Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

Numerical solution of a Fredholm integro-differential equation modelling neural networks

We compare piecewise linear and polynomial collocation approaches for the numerical solution of a Fredholm integro-differential equations modelling neural networks. Both approaches combine the use of Gaussian quadrature rules on an infinite interval of integration with interpolation to a uniformly distributed grid on a bounded interval. These methods are illustrated by numerical experiments on neural networks equations.     

A series solution of the nonlinear Volterra and Fredholm integro-differential equations

In this paper, the HAM is applied to obtained the series solution of the high-order nonlinear Volterra and Fredholm integro-differential problems with power-law nonlinearity. Two cases are considered, in the first case the set of base functions is introduced to represent solution of given nonlinear problem and in the other case, the set of base functions is not introduced. However, in both cases, the convergence-parameter provides us with a simple way to adjust and control the convergence region of solution series.     

Asymptotic solution of a system of integro-differential equations

An asymptotic solution of a system of inegro-differential equations is constructed for the case where turning points are present. Vinnychenko Kirovograd Pedagogic Institute, Kirovograd. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 49, No. 12, pp. 1617–1623, December, 1997     

Global smooth solution for a coupled non-linear wave equations

In this paper, the global existence and uniqueness of smooth solution to the initial-value problem for coupled non-linear wave equations are studied using the method of a priori estimates.     

An algorithm for solving Fredholm integro-differential equations

The aim of this paper is to present an efficient numerical procedure for solving linear second order Fredholm integro-differential equations. The scheme is based on B-spline collocation and cubature formulas. The analysis is accompanied by numerical examples. The results demonstrate reliability and efficiency of the proposed algorithm.      

Numerical solution of non-linear Volterra integral equations

This paper deals with non-linear Volterra integral equations of the type $\text{y(x)}\text{=}\text{f(x)}\text{+ ?}{}_0{}^{\mathrm{<mtext>x</mtext>}}\text{H[t, x, y (t), y (x)] dt}$. Convergence criteria are given (in the same sense of the maximum and C^a norms) for the numerical solution of this type of Volterra integral equation. Several numerical methods are compared.     

Numerical solution of Volterra integro-differential equations modelling thalamo-cortical systems

Our study concerns thalamo-cortical systems which are modelled by nonlinear systems of Volterra integro-differential equations of convolution type. The thalamo-cortical systems describe a new architecture for a neurocomputer. Such a computer employs principles of human brain. It consists of oscillators which have different frequencies and are weakly connected via a common medium forced by an external input. Since a neurocomputer consists of many interconnected oscillators (referred also as neurons), the thalamo-cortical systems include large numbers of Volterra integro-differential equations. Solving such systems numerically is expensive not only because of their large dimensions but also because of many kernel evaluations which are needed over the whole interval from the initial point, where the initial condition is imposed, up to the present point, where the computations are currently executed. Moreover, the whole computed history of the solution has to be stored in the memory of the computing machine. Therefore, robust and efficient numerical algorithms are needed for computer simulations for the solutions to the thalamocortical systems. In this paper, we illustrate an iteration technique to solve the thalamo-cortical systems. The proposed successive iterates are vector functions of time, which change the original problems into systems of easier and separated equations. Such separated equations can then be solved in parallel computing environments. Results of numerical experiments are presented for large numbers of oscillators. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 20–27.     

Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW

Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.     

Interpolation correction for collocation solutions of Fredholm integro-differential equations

In this paper we discuss the collocation method for a large class of Fredholm linear integro-differential equations. It will be shown that, when a certain higher order interpolation operation is added to the collocation solution of this equation, the new approximations will, under suitable assumptions, admit a multiterm error expansion in even powers of the step-size . Based on this expansion, ideal multilevel correction results of this collocation solution are obtained.

     

Construction of a nontrivial solution of a system of nonlinear integro-differential equations

The paper studies a system of nonlinear, vector integro-differential equations with sum and difference kernel and possessing the trivial solution. A nontrivial solution of the system in the Sobolev spaceW _∞,2^×n (0,+∞) is constructed.     

The numerical solution of the non-linear integro-differential equations based on the meshless method

This article investigates the numerical solution of the nonlinear integro-differential equations. The numerical scheme developed in the current paper is based on the moving least square method. The moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. It consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its n closet points. Hence the method is a meshless method and does not need any background mesh or cell structures. The error analysis of the proposed method is provided. The validity and efficiency of the new method are demonstrated through several tests.     

Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using Homotopy Analysis Method

The use of homotopy analysis method to approximate the solution of nonlinear Volterra-Fredholm integro-differential equation is proposed in this paper. In this case, the existence and uniqueness of the obtained solution and convergence of the method are proved. The accuracy of the proposed numerical scheme is examined by comparing with the modified Adomian decomposition method and Taylor polynomial method in the example. Also, the cost of operations in the algorithms are obtained to demonstrate the efficiency of the presented method.