Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.     

Approximate solution of dual integral equations using Chebyshev polynomials

The aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy-type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy-type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.     

A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions

In this paper, we use hat basis functions to solve the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs) of the second kind. This method converts the system of integral equations into a linear or nonlinear system of algebraic equations. Also, we consider the order of convergence of the method and show that it is O(h²). Application of the method on some examples show its accuracy and efficiency.     

Approximate solution of systems of Volterra integral equations with error analysis

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(?c N ^1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.     

Using the Fredholm resolvent for computing the ℋ2-norm of linear periodic systems

This article presents a method for the ?₂-norm computation of single-loop finite-dimensional linear continuous periodic systems, which is based on the parametric transfer function concept and the calculus of Fredholm integral equations of the second kind. General formulae for the computation of the mean output variance are provided as relations of two integral functions of an auxiliary parameter. On the basis of the general solution, an approximate computation method is derived, and the deviation of the approximate solution is estimated. In principle, this approach allows one to perform the computation with any desired accuracy.     

A Method for Solution of Systems of Linear Algebraic Equations with m-Dimensional @lambda;-Matrices

A system of linear algebraic equations with m-dimensional @lambda;-matrices is considered. The proposed method of searching for the solution of this system lies in reducing it to a numerical system of a special kind.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions

This article proposes a simple efficient method for solving a Volterra integral equations system of the first kind. By using block pulse functions and their operational matrix of integration, a first kind integral equations system can be reduced to a linear system of algebraic equations. The coefficient matrix of this system is a block matrix with lower triangular blocks. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Artificial neural networks based modeling for solving Volterra integral equations system

Properly designing an artificial neural network is very important for achieving the optimal performance. This study aims to utilize an architecture of these networks together with the Taylor polynomials, to achieve the approximate solution of second kind linear Volterra integral equations system. For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system. Then we apply the suggested neural net for adjusting the numerical coefficients of given expansions in resulting system. Consequently, the reported architecture using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. The proposed method was illustrated by several examples with computer simulations. Subsequently, performance comparisons with other developed methods was made. The comparative experimental results showed that this approach is more effective and robust.     

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; ?. Nas, S. Yalçinba?, 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. Yalçinba?, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinba? 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.     

Numerical solution of laplace's equation on non-simply connected regions on R 2

We consider the interior Dirichlet problem for Laplace's equation on a non-simply connected two-dimensional regions with smooth boundaries.The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral.The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind.Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “Modified Dirichlet problem”.The Nystrom's method coupled with the trapezoidal rule is used as numerical integration scheme.The linear system derived from the integral equation is solved using the conjugate gradient applied to the normal equation.Theoretical and computational details of the method are presented.     

B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

Some schwarz methods for integral equations on surfaces-h and p versions

Abstact We present new results from 11, 7, 12 on various Schwarz methods for the h and p versions of the boundary element methods applied to prototype first kind integral equations on surfaces. When those integral equations (weakly/hypersingular) are solved numerically by the Galerkin boundary element method, the resulting matrices become ill-conditioned. Hence, for an efficient solution procedure appropriate preconditioners are necessary to reduce the numbers of CG-iterations. In the p version where accuracy of the Galerkin solution is achieved by increasing the polynomial degree the use of suitable Schwarz preconditioners (presented in the paper) leads to only polylogarithmically growing condition numbers. For the h version where accuracy is achieved by reducing the mesh size we present a multi-level additive Schwarz method which is competitive with the multigrid method. Communicated by: U. Langer Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

A parameter choice strategy for the regularized approximation of Fredholm integral equations of the first kind

Recently, Rajan [A Modified convergence analyis for solving Fredholm integral equations of the first kind, Integral Equ. Oper. Theory, 49 (2004), pp. 511–516.] suggested a modified convergence analysis for solving Fredholm integral equations of the first kind and also considered an a priori parameter choice strategy of choosing the regularization parameter. In this article, as an a posteriori parameter choice strategy for choosing the regularization parameter, a class of discrepancy principle is proposed under the Hilbert space setting and is illustrated numerically.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

A reliable computational approach for approximate solution of Hammerstein integral equations of mixed type

The aim of this article is to present an efficient analytical and numerical procedure for solving the nonlinear Hammerstein integral equations of mixed type. Our method mainly depends on a Taylor expansion approach. Also, we obtain the approximate solution of the nonlinear Volterra–Hammerstein integral equations of mixed type in terms of the Taylor polynomials. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.     

Approximate solution of a class of linear integro-differential equations by Taylor expansion method

In this paper, a simple and effective Taylor expansion method is presented for solving a class of linear integro-differential equations including those of Fredholm and of Volterra types. By means of the nth-order Taylor expansion of an unknown function at an arbitrary point, a linear integro-differential equation can be converted approximately to a system of linear equations for the unknown function itself and its first n derivatives under initial conditions. The nth-order approximate solution is exact for a polynomial of degree equal to or less than n. Some examples are given to illustrate the accuracy of this method.     

Extrapolation for solving a system of weakly singular nonlinear Volterra integral equations of the second kind

This article discusses an extrapolation method for solving a system of weakly singular nonlinear Volterra integral equations of the second kind. Based on a generalization of the discrete Gronwall inequality and Navot's quadrature rule, the modified trapeziform quadrature algorithm is presented. The iterative algorithm for solving the discrete system possesses a high accuracy order O(h ^2+α). After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using extrapolation. An a posteriori error estimation is provided. Some numerical results are presented to illustrate the efficiency of our methods.