On the local superconvergence of the fully discretized multiprojection method for weakly singular Volterra integral equations of the second kind (original) (raw)
Related papers
Superconvergence of collocation methods for a class of weakly singular Volterra integral equations
Journal of Computational and Applied Mathematics, 2008
We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.
1993
DISCLAIMER X',ork performed under the auspices of the U.S. Departmeat of Energy by Lawrence i+hermore National Labora-tor_ under contract number %%'-7405-['NG-48. ]his document was prepared as an account of _ork SlUm_Jred b_ an agency of the United States Government. Neither the United States Government nor the University of California nor ant' of their emplo.vees, makes ant _arranty, expr_,ss or implied, or assumes an) le_zal liability or responsibilit._ for the accuraet, completeness, or usefulness of ant' information, apparatus, product, or process disclosed, or repr_:.,cnts that its use would not infringe privately o_ned right,,. Ref_.rence herein to ant' specific commercial products. process, or service by trade name. trademark, manufacturer, or otherwise, does not necessaril_ constitute or imply its endorsement, recommendation, or favoring bt the United Statt, s Government or the Universit._ of California. The vie_,_ and opinions of authors expressed herein do not necessaril._ _tate or reflect those of the United States Government or the I, ni_ersitt r of California, and shall not be used for ad_erti,_ing _,r product endor_ment porl_Ses. \ MULTIQUADRIC COLLOCATION METHODS IN THE NUMER.ICAL SOLUTION OF VOLTERRA INTEGRAL
Comparative study of numerical methods for a nonlinear weakly singular Volterra integral equation
HERMIS J. v7, 2006
This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. We investigate the application of product integration methods and a detailed analysis of the Trapezoidal method is given. In order to improve the numerical results we consider extrapolation procedures and collocation methods based on graded meshes. Several examples are presented illustrating the performance of the methods.
A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels
SIAM journal on numerical analysis, 2003
The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving (nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.
A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel
Journal of Computational and Applied Mathematics, 2010
This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.
A numerical method for the weakly singular Volterra integral equations
In this paper the method that presented by L.Tao and H.Young [1] for the nonlinear weakly singular Volterra integral equations of the second kind , have been considered and The convergence and error estimation have been presented. Then this method have been developed for the system of weakly singular Volterra integral equations. A convergence theorem and an error estimation is given.
Numerical solution of linear Volterra integral equations of the second kind with sharp gradients
Journal of Computational and Applied Mathematics, 2011
Collocation methods are a well developed approach for the numerical solution of smooth and weakly-singular Volterra integral equations. In this paper we extend these methods, through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar, Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous time random-walk in three-dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods where quadrature approximation and solution approximation are coupled.
A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations
Journal of Scientific Computing, 2019
In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel (x − s) −µ , 0 < µ < 1. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both L ∞-and weighted L 2-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change x → x 1/λ for a suitable real number λ. Finally a series of numerical examples are presented to demonstrate the efficiency of the method.