On a symptotic methods for Fredholm–Volterra integral equation of the second kind in contact problems

A method is used to obtain the general solution of Fredholm–Volterra integral equation of the second kind in the space $\text{L}{}_2\text{(Ω)×C(0,T),}\text{0⩽t⩽T<∞;Ω}$ is the domain of integrations.The kernel of the Fredholm integral term belong to $\text{C([Ω]×[Ω])}$ and has a singular term and a smooth term. The kernel of Volterra integral term is a positive continuous in the class C(0,T), while $\text{Ω}$ is the domain of integration with respect to the Fredholm integral term.Besides the separation method, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kernel which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite algebraic system is obtained.     

Hybrid function method for solving Fredholm and Volterra integral equations of the second kind

Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix   with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n$n$ and m$m$ are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n$n$ and m$m$ can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269–C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12–20] for these problems.     

PQWs in complex plane: Application to Fredholm integral equations

In this paper, we use a numerical procedure for solving Fredholm integral equations of the second kind in complex plane. The periodic quasi-wavelets (PQWs) constructed on [0,2π]$[0\text{,}2\pi ]$ are utilized as a basis in collocation method to reduce the solution of linear integral equations to a system of algebraic equations. Convergence analysis is derived and we used some numerical examples to illustrate the accuracy and the implementation of the method.     

A new numerical approach for solving a class of singular two-point boundary value problems

In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x ?? (0, 1]

$$\begin{array}{@{}rcl@{}} (g(x)y^{\prime})^{\prime}=g(x)f(x,y),\\ y^{\prime}(0)=0,\mu y(1)+\sigma y^{\prime}(1)=B. \end{array} $$

A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

Convergence of approximate solution of system of Fredholm integral equations

In this paper numerical solution of system of linear Fredholm integral equations by means of the Sinc-collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The exponential convergence rate of the method is proved. The method is applied to a few test examples with continuous kernels to illustrate the accuracy and the implementation of the method.     

Fast solution methods for fredholm integral equations of the second kind

Summary The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [–1, 1]. We discretize by a Nyström method with nodes {cos(j/N)} _j ^=0/N . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by , a low-rank modification of the identity matrix. ReplacingA by in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution , can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then is shown to converge optimally to the solution of the integral equation asN increases.We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andÃ. They yield the solution inO (N ²) arithmetic operations under mild restrictions on the kernel and the right-hand side function.Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nyström method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.     

Spectral collocation method for linear fractional integro-differential equations

In this paper, we propose and analyze a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. First, we use some function and variable transformations to change the equation into a Volterra integral equation defined on the standard interval [-1,1]$[-1\text{,}1]$. Then the Jacobi–Gauss points are used as collocation nodes and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence order of the proposed method is investigated in the infinity norm. Finally, some numerical results are given to demonstrate the effectiveness of the proposed method.     

Computing narrow inclusions for the solution of integral equations

Interval analysis in connection with modified fixed point theorems is applied to compute close and guaranteed error bounds for he solution of Fredholm and Volterra integral equations of the second kind. The bounds can be obtained automatically by employing interval tools and using inclusions of the kernels and inhomogeneities.

The theory of interval analysis is sketched briefly, then Fredholm and Volterra equations aretreated. At last some numerical examples are given.     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

A discrete Galerkin method for a hyperslngular boundary integral equation

Received on 14 August 1995. Revised on 20 August 1996. Consider solving the interior Neumann problem with D a simply-connected planar region and S=D a smooth curve.A double-layer potential is used to represent the solution,and it leads to the problem of solving a hypersingular integralequation. This integral equation is reformulated as a Cauchysingular integral equation. A discrete Galerkin method withtrigonometric polynomials is then given for its solution. Anerror analysis is given, and numerical examples complete thepaper.     

The singular integral equation on a closed piecewise smooth manifold inCn

In this paper, we obtain the general permutation formulas and composition formulas of singular integral of the Bochner-Martinelli type on a closed piecewiseC ⁽¹⁾ smooth manifold. As an application, we consider the corresponding singular integral equation of linear variable coefficients, and prove that the singular integral equation can be transformed to an equivalent Fredholm equation, whose characteristic equation has a unique solution in , here denotes the function set which satisfies the Hölder condition on D and holomorphically expands to domainD.Corresponding author. Project supported in part by the Mathematical Tian Yuan Foundation of China (Grant No. TY10126033).     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations

A numerical method based on an m-set of general, orthogonal triangular functions (TF) is proposed to approximate the solution of nonlinear Volterra–Fredholm integral equations. The orthogonal triangular functions are utilized as a basis in collocation method to reduce the solution of nonlinear Volterra–Fredholm integral equations to the solution of algebraic equations. Also a theorem is proved for convergence analysis. Some numerical examples illustrate the proposed method.     

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an indicator of the quality of approximation.     

Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.