scholarly journals Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Yin Yang ◽  
Yunqing Huang
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Integrodifferential Equations ◽  
Collocation Methods ◽  
Volterra Type ◽  
Numerical Examples ◽  
Rigorous Error ◽  
Spectral Collocation Methods ◽  
Theoretical Results

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

Convergence Analysis of Legendre-Collocation Methods for Nonlinear Volterra Type Integro Equations

2015 ◽  
Vol 7 (1) ◽  
pp. 74-88 ◽  
Author(s):  
Yin Yang ◽  
Yanping Chen ◽  
Yunqing Huang ◽  
Wei Yang
Keyword(s):  
Error Analysis ◽  
Theoretical Prediction ◽  
Rate Of Convergence ◽  
Collocation Methods ◽  
Exponential Rate ◽  
Volterra Type ◽  
Numerical Errors ◽  
Legendre Collocation Method ◽  
Exponential Rate Of Convergence ◽  
Rigorous Error

AbstractA Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors inL2-norm andL∞-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.

scholarly journals Approximate Solution of Urysohn Integral Equations Using the Adomian Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Randhir Singh ◽  
Gnaneshwar Nelakanti ◽  
Jitendra Kumar
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Integral Equations ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Numerical Examples ◽  
Recursive Scheme ◽  
Urysohn Integral Equations ◽  
Adomian Decomposition

We apply Adomian decomposition method (ADM) for obtaining approximate series solution of Urysohn integral equations. The ADM provides a direct recursive scheme for solving such problems approximately. The approximations of the solution are obtained in the form of series with easily calculable components. Furthermore, we also discuss the convergence and error analysis of the ADM. Moreover, three numerical examples are included to demonstrate the accuracy and applicability of the method.

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 solution method for integro-differential equations of conformable fractional derivative

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 7-14 ◽  
Author(s):  
Mustafa Bayram ◽  
Veysel Hatipoglu ◽  
Sertan Alkan ◽  
Sebahat Das
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Solution Method ◽  
Approximate Solutions ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Numerical Examples

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

scholarly journals Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
F. Ghomanjani ◽  
A. Kılıçman ◽  
S. Effati
Keyword(s):  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Control Points ◽  
Volterra Type ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Numerical Examples ◽  
Piecewise Polynomials

A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using Bezier curves. The Bezier curves are chosen as piecewise polynomials of degreen, and Bezier curves are determined on[t0, tf ]by n+1control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.

scholarly journals On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 286
Author(s):  
Fairouz Tchier ◽  
Ioannis Dassios ◽  
Ferdous Tawfiq ◽  
Lakhdar Ragoub
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Efficient Method ◽  
Pseudospectral Method ◽  
Numerical Examples ◽  
Expansion Coefficients ◽  
Cardinal Functions ◽  
Theoretical Results ◽  
Chebyshev Functions

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

scholarly journals An Approximation of Ultra-Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Serhat Yılmaz
Keyword(s):  
Approximate Solution ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
Accuracy Difference ◽  
Initial Boundary Value ◽  
Theoretical Results

The first and second order of accuracy difference schemes for the approximate solution of the initial boundary value problem for ultra-parabolic equations are presented. Stability of these difference schemes is established. Theoretical results are supported by the result of numerical examples.

scholarly journals Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jieheng Wu ◽  
Guo Jiang ◽  
Xiaoyan Sang
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Numerical Solution ◽  
Numerical Method ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Haar Wavelets ◽  
Stochastic Integration ◽  
Numerical Examples

AbstractIn this paper, an efficient numerical method is presented for solving nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets. By the properties of Haar wavelets and stochastic integration operational matrixes, the approximate solution of nonlinear stochastic Itô–Volterra integral equations can be found. At the same time, the error analysis is established. Finally, two numerical examples are offered to testify the validity and precision of the presented method.

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