High-order Linear Volterra Research Articles

Numerical solution of high-order linear Volterra integro-differential equations by using Taylor collocation method

ABSTRACTThe main purpose of this work is to provide a new direct numerical method for high-order linear Volterra integro-differential equations (VIDEs). An algorithm based on the use of Taylor polynomials is developed for the numerical solution of high-order linear VIDEs. It is shown that this algorithm is convergent. Numerical results are presented and comparisons are made with well-known numerical methods to prove the effectiveness of the presented algorithm.

A Numerical Approach for Solving High-Order Linear Delay Volterra Integro-Differential Equations

In this study, a numerical method is proposed to solve high-order linear Volterra delay integro-differential equations. In this approach, we assume that the exact solution can be expressed as a power series, which we truncate after the [Formula: see text]-st term so that it becomes a polynomial of degree [Formula: see text]. Substituting the unknown function, its derivatives and the integrals by their matrix counterparts, we obtain a vector equivalent of the equation in question. Applying inner product to this vector with a set of monomials, we are left with a linear algebraic equation system of [Formula: see text] unknowns. The approximate solution of the problem is then computed from the solution of the resulting linear system. In addition, the technique of residual correction, whose aim is to increase the accuracy of the approximate solutions by estimating the error of those solutions, is discussed briefly. Both the method and this technique are illustrated with several examples. Finally, comparison of the present scheme with other methods is made wherever possible.

Numerical solution of high-order Volterra–Fredholm integro-differential equations by using Legendre collocation method

The main purpose of this paper is to use the Legendre collocation spectral method for solving the high-order linear Volterra–Fredholm integro-differential equations under the mixed conditions. Avoiding integration of both sides of the equation, we expressed mixed conditions as equivalent integral equations, by adding the neutral term to the equation. Error analysis for approximate solution and approximate derivatives up to order k of the solution is obtained in both L2 norm and L∞ norm. To illustrate the accuracy of the spectral method, some numerical examples are presented.

A new collocation approach for solving systems of high-order linear Volterra integro-differential equations with variable coefficients

This paper contributes an efficient numerical approach for solving the systems of high-order linear Volterra integro-differential equations with variable coefficients under the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Fibonacci polynomials and their derivatives by means of collocations. In addition, the method is presented with error. Numerical results with comparisons are given to demonstrate the applicability, efficiency and accuracy of the proposed method. The results of the examples indicated that the method is simple and effective, and could provide an approximate solution with high accuracy or exact solution of the system of high-order linear Volterra integro-differential equations.

Improved Bessel collocation method for linear Volterra integro-differential equations with piecewise intervals and application of a Volterra population model

In this paper, an improved Bessel collocation method (IBCM) based on the residual correction technique is presented to solve the high-order linear Volterra integro-differential equation (VIDE) which involves a population model with piecewise intervals. Correction of the approximated solution is obtained using the residual function of the operator equation. The error differential equation, gained by residual function, is solved by the Bessel collocation method (BCM). By summing the approximate solution of the error differential equation with the approximate solution of VIDE with piecewise intervals, a better approximare solution is obtained. Also, an upper bound is given for the corrected approximate solutions. Numerical examples are given to demonstrate the efficiency of the method and the numerical results are compared by existing results. In addition, an application of the population model is given for the method.

Bessel polynomial solutions of high-order linear Volterra integro-differential equations

In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).

Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; Ş. Nas, S. Yalçinbaş, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.

The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials

In the present paper, a Taylor method is developed to find the approximate solution of high-order linear Volterra–Fredholm integro-differential equations under the mixed conditions in terms of Taylor polynomials about any point. In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed.