Exact solution of a simple time-dependent integro-differential equation by the method of Laplace transform and the theory of linear singular operators

1980 ◽  
Vol 71 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Rabindra Nath Das
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Laplace Transform ◽  
Time Dependent ◽  
Integro Differential Equation ◽  
Singular Operators

scholarly journals Successive Approximation Method of Integro–Differential Equation With Applications

2019 ◽  
Vol 8 (3) ◽  
pp. 96
Author(s):  
Samir H. Abbas ◽  
Younis M. Younis
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Successive Approximation ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Successive Approximations ◽  
Successive Approximation Method ◽  
Integro Differential Equation

The aim of this paper is studying the existence and uniqueness solution of integro- differential equations by using Successive approximations method of picard. The results of written program in Mat-Lab show that the method is very interested and efficient with comparison the exact solution for solving of integro-differential equation.

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.

scholarly journals Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2181
Author(s):  
Daniela Inoan ◽  
Daniela Marian
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Convolution Type ◽  
Convolution Product ◽  
Polynomial Functions ◽  
Integro Differential Equation ◽  
The Laplace Transform ◽  
The Stability ◽  
Rassias Stability

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

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