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.

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 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 Method for Second-Kind Volterra Integral Equations with a Weakly Singular Kernel

2012 ◽  
Vol 263-266 ◽  
pp. 3313-3316
Author(s):  
Yiao Yong Zhang ◽  
Hua Feng Wu
Keyword(s):  
Integral Equations ◽  
Source Function ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
L2 Norm ◽  
Spectral Galerkin Method ◽  
Rigorous Error ◽  
Theoretical Results

The Legendre spectral Galerkin method for Volterra integral equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L2 -norm and the L∞ -norm ) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results.

scholarly journals A Hybrid Function Approach to Solving a Class of Fredholm and Volterra Integro-Differential Equations

2020 ◽  
Vol 25 (2) ◽  
pp. 30
Author(s):  
Aline Hosry ◽  
Roger Nakad ◽  
Sachin Bhalekar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Function Approach ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Hybrid Function ◽  
Derivatives Of

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing the efficiency and validity of the method that we introduce.

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 Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Rezvan Ghoochani-Shirvan ◽  
Jafar Saberi-Nadjafi ◽  
Morteza Gachpazan
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Transform Method ◽  
Weakly Singular ◽  
Differential Transform ◽  
Fractional Differential

An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.

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.

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