Iterated Galerkin Method Research Articles

Superconvergent scheme for a system of green Fredholm integral equations

In this study, a numerical scheme to a system of second-kind linear Fredholm integral equations featuring a Green's kernel function is proposed. This involves introducing Galerkin and iterated Galerkin (IG) methods based on piecewise polynomials to tackle the integral model. A thorough analysis of convergence and error for these proposed methods is carried out. Firstly, the existence and uniqueness of solutions for the Galerkin and iterated Galerkin methods are established. Later, the order of convergence is derived using tools from functional analysis and the boundedness property of Green's kernel. The Galerkin scheme has O(hα) order of convergence. Next, the superconvergence of the iterated Galerkin (IG) method is established. The IG method exhibits O(hα+α⁎) order of convergence. Theoretical findings are validated through extensive numerical experiments.

Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.

APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH

In this article, we address the approximation solution of VolterraUrysohn integral equations which involves weakly singular kernels. In order to get better convergence rates, projection methods namely Galerkin and multi Galerkin methods, along with their iterated versions are used in the space of piecewise polynomials subspaces based on the graded mesh. In addition, we compute the superconvergence results for the proposed integral equation and show that iterated Galerkin method outperforms Galerkin method in terms of order of convergence. Further, we demonstrate numerical examples to verify the proposed theoretical framework.

Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels

We consider a Urysohn integral operator with the kernel of the type of Green's function. For , a space of piecewise polynomials of degree with respect to a uniform partition is chosen to be the approximating space, and the projection is chosen to be the orthogonal projection. The iterated Galerkin method is applied to the integral equation . It is known that the order of convergence of the iterated Galerkin solution is r + 2 and is at the partition points. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

Projection and multi-projection methods for second kind Volterra-Hammerstein integral equation

In this article, we discuss the piecewise polynomial based Galerkin method to approximate the solutions of second kind Volterra-Hammerstein integral equations. We discuss the convergence of the approximate solutions to the exact solutions and obtain the orders of convergence $mathcal O(h^{r})$ and $mathcal O(h^{2r}),$ respectively, for Galerkin and its iterated Galerkin methods in uniform norm, where $h, ~r$ denotes the norm of the partition and smoothness of the kernel, respectively. We also obtain the superconvergence results for multi-Galerkin and iterated multi-Galerkin methods. We show that iterated multi-Galerkin method has the order of convergence $mathcal O(h^{3r})$ in the uniform norm. Numerical results are provided to demonstrate the theoretical results.

Convergence analysis of Galerkin and multi-Galerkin methods for nonlinear-Hammerstein integral equations on the half-line using Laguerre polynomials

In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. We obtain optimal convergence results in iterated-Galerkin method in both infinity and weighted -norms. We also obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods, respectively, in weighted -norm. Numerical results are presented to validate the theoretical results.

Approximation methods for system of linear Fredholm integral equations of second kind

In this paper, Galerkin, multi-Galerkin methods and their iterated versions are developed for solving the system of linear Fredholm integral equations of the second kind for both smooth and weakly singular algebraic and logarithmic type kernels. Here first we develop the theoretical framework for Galerkin and iterated Galerkin methods to solve the system of linear second kind Fredholm integral equations using piecewise polynomials as basis functions and then obtain the superconvergence results similar to that of single linear Fredholm integral equation of the second kind. We show that iterated Galerkin approximation yields better convergence rates than Galerkin approximate solution. Further we enhance these results by using multi-Galerkin and iterated-multi-Galerkin methods and show that the iterated multi-Galerkin approximation yields improved superconvergence rates over iterated Galerkin and multi-Galerkin approximations. The theoretical results are justified by the numerical results.

Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green's kernel

We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.

Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence $${\mathcal {O}}(n^{-m})$$, whereas iterated Galerkin method converges with the order $${\mathcal {O}}(n^{-2m})$$ in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is $${\mathcal {O}}(n^{-3m}) $$. Numerical results are provided to justify the theoretical results.

Convergence analysis of Galerkin and multi-Galerkin methods on unbounded interval using Hermite polynomials

We consider in this paper the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the real line with a sufficiently smooth kernels, using Hermite polynomials as basis functions. We obtain optimal convergence results in iterated Galerkin method in weighted L2−norm. We also discuss multi-Galerkin methods and we obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods. Numerical results are presented to confirm the theoretical results.

