Linear Fredholm Volterra Integro-differential Equations Research Articles

On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given.

Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams

Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for nth order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.

A Galerkin-type approach to solve systems of linear Volterra-Fredholm integro-differential equations

The main interest of this paper is to propose a numerical scheme in order to solve linear systems of Volterra-Fredholm integro-differential equations given with mixed conditions. The proposed method is a weighted residual scheme which uses monomials up to a prescribed degree $N$ as the basis functions. By taking inner product of the equation system with the elements of this basis set in a Galerkin-like fashion, the original problem is transformed into a linear algebraic equation system. After a suitable incorporation of the mixed conditions, a final algebraic system is obtained, from which the approximate solutions of the problem are computed. The proposed numerical scheme is illustrated with example problems taken from the literature.

Euler wavelets method for solving fractional-order linear Volterra–Fredholm integro-differential equations with weakly singular kernels

Euler wavelets method for solving fractional-order linear Volterra–Fredholm integro-differential equations with weakly singular kernels

An operational matrix method to solve linear Fredholm–Volterra integro-differential equations with piecewise intervals

In recent times, operational matrix methods become overmuch popular. Actually, we have many more operational matrix methods. In this study, a new remodeled method is offered to solve linear Fredholm–Volterra integro-differential equations (FVIDEs) with piecewise intervals using Chebyshev operational matrix method. Using the properties of the Chebyshev polynomials, the Chebyshev operational matrix method is used to reduce FVIDEs into a linear algebraic equations. Some numerical examples are solved to show the accuracy and validity of the proposed method. Moreover, the numerical results are compared with some numerical algorithm.

A Combination of the Orthogonal Polynomials with Least –Squares Method for Solving High-Orders Fredholm-Volterra Integro-Differential Equations

This study introduced new technique which is based on a combination of the least-squares technique (LST) with Chebyshev and Legendre polynomials for finding the approximate solutions of higher-order linear Fredholm-Volterra integro-differential equations (FVIDEs) subject to the mixed conditions. Two examples of second and third-order linear FVIDEs are considered to illustrate the proposed method, the numerical results are comprised to demonstrate the validity and applicability of this technique, and comparisons with the exact solution are made. These results have shown that the competence and accuracy of the present technique.

Pell-Lucas collocation method to solve high-order linear Fredholm-Volterraintegro-differential equations and residual correction

In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations FVIDE . The approximate solutions are assumed in form of the truncated Pell-Lucas polynomial series. By using Pell-Lucas polynomials and relations of their derivatives, the solution form and its derivatives are brought to matrix forms. By applying the collocation method based on equally spaced collocation points, the method reduces the problem to a system of linear algebraic equations. Solution of this system determines the coefficients of assumed solution. Error estimation is made and also a method with the help of the obtained approximate solution is developed that finds approximate solution with better results. Then, the applications are made on five examples to show that the method is successful. In addition, the results are supported by tables and graphs and the comparisons are made with other methods available in the literature. All calculations in this study have been made using codes written in Matlab.

A projection method for linear Fredholm–Volterra integro-differential equations

ABSTRACTIn this study, a collocation method, one of the type of projection methods based on the generalized Bernstein polynomials, is developed for the solution of high-order linear Fredholm–Volterra integro-differential equations containing derivatives of unknown function in the integral part. The method is valid for the mixed conditions. The convergence analysis and error bounds of the method are also given. Besides, six examples are presented to demonstrate the applicability and validity of the method.

Solving multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type using Bernstein polynomials method

In this paper, the Bernstein polynomials method is proposed for the numerical solution of a class of multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type. The main characteristic behind this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of Bernstein polynomials method for solving these equations are high, which will be shown by preparing some theorems. Also, the existence and uniqueness of solution have been proved. Finally, the numerical experiments are presented to show the excellent behavior and high accuracy of proposed scheme in comparison with some other well-known methods.

An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement

In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm-Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods.

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.

An operational matrix method for solving linear Fredholm--Volterra integro-differential equations

The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm‒Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational matrices of Taylor polynomials in order to solve linear FVIDEs. By using the operational matrices of integration and product for the Taylor polynomials, the problem for linear FVIDEs is transformed into a system of linear algebraic equations. The solution of the problem is obtained from this linear system after the incorporation of initial conditions. Numerical examples are presented to show the applicability and the efficiency of the method. Wherever possible, the results of our method are compared with those yielded by some other methods.

Approximation Solution to Solving Linear Volterra-Fredholm Integro-Differential Equations of the Second Kind by Using Bernstein Polynomials Method

The aim of this study is to present numerical method for solving the linear Volterra-Fredholm integro-differential equations of the second kind. This method is called the Bernestein polynomials method. This technique transforms the integro-differential equations to the system of algebraic equations. Some numerical results are presented to illustrate the efficiency and accuracy of this method.

The backward substitution method for multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type

The paper presents a new numerical method for solving multipoint boundary value problems for Volterra–Fredholm integro-differential equations with linear functional arguments. The method consists of replacing the initial equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. Numerical results show that the proposed method is of high accuracy and is efficient for solving a wide class of the functional integro-differential equations of the most general form.

Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation

In this study, the Bessel collocation method is presented for the solutions of system of linear Fredholm–Volterra integro-differential equations which includes the derivatives of unknown functions in integral parts. The Bessel collocation method transforms the problem into a system of linear algebraic equations by means of the Bessel functions of first kind, the collocation points and the matrix relations. Also, an error estimation is given for the considered problem and the method. Illustrative examples are presented to show efficiency of method and the comparisons are made with the results of other methods. All of numerical calculations have been made on a computer using a program written in Matlab.

A Bernstein operational matrix approach for solving a system of high order linear Volterra–Fredholm integro-differential equations

In this paper, we present some efficient direct solvers for solving a system of high order linear Volterra–Fredholm integro-differential equations (VFIDEs). A new approach implementing a collocation method in combination with operational matrices of Bernstein polynomials for the numerical solution of VFIDEs is introduced. The main characteristic behind this approach is that it reduces such problems to ones of solving systems of algebraic equations. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving systems of high order linear VFIDEs.

A collocation approach for solving high-order linear Fredholm–Volterra integro-differential equations

In this study, a collocation method based on the Bessel polynomials is introduced for the approximate solutions of high-order linear Fredholm–Volterra integro-differential equations (FVIDEs) under mixed conditions. In addition, the method is presented with error and stability analysis. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results.

Spectral Method for Solving The General Form Linear Fredholm-Volterra Integro Differential Equations Based on Chebyshev Polynomials

The main aim of this paper reports a pseudo spectral method based on integrated Chebyshev polynomials for numerically solving higher order linear Fredholm Volterra integro differential equations (FVIDE) in the most general form. This method transforms FVIDE to matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods.

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.

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

AbstractThis article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011