The Numerical Validation of the Adomian Decomposition Method for Solving Volterra Integral Equation with Discontinuous Kernels Using the CESTAC Method

The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to sh...

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Bibliographic Details
Published in:Mathematics (Basel) 2021, Vol.9 (3), p.260
Main Authors: Noeiaghdam, Samad, Sidorov, Denis, Wazwaz, Abdul-Majid, Sidorov, Nikolai, Sizikov, Valery
Format: Article
Language:English
Subjects:
Adomian decomposition method
Algorithms
Approximation
CADNA library
CESTAC method
Computer engineering
Computers
Decomposition
discontinuous kernel
Errors
Floating point arithmetic
Food science
Integral equations
Iterative methods
Kernels
Libraries
Mathematical problems
Methods
Partial differential equations
Random variables
Theorems
volterra integral equation
Volterra integral equations
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Summary:The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9030260