Error Analysis of Jacobi Spectral Galerkin and Multi-Galerkin Methods for Weakly Singular Volterra Integral Equations

In this article, a Jacobi spectral Galerkin method is developed for weakly singular Volterra integral equations of the second kind. To obtain the superconvergence results, we transform the domain of integration of Volterra integral equation to the standard interval $$[-1, 1]$$ using variable transformation and function transformation. We obtain the convergence rates both in infinity and weighted $$L^2$$-norm. We prove that the Jacobi spectral iterated Galerkin method shows improvement over the Jacobi spectral Galerkin method. We improve these results further by considering the Jacobi spectral iterated multi-Galerkin method. Theoretical results are justified by the numerical results.

Convergence analysis for derivative dependent Fredholm-Hammerstein integral equations with Green’s kernel

In this article, we consider a class of derivative dependent Fredholm-Hammerstein integral equations i.e., the integral equation, where the nonlinear function inside the integral sign is dependent on derivative and the kernel function is of Green’s type. We propose the piecewise polynomial based Galerkin and iterated Galerkin methods to solve these type of derivative dependent Fredholm-Hammerstein integral equations. We discuss the convergence and error analysis of the proposed methods and also obtain the superconvergence results for iterated Galerkin approximations. Some numerical results are given to illustrate this improvement.

Approximation methods for second kind weakly singular Volterra integral equations

Projection methods are applied to obtain the convergence rates for Volterra integral equations with weakly singular kernels. We consider Galerkin and multi Galerkin methods and their iterated versions to solve Volterra integral equations with weakly singular kernels, in the space of piecewise polynomials subspaces based on graded mesh. We will show that the iterated multi-Galerkin method improves over iterated Galerkin method. In fact, we show that iterated multi-Galerkin solution converges with the convergence rates O(n−3m) and O(n−3m(logn)2), for algebraic and logarithmic type kernels, respectively. We prove that iterated Galerkin method, for algebraic kernel, converges with the convergence rate O(n−2m) and for logarithmic type kernel converges with the convergence rate O(n−2mlogn), where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations. Theoretical results are justified by the numerical results.

Error Estimates of Projection Type Methods for Solving Weakly Singular Integral Equations

We study projection type methods for numerical solution of one class of weakly singular integral Fredholm equations of the second kind including the integral radiative transfer equation. We consider Galerkin type methods and their modifications: the iterated Galerkin method (the Sloan method), the Kantorovich method, and the iterated Kantorovich method. For the approximating space we take the space of piecewise constant functions or the space of piecewise linear functions related to an arbitrary nonuniform grid. Error estimates are obtained.

Jacobi Spectral Galerkin and Iterated Methods for Nonlinear Volterra Integral Equation

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Modified projection method for Urysohn integral equations with non-smooth kernels

Consider a nonlinear operator equation x−K(x)=f, where K is a Urysohn integral operator with a Green’s function type kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials, previous authors have investigated approximate solution of this equation using the Galerkin and the iterated Galerkin methods. They have shown that the iterated Galerkin solution is superconvergent. In this paper, a solution obtained using the iterated modified projection method is shown to converge faster than the iterated Galerkin solution. The improvement in the order of convergence is achieved by retaining the size of the system of equations same as for the Galerkin method. Numerical results are given to illustrate the improvement in the order of convergence.

A Note on Asymptotic Expansions to Approximate Eigenvalues of Integral Operators with Green's Kernels using the Iterated Galerkin Method

In this article, we consider approximation of eigenvalues of integral operators with Green's function-type kernels using the iterated Galerkin method. We obtain asymptotic expansions for approximate eigenvalues. The Richardson extrapolation is used to obtain eigenvalue approximations of higher order. A numerical example is considered in order to illustrate our theoretical results.

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and nonconforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

Superconvergence of Some Projection Approximations for Weakly Singular Integral Equations Using General Grids

This paper deals with superconvergence phenomena in general grids when projection-based approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme.

Numerical solution of Urysohn integral equations using the iterated collocation method

In this paper, we analyse the iterated collocation method for the nonlinear operator equation x = y+K(x) with K a smooth kernel. The paper expands the study begun by H. Kaneko and Y. Xu concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. Let x* denote an isolated fixed point of K. Let X n , n≥1, denote a sequence of finite-dimensional approximating subspaces, and let P n be a projection of X onto X n . The projection method for solving x = y+K(x) is given by x n = P n y+P n K(x n ), and the iterated projection solution is defined as . We analyse the convergence of {x n } and {} to x*, giving a general analysis that includes the collocation method. A detailed analysis is then given for a large class of Urysohn integral operators in one variable, showing the superconvergence of {} to x*.