasibility of the proposed numerical algorithm is demonstrated by applying examples in measuring the maximum errors with exact solutions at every computed nodes and calculating the corresponding numerical rates of convergence thereafter.

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.

Optimization in Transition between Two Dynamic Systems Governed by a Class of Weakly Singular Integro-Differential Equations

This study presents numerical methods for solving the minimum energies that satisfy typical optimal requirements in the transition between two dynamic systems where each system is governed by a different kind of weakly singular integro-differential equation. The class of weakly singular integro-differential equations originates from mathematical models in aeroelasticity. The proposed numerical methods are based on earlier reported approximation schemes for the equations of the first kind and the second kind. The main result of this study is the development of numerical techniques for determining the stability between two dynamic systems in the minimum energy sense.

On the Well-Posedness of a Class of Hybrid Weakly Singular Integro-Differential Equations

In this study, a revised version of some numerical methods for a class of hybrid integro-differential equations with weakly singular kernels (Abel types) is presented. These equations were developed from a class of integro-differential equations of first kind originating from an aeroelasticity problem. By manipulating the bounds of initial conditions with random variations, this study numerically demonstrated the well-posedness properties of the equations. Finally, an assumption of separating variables, allowed for linear splines to be chosen as a basis and for the differentiation and integration of the integro-differential part to be interchanged;hence, a numerical scheme was constructed.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.

SOLUTIONS FOR SECOND ORDER IMPULSIVE INTEGRO- DIFFERENTIAL EQUATION ON UNBOUNDED DOMAINS IN BANACH SPACES

Under loose conditions, the existence of solutions to initial value problem are studied for second order impulsive integro-differential equation with infinite moments of impulse effect on the positive half real axis in Banach spaces. By the use of recurrence method, Tonelii sequence and the locally convex topology, the new existence theorems are achieved, which improve the related results obtained by Guo Da-jun.

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method.

BOUNDARY VALUE PROBLEM FOR NONLINEAR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATION IN BANACH SPACE

In this paper, by establishing a new comparison result and using the monotone iterative technique, the existence of maximal and minimal solutions of the boundary value problem for second-order impulsive differential equation which depends on x’ in Banach space is obtained.

GLOBAL EXPONENTIAL STABILITY FOR AN IMPULSIVE DELAYED INTEGRO-DIFFERENTIAL EQUATION

In this paper, the stability of an impulsive integro-differential equation with finite and infinite delays is investigated. By applying the Lyapunov-Razumikin method, sufficient condition for global exponential stability of such equation is obtained.

INTEGRAL BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

By developing a comparison result and using the monotone iterative technique, we obtain the existence of the minimal and the maximal solutions to an integral boundary value problem for first order impulsive integro-differential equations.

APPROXIMATE CONTROLLABILITY OF FRACTIONAL IMPULSIVE NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND INFINITE DELAY

This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. By the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results.

The Singularly Perturbed Problem of Vector Integro-differential Equations

The singularly perturbed boundary value problem of scalar integro-differential equations has been studied extensively by the differential inequality method . However, it does not seem possible to carry this method over to a corresponding nonlinear vector integro-differential equation. Therefore , for n-dimensional vector integro-differential equations the problem has not been solved fully. Here, we study this nonlinear vector problem and obtain some results. The approach in this paper is to transform the appropriate integro-differential equations into a canonical or diagonalized system of two first-order equations.

NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID

In this paper,we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay.This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived.Based on the a priori error bound and mesh equidistribution principle,we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter.The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm.Furthermore,we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations.Numerical results are provided to demonstrate the effectiveness of our presented monitor function.Meanwhile,it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.