Collocation methods for fractional integro-differential equations with weakly singular kernels

In this paper, the piecewise polynomial collocation methods are used for solving the fractional integro-differential equations with weakly singular kernels. We present that a suitable transformation can convert fractional integro-differential equations to one type of second kind Volterra integral equations (VIEs) with weakly singular kernels. Then we solve the VIEs by standard piecewise polynomial collocation methods. It is shown that such kinds of methods are able to yield optimal convergence rate. Finally, some numerical experiments are given to show that the numerical results are consistent with the theoretical results.     

Exact reachability for second-order integro-differential equations

In this Note we analyze a reachability problem for an integro-differential equation by using a harmonic analysis approach. To cite this article: P. Loreti, D. Sforza, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.     

PBVP for integro-differential equations of volterra type in B.S.

In this paper, we study the PBVP for integro-differential equations of Volterra type in Banach spaces. By developing monotone iterative technique for the PBVP, we get some results concerning the existence of extremal solutions, which are the limits of monotone sequences.The project supported by the Natural Science Foundation of Shandong Province.     

Periodic boundary value problems for second-order impulsive integro-differential equations

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second-order nonlinear impulsive integro-differential equations of mixed type.     

Periodic boundary value problems for second order impulsive integro-differential equations of volterra type

The existence of solutions of periodic boundary value problems for second order impulsive integro-differential equations of Volterra type is investigated. By using the method of upper and lower solutions, it is proved that the problem in whi h impulses occur at fixed times has a solution. Some impulsive integro-differential inequalities related to such problem are also established.     

A Sinc–Collocation method for the linear Fredholm integro-differential equations

A Sinc–Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is O( e ^{- k?N} )O{\left( {e^{{ - k{\sqrt N }}} } \right)} . Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.     

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.     

On spectral methods for Volterra-type integro-differential equations

This paper considers the spectral methods for a Volterra-type integro-differential equation. Firstly, the Volterra-type integro-differential equation is equivalently restated as two integral equations of the second kind. Secondly, a Legendre-collocation method is used to solve them. Then the error analysis is conducted based on the _L∞$L_{\infty }$-norm. In addition, numerical results are presented to confirm our analysis.     

Callocation methods for volterra integrodifferential equations with singular kernels

A collocation method which uses Hermite cubic elements is proposed for the solution of Volterra integrodifferential equations with singular kernels. Optimum error estimates in the uniform norm are obtained by means of interpolation operators. We also report on results of numerical comparisons with one well established method and another new “modified collocation” scheme.     

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.     

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.     

Collocation methods for differential-algebraic equations of index 3

Summary This article give sharp convergence results for stiffly accurate collocation methods as applied to differential-algebraic equations (DAE's) of index 3 in Hessenberg form, proving partially a conjecture of Hairer, Lubich, and Roche.     

Dissipativity of multistep runge-kutta methods for nonlinear volterra delay-integro-differential equations

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.We investigate the dissipativity properties of (k,l)algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.     

Iterative and non-iterative methods for non-linear Volterra integro-differential equations

Iterative and non-iterative methods for the solution of nonlinear Volterra integro-differential equations are presented and their local convergence is proved. The iterative methods provide a sequence solution and make use of fixed-point theory, whereas the non-iterative ones result in series solutions and also make use of fixed-point principles. By means of integration by parts and use of certain integral identities, it is shown that the initial conditions that appear in the iterative methods presented here can be eliminated and the resulting iterative technique is identical to the variational iteration method which is derived here without making any use at all of Lagrange multipliers and constrained variations. It is also shown that the formulation presented here can be applied to initial-value problems in ordinary differential, Volterra’s integral and integro-differential, pantograph, and nonlinear and linear algebraic equations. A technique for improving/accelerating the convergence of the iterative methods presented here is also presented and results in a Lipschitz constant that may be varied as the iteration progresses. It is shown that this acceleration technique is related to preconditioning methods for the solution of linear algebraic equations. It is also argued that the non-iterative methods presented in this paper may not competitive with iterative ones because of possible cancellation errors, if implemented numerically. An analytical continuation procedure based on dividing the interval of integration into disjoint subintervals is also presented and its limitations are discussed.     

Exponentially-fitted Obrechkoff methods for second-order differential equations

We develop two-step exponentially-fitted Obrechkoff methods. The combination of exponential fitting and methods of Obrechkoff type is discussed here for the first time. We shall construct such methods of different orders and study their linear stability along the line of the Coleman/Ixaru definition. Some numerical results are introduced to show the applicability of such methods.     

Boundary value methods for Volterra integral and integro-differential equations

New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.