scholarly journals A numerical study on the non-smooth solutions of the nonlinear weakly singular fractional integro-differential equations

2021 ◽  
Author(s):  
Sayed Arsalan Sajjadi ◽  
Hashem Saberi Najafi ◽  
Hossein Aminikhah
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Caputo Derivative ◽  
Numerical Study ◽  
Singular Kernel ◽  
Smooth Solutions ◽  
Weakly Singular Kernel ◽  
Numerical Examples ◽  
Weakly Singular

The solutions of weakly singular fractional integro-differential equations involving the Caputo derivative have singularity at the lower bound of the domain of integration. In this paper, we design an algorithm to prevail on this non-smooth behaviour of solutions of the nonlinear fractional integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

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 Second-order time discretization with finite-element method for partial integro-differential equations with a weakly singular kernel

1999 ◽  
Vol 40 (4) ◽  
pp. 513-524
Author(s):  
Chang Ho Kim ◽  
U Jin Choi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Time Discretization ◽  
Second Order ◽  
Singular Kernel ◽  
Element Approximation ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Element Method

AbstractWe propose the second-order time discretization scheme with the finite-element approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the finite element method and the time discretization is based on the Crank-Nicolson scheme with a graded mesh. We show the stability of the scheme and obtain the second-order convergence result for the fully discretized scheme.

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