scholarly journals Advantages of the Discrete Stochastic Arithmetic to Validate the Results of the Taylor Expansion Method to Solve the Generalized Abel’s Integral Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1370
Author(s):  
Eisa Zarei ◽  
Samad Noeiaghdam
Keyword(s):  
Integral Equation ◽  
Taylor Expansion ◽  
Expansion Method ◽  
Absolute Error ◽  
Optimal Error ◽  
Cestac Method ◽  
Stochastic Arithmetic ◽  
Cadna Library ◽  
Numerical Instabilities ◽  
Termination Criterion

The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, we use the CESTAC method and the CADNA library. Using this method, we can find the optimal step of the method, the optimal approximation, the optimal error, and some of numerical