Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

2016 ◽  
Vol 8 (4) ◽  
pp. 648-669 ◽  
Author(s):  
Xiulian Shi ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Approximate Solutions ◽  
Weakly Singular Kernels ◽  
Numerical Examples ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Solutions Of Equations ◽  
Rigorous Error ◽  
Special Case

AbstractA spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially inL∞norm and weightedL2norm. Finally, two numerical examples are presented to demonstrate our error analysis.

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.

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 High Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second Kind

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Lu Pan ◽  
Xiaoming He ◽  
Tao Lü
Keyword(s):  
Differential Equations ◽  
Computational Cost ◽  
Posteriori Error ◽  
High Accuracy ◽  
Combination Method ◽  
Weakly Singular Kernels ◽  
A Posteriori Error ◽  
Weakly Singular ◽  
Accuracy Order ◽  
Singular Kernels

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.

scholarly journals Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

2021 ◽  
Vol 5 (3) ◽  
pp. 90
Author(s):  
Arvet Pedas ◽  
Mikk Vikerpuur
Keyword(s):  
Differential Equations ◽  
Special Choice ◽  
Spline Collocation ◽  
Numerical Illustration ◽  
Weakly Singular Kernels ◽  
Local Boundary ◽  
Weakly Singular ◽  
Regularity Properties ◽  
Singular Kernels ◽  
Non Local

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

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