scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRO‐DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 15 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Raul Kangro ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Approximate Solutions ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Fully Discrete ◽  
Weakly Singular ◽  
Good Accordance ◽  
Theoretical Results ◽  
Simple Quadrature

In order to find approximate solutions of Volterra and Fredholm integro‐differential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non‐smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRAL EQUATIONS

2009 ◽  
Vol 14 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Raul Kangro ◽  
Inga Kangro
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Nonuniform Grids ◽  
Fully Discrete ◽  
Straightforward Application ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations

A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

2012 ◽  
Vol 4 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Yunxia Wei ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Spectral Methods ◽  
Approximate Solutions ◽  
Smooth Solutions ◽  
Weakly Singular Kernel ◽  
Numerical Examples ◽  
Weakly Singular ◽  
Rigorous Error ◽  
Derivatives Of ◽  
Theoretical Results

AbstractThe theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel (t - s)->* with 0< μ <1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially inL°°-norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

scholarly journals Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

2021 ◽  
Author(s):  
Dũng Đinh
Keyword(s):  
Convergence Rates ◽  
Polynomial Interpolation ◽  
Hermite Polynomials ◽  
Adaptive Methods ◽  
Sparse Grid ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Elliptic Pdes ◽  
Fully Discrete ◽  
Weighted Quadrature

By combining a certain  approximation property in the spatial domain, and weighted summability  of the Hermite polynomial expansion coefficients  in the parametric domain, we investigate  linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We explicitly construct  such methods and prove convergence rates of the approximations by them.  The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are  certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of  successive differences of  their parametric Lagrange interpolating polynomials. The  Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the  corresponding integration can be estimated via the error in Bochner space.  We also briefly  consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and  problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

scholarly journals Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

2021 ◽  
Vol 5 (3) ◽  
pp. 70
Author(s):  
Esmail Bargamadi ◽  
Leila Torkzadeh ◽  
Kazem Nouri ◽  
Amin Jajarmi
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Algebraic System ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Chebyshev Wavelets ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Chebyshev Wavelet

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

scholarly journals PRODUCT INTEGRATION FOR WEAKLY SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 153-172 ◽  
Author(s):  
Arvet Pedas ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Discrete Version ◽  
Product Integration ◽  
Weakly Singular ◽  
Integration Techniques ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

scholarly journals Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

2017 ◽  
Vol 17 (3) ◽  
pp. 479-498 ◽  
Author(s):  
Raphael Kruse ◽  
Yue Wu
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Quadrature Rule ◽  
Time Variable ◽  
Important Ingredient ◽  
Riemann Sum ◽  
Runge Kutta ◽  
Theoretical Results

AbstractThis paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the {L^{p}(\Omega;{\mathbb{R}}^{d})}-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

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