Iterated Collocation Method Research Articles

A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t−s)−α(t) with 0≤α(t)<1. A Nyström method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, superconvergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nyström method.

Collocation and modified collocation methods for solving second kind Fredholm integral equations in weighted spaces

For the numerical solution of Fredholm integral equations on [−1,1] whose integrands have endpoint algebraic singularities, we investigate in this paper a modified collocation method based on the zeros of the Jacobi polynomials in appropriate weighted spaces. The proposed method converges faster than the standard collocation scheme, and the Sloan iteration can be used to make the solution even more accurate. The iterated collocation method is also defined in this paper. To the best of our knowledge, this work is the first to investigate superconvergent methods for such integral equations. Some numerical tests are presented to show the effectiveness of the suggested methods.

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays

Super-convergence analysis of collocation methods for linear and nonlinear third-kind Volterra integral equations with non-compact operators

This paper concerns iterated collocation methods for linear and nonlinear Volterra integral equations (VIEs) of the third kind with noncompact operators. The optimal global and local super-convergence orders are determined under a new mesh. Finally, some numerical examples are provided to verify the convergence results.

The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations

It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time $$t = 0$$ , only $$1 - \alpha $$ global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for $$m = 1$$ at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.

Approximate solution of 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 kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min \{ r, 1 \}$, respectively. We show that the order of convergence in the iterated modified projection method is $4$ if $ r = 0$ and is $ 2 r + 3$ if $ r \geq 1$. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples.

Composite quadrature rules for a class of weakly singular Volterra integral equations with noncompact kernels

A special class of weakly singular Volterra integral equations with noncompact kernels is considered. We consider a representation of the unique smooth solution of the equation and present a novel class of numerical approximations based on Gaussian quadrature rules. It is shown that the method of this type has a(n) (nearly) optimal rate of convergence under a specific condition which is very practical and easy to check. In some cases, the superconvergence property is also achieved. A stability analysis of the method is also provided. The method may be preferred to the iterated collocation method which is superconvergent under many conditions on the unknown solution. Some numerical examples are presented which are in accordance with the theoretical results.

Asymptotic Expansions for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

We consider approximation of eigenvalues of integral operators with Green's function kernels using the Nyström method and the iterated collocation method and obtain asymptotic expansions for approximate eigenvalues. We show that the Richardson extrapolation is applicable to find eigenvalue approximations of higher order and illustrate our results by numerical examples.

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.

The discrete collocation method for weakly singular Urysohn equations

In this paper, we analyse the iterated collocation method for the nonlinear Urysohn operator equation x=y+K(x) with K a singular kernel. The paper extends the study [H. Kaneko, R.D. Noren, and P.A. Padilla, J. Comput. Appl. Math. 80 (1997), pp. 335–349] in which the convergence of the iterated collocation method for Urysohn equations is considered.

Superconvergence of interpolated collocation solutions for Hammerstein equations

In this article, we discuss the superconvergence of the interpolated collocation solutions for Hammerstein equations. Applying this new interpolation postprocessing to the collocation approximation xh, we get a higher accuracy approximation I xh, whose convergence order is the same as that of the iterated collocation method. Such an interpolation postprocessing method is much simpler. Also, numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Closed-form error estimates for the numerical solution of Fredholm integral equations of the second kind

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel orthogonalpolynomial approximation of the exact kernel. It is proved that, under suitable conditions, the discrepancy between our theoretically predicted error and the actual numerical error tends to zero at the rate ‖K − KM‖, where M is the rank of the degenerate-kernel approximation. A leading-order error analysis is validated on three increasingly accurate projection methods applied to both smooth and sharply peaked kernels, our error predictions being demonstrated to be exponentially convergent to experimentally obtained global numerical errors. The new method is demonstrated to be cost-effective relative to standard extrapolation.

Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations

We discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular Volterra integral equation associated with certain heat conduction problems. This work completes the previous studies of numerical methods for this type of equations with noncompact kernel. In particular, a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces. Some numerical examples illustrate the theoretical results.

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*.

A new superconvergent collocation method for eigenvalue problems

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1 \leq r-1 , we show that the proposed method exhibits an error of the order of 4 r 4r for eigenvalue approximation and of the order of 3 r 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4 r 4r can be obtained. This improves upon the order 2 r 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r r and 2 r 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.

Approximate Solution of Multivariable Integral Equations of the Second Kind

For a multidimensional integral equation of the second kind with a smooth kernel, using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree r, Atkinson has established an order r + 1 convergence for the Galerkin solution and an order 2r+2 convergence for the iterated Galerkin solution. In a recent paper [15], a new method based on projections has been shown to give a 4r + 4 convergence for one-dimensional second kind integral equations. The size of the system of equations that must be solved in implementing this method remains the same as for the Galerkin method. In this paper, this method is extended to multi-dimensional second kind equations and is shown to have convergence of order 4r + 4. For interpolatory projections onto a space of piecewise polynomials, it is shown that the order of convergence of the new method improves on the previously established orders of convergence for the collocation and the iterated collocation methods. A two-grid norm convergent method based on the new method is also defined.

Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations

In this paper, we study the numerical solution of two-dimensional nonlinear Volterra integral equation by collocation and iterated collocation method. Asymptotic error expansion of iterated collocation solution is obtained. Using Richardson's extrapolation, we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly. Some numerical examples are given to illustrate this theory.

The Nyström method for solving a class of singular integral equations and applications in 3D-plate elasticity

The paper consists of two parts. In the first part we investigate a Nystrom- or product integration method for second kind singular integral equations. We prove an asymptotically optimal error estimate in the scale of Sobolev Hilbert spaces. Although the result can also be obtained as a special case of a discrete iterated collocation method our proof is more direct and uses the Nystrom interpolation. In the second part of this paper we consider the Dirichlet problem for thin elastic plates with transverse shear deformation. The boundary value problem is transformed into a 3 x 3 system of singular Fredholm integral equations of second kind. After discussing existence and uniqueness of the solution to the integral equations in a Sobolev space setting, we apply the Nystrom method to solve the integral equations numerically.

On mixed collocation methods for Volterra integral equations with periodic solution

In this paper we extend recent results for the numerical solution of Volterra integro-differential equations with periodic solution (Brunner, Makroglou and Miller, 1996) to the numerical solution of Volterra integral equations with periodic solution. We apply collocation and iterated collocation methods based on the mixed interpolation methods of De Meyer, Vanthournout and Vanden Berghe (1990). The existence of unique continuous periodic solutions is examined and a convergence analysis of the numerical method is given. Numerical results are included for comparison with the polynomial collocation methods based on Lobatto and Gauss points as collocation parameters.

Superconvergence of the iterated collocation methods for Hammerstein equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].