Solving Volterra Integral Equations Research Articles

Implementation of adomian polynomials in variational iteration method for solving volterra integral equations

In this paper Adomian polynomials are employed is solving Variational Iteration Method for finding Volterra integral equations. The proposed technique involves Adomian Polynomials in the correction functional equation. The solutions have been found by suggested iterative scheme without any discretization, linearization, or restrictive assumptions, and it is quite efficient and is practically well suited. Two examples are given to verify the reliability and the efficiency of this method. Keywords : Variational Iteration Method, Adomian Polynomials, Volterra Integral Equations, Functional Equation.

Optimal homotopy asymptotic method for solving Volterra integral equation of first kind

In this paper, authors demonstrate the efficiency of optimal homotopy asymptotic method (OHAM). This is done by solving nonlinear Volterra integral equation of first kind. OHAM is applied to Volterra integral equations which involves exponential, trigonometric function as their kernels. It is observed that solution obtained by OHAM is more accurate than existing techniques, which proves its validity and stability for solving Volterra integral equation of first kind.

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].

Features of behavior of numerical methods for solving Volterra integral equations of the second kind

Systems of second-kind Volterra integral equations with stiff and oscillating components are considered. An implicit noniterative method of the second order is proposed for the numerical solution of such problems. The efficiency of the method is demonstrated using several examples.

Product Block by Block for the Second Kind Volterra Integral Equations with Singular Kernel

The block-by -block method is a good self-starting method for solving Volterra Integral equations of the second kind with continuous kernel. In this paper two modifications are suggested to develop block by block method to solve Volterra integral equations with weakly singular kernels. The idea of the product technique with higher order is used for block-by block method, which is the first modification, while the second modification is using the method on the graded nodes. Implementation and testing of the considered algorithms have been done. The results showed that this method gives good results compared with others methods on equal spaced nodes. AMS Subject Classification: 65L05, 65R20 Keywords: VOLTERRA INTEGRAL EQUATION, PRODUCT INTEGRATION, SINGULAR KERNEL, BLOCK -BY-BLOCK METHOD, BOOL’S QUADRATURE.

Solving Volterra integral equations of the second kind by sigmoidal functions approximation

In this paper, a numerical collocation method is developed for solving linear and nonlinear Volterra integral equations of the second kind. The method is based on the approximation of the (exact) solution by a superposition of sigmoidal functions and allows one to solve a large class of integral equations having either continuous or $L^p$ solutions. Special computational advantages are obtained using unit step functions, and analytical approximations of the solution are also at hand. The numerical errors are discussed, and a priori as well as a posteriori estimates are derived for them. Numerical examples are given for the purpose of illustration.

Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.

Parallel in Time Algorithm with Spectral-Subdomain Enhancement for Volterra Integral Equations

This paper proposes a parallel in time (also called time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the spirit of the domain decomposition that consists of breaking the domain of computation into subdomains and solving iteratively the subproblems in a parallel way. To obtain a high order of accuracy, a spectral collocation accuracy enhancement in subdomains will be employed. Our main contributions in this work are twofold: (i) A time parareal method is designed for the integral equations, which to our knowledge is the first of its kind. The new method is an iterative process combining a coarse prediction in the whole domain with fine corrections in subdomains by using spectral approximation, leading to an algorithm of very high accuracy. (ii) A rigorous convergence analysis of the overall method is provided. The numerical experiment confirms that the overall computational cost is considerably reduced while the desired spectral rate of convergence can be obtained.

LS-SVR-based solving Volterra integral equations

In this paper, a novel hybrid method is presented for solving the second kind linear Volterra integral equations. Due to the powerful regression ability of least squares support vector regression (LS-SVR), we approximate the unknown function of integral equations by using LS-SVR in intervals with known numerical solutions. The trapezoid quadrature is used to approximate subsequent integrations in intervals with unknown numerical solutions. The feasibility of the proposed method is examined on some integral equations. Experimental results of comparison with analytic and repeated modified trapezoid quadrature method’s solutions show that the proposed algorithm could reach a very high accuracy. The proposed algorithm could be a good tool for solving the second kind linear Volterra integral equations.

Solving Volterra integral equations of the second kind by wavelet-Galerkin scheme

In this paper, we apply the wavelet-Galerkin method to obtain approximate solutions to linear Volterra integral equations (VIEs) of the second kind. Daubechies wavelets are used to find such approximations. In this approach, we introduce some new connection coefficients and discuss their properties and propose algorithms to evaluate them. These coefficients can be computed just once and applied for solving every linear VIE of the second kind. Convergence and error analysis are discussed and numerical examples illustrate the efficiency of the method.

An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function

This article presents an R package that implements a general heuristic strategy for the efficient application of numerical schemes for solving Volterra integral equations which have as solution first-passage-time density functions associated to certain stochastic processes. Such a strategy is based on the information provided by the First-Passage-Time Location (FPTL) function about the location of the first-passage-time variable, and it is valid for general situations that expand on the particular cases considered in Román et al. (2008) [10]. Numerical applications are shown to illustrate the validity of the strategy as well as its computational advantages.

Iterated Block-Pulse Method for Solving Volterra Integral Equations

In this paper, an iterated method is presented to determine the numerical solution of linear Volterra integral equations of the second kind (VIEs2). This method initially uses the solution of the direct method to obtain the more accurate solution. The convergence and error analysis of this method are given. Finally, numerical examples illustrate efficiency and accuracy of the proposed method. Also, the numerical results of this method are compared with the results of direct method, collocation method and iterated collocation method.

Multistep Hermite collocation methods for solving Volterra Integral Equations

In this paper, we propose a new class of multistep collocation methods for solving nonlinear Volterra Integral Equations, based on Hermite interpolation. These methods furnish an approximation of the solution in each subinterval by using approximated values of the solution, as well as its first derivative, in the r previous steps and m collocation points. Convergence order of the new methods is determined and their linear stability is analyzed. Some numerical examples show efficiency of the methods.

Differential transform method for systems of Volterra integral equations of the second kind and comparison with homotopy perturbation method

In this article, differential transform method was employed to solve Volterra integral equations of the second kind. The procedure of the method is illustrated systematically. In this paper some results useful have been proved for the first time, and they have been used in the solution of these systems. To show the power of this method, some examples have been created (Examples 2 and 3) and solved. The results of them are compared with the results of homotopy perturbation method which has been used to solve the same systems (Biazar and Ghazvini, 2009) by plots. Results of the comparison reveal that, this method is more effective and promising than homotopy perturbation method for Volterra integral equations of the second kind. Key words: Differential transform method, homotopy perturbation method, systems of Volterra integral equations of the second kind.

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.

A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation

In this paper, we present a numerical method for solving Volterra integral equations of the second kind (VK2), first kind (VK1) and even singular type of these equations. The proposed method is based on approximating unknown function with Bernstein’s approximation. This method using simple computation with quite acceptable approximate solution. Furthermore we get an estimation of error bound for this method. For showing efficiency of this method we use several examples.

An expansion–iterative method for numerically solving Volterra integral equation of the first kind

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion–iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy.

Differential transform method for solving Volterra integral equation with separable kernels

In this paper, Volterra integral equations with separable kerenels are solved using the differential transform method. The approximate solution of this equation is calculated in the form of a series with easily computable terms. Exact solutions of linear and nonlinear integral equations have been investigated and the results illustrate the reliability and the performance of the differential transform method.

Solving integral equations with logarithmic kernels by Chebyshev polynomials

In this paper, a finite Chebyshev expansion is developed to solve Volterra integral equations with logarithmic singularities in their kernels. The error analysis is derived. Numerical results are given showing a marked improvement in comparison with the piecewise polynomial collocation method given in literature.

Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions

This article proposes a simple efficient direct method for solving Volterra integral equation of the first kind. By using block-pulse functions and their operational matrix of integration, first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. Some examples are presented to illustrate efficiency and accuracy of the proposed method.