Error estimates for semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth data

CALCOLO ◽  
2000 ◽  
Vol 37 (4) ◽  
pp. 181-205 ◽  
Author(s):  
A. K. Pani ◽  
R. K. Sinha
Keyword(s):  
Differential Equation ◽  
Error Estimates ◽  
Galerkin Approximation ◽  
Time Dependent ◽  
Nonsmooth Data ◽  
Integro Differential Equation

scholarly journals Monte-Carlo Galerkin Approximation of Fractional Stochastic Integro-Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Abdallah Ali Badr ◽  
Hanan Salem El-Hoety
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Monte Carlo ◽  
Stochastic Differential Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Galerkin Approximation ◽  
Integro Differential Equation ◽  
Uniqueness Of The Solution ◽  
Time Continuum

A stochastic differential equation, SDE, describes the dynamics of a stochastic process defined on a space-time continuum. This paper reformulates the fractional stochastic integro-differential equation as a SDE. Existence and uniqueness of the solution to this equation is discussed. A numerical method for solving SDEs based on the Monte-Carlo Galerkin method is presented.

scholarly journals Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 497 ◽  
Author(s):  
Ratinan Boonklurb ◽  
Ampol Duangpan ◽  
Phansphitcha Gugaew
Keyword(s):  
Differential Equation ◽  
Inverse Problems ◽  
Chebyshev Polynomials ◽  
Integration Method ◽  
Computational Cost ◽  
Main Tool ◽  
Time Dependent ◽  
Tikhonov Regularization Method ◽  
Direct And Inverse Problems ◽  
Integro Differential Equation

In this article, the direct and inverse problems for the one-dimensional time-dependent Volterra integro-differential equation involving two integration terms of the unknown function (i.e., with respect to time and space) are considered. In order to acquire accurate numerical results, we apply the finite integration method based on shifted Chebyshev polynomials (FIM-SCP) to handle the spatial variable. These shifted Chebyshev polynomials are symmetric (either with respect to the point x = L 2 or the vertical line x = L 2 depending on their degree) over [ 0 , L ] , and their zeros in the interval are distributed symmetrically. We use these zeros to construct the main tool of FIM-SCP: the Chebyshev integration matrix. The forward difference quotient is used to deal with the temporal variable. Then, we obtain efficient numerical algorithms for solving both the direct and inverse problems. However, the ill-posedness of the inverse problem causes instability in the solution and, so, the Tikhonov regularization method is utilized to stabilize the solution. Furthermore, several direct and inverse numerical experiments are illustrated. Evidently, our proposed algorithms for both the direct and inverse problems give a highly accurate result with low computational cost, due to the small number of iterations and discretization.

Some numerical graphs with successive approximation method of fractional integro–differential equation

2019 ◽  
Vol 8 (4) ◽  
pp. 36
Author(s):  
Samir H. Abbas
Keyword(s):  
Differential Equation ◽  
Successive Approximation ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Successive Approximation Method ◽  
Integro Differential Equation

This paper studies the existence and uniqueness solution of fractional integro-differential equation, by using some numerical graphs with successive approximation method of fractional integro –differential equation. The results of written new program in Mat-Lab show that the method is very interested and efficient. Also we extend the results of Butris [3].

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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