Class Of Volterra Integro-differential Equations Research Articles

New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme

<p style='text-indent:20px;'>In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators so-called fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.

On the stability of perturbed Volterra integrodifferential equations

The asymptotic behavior of solutions of a class of Volterra integrodifferential equations is studied. Some new sufficient conditions are obtained in the presence of perturbation term. Our approach is based on some estimations on the solutions of the perturbed equation with respect to the solutions of the original unperturbed Volterra integrodifferential equations. We prove that the convergence of solutions to the equilibrium point can be studied provided that the perturbation term is bounded by a suitable function. The idea for this approach is based on a new integral inequality of Gronwall type.

Collocation method for approximate solution of Volterra integro-differential equations of the third-kind

In this paper, we have adapted the collocation method for solving a special class of Volterra integro-differential equations of the third-kind. Structure of the method and its applicability for proposed equations have been presented. Moreover we have investigated the solvability and convergence analysis of the method. Also some numerical examples are provided to illustrate the theoretical results.

Continuous linear multistep method for the general solution of first order initial value problems for Volterra integro-differential equations

Purpose The purpose of this paper is to develop a block method of order five for the general solution of the first-order initial value problems for Volterra integro-differential equations (VIDEs). Design/methodology/approach A collocation approximation method is adopted using the shifted Legendre polynomial as the basis function, and the developed method is applied as simultaneous integrators on the first-order VIDEs. Findings The new block method possessed the desirable feature of the Runge–Kutta method of being self-starting, hence eliminating the use of predictors. Originality/value In this paper, some information about solving VIDEs is provided. The authors have presented and illustrated the collocation approximation method using the shifted Legendre polynomial as the basis function to investigate solving an initial value problem in the class of VIDEs, which are very difficult, if not impossible, to solve analytically. With the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in the predictor corrector method where additional equations are supplied from a different formulation, all the additional equations are from the same continuous formulation which shows the beauty of the method. However, the absolute stability region showed that the method is A-stable, and the application of this method to practical problems revealed that the method is more accurate than earlier methods.

Efficient general linear methods for a class of Volterra integro-differential equations

This paper concerns on the introduction of a method for solving a class of Volterra integro-differential equations (VIDEs) of the second kind. It is based on the combination of special general linear methods for ordinary differential equations and Gregory quadrature rule and equipped with a starting procedure. The convergence and stability of the method are analyzed. Some numerical experiments are given to illustrate the agreement of our implementation with the theoretical convergence orders and show the capability of the method in solving nonlinear VIDEs.

On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates aC0-semigroup. Next, a necessary and sufficient condition forLp-admissibilityp∈1,∞of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

Spectral collocation methods for Volterra-integro differential equations with noncompact kernels

The aim of this paper is to analyze spectral collocation methods for a class of Volterra-integro differential equations with noncompact kernels. The spectral collocation method is proposed to solve the equations after the results on existence, uniqueness and regularity of the solutions are established. The convergence rates of the methods are proven and a numerical example is provided to verify the theoretical findings.

Some remarks on a new DQ-based method for solving a class of Volterra integro-differential equations

In this paper a numerical Picard-like method, which combines successive approximations with integral and differential quadrature, is considered to solve some Volterra integro-differential equations where rational functions are involved. Under certain conditions, the method provides solutions in explicit form, without any recursive computation, and for some cases it is possible to give a formula to compute the related error. Several numerical examples are proposed to illustrate the behaviour of the method, by comparing it, where possible, with some existing methods.

Approximations for a class of volterra integro-differential equations

We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. Numerical examples are presented to illustrate the methods and in the cases where both methods are applicable comparisions of the results are included.

Asymptotic behavior of solutions to a class of Volterra integrodifferential equations

Asymptotic behavior of solutions to a class of Volterra integrodifferential equations

Fractional order Volterra equations with applications to elasticity

A class of Volterra integro-differential equations with singular kernels is studied. It is chosen so that as the order of singularity approaches zero or one the equation approaches an analog of the equation for viscoelasticity or a Kelvin-Voigt solid. Qualitative results are obtained and estimates given for the effect of the order of singularity on errors in Galerkin's method. The role of the quasi-static approximation is studied.

Weak solutions for a class of parabolic Volterra integrodifferential equations

We define weak solutions for a class of Volterra integrodifferential equations of the form u ′( t ) + Au ( t ) = ∝ 0 t a ( t , s ) g ( s , u ( s )) ds + f ( t , u ( t )), t ⩾ 0, u (0) = 0. The operator A is the negative infinitesimal generator of an analytic semigroup in a Banach space X . The operator g ( t , u ) is related to A by a special form g(t, u) = A 1 2 q(t, u) , where q ( t , u ) is an appropriate “lower order” operator. We show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions. Using our results we study the asymptotic behavior, as time goes to infinity, of strong solutions of a second order initial-boundary value problem.

A New Error Analysis for a Cubic Spline Approximate Solution of a Class of Volterra Integro-Differential Equations

A New Error Analysis for a Cubic Spline Approximate Solution of a Class of Volterra Integro-Differential Equations

A new error analysis for a cubic spline approximate solution of a class of Volterra integro-differential equations

In this paper a third-order numerical method is considered which utilizes a twice continuously differentiable third degree spline to approximate the solution of \[ x ˙ ( t ) = F ( t , x ( t ) , ∫ a t K ( t , u , x ( u ) ) d u ) , x ( a ) = x 0 , \begin {array}{*{20}{c}} {\dot x(t) = F\left ( {t,x(t),\int _a^t {K(t,u,x(u))\;du} } \right ),} \hfill \\ {x(a) = {x_0},} \hfill \\ \end {array} \] at discrete points in the interval [a, b]. The error analysis uses a technique usually associated with linear multistep methods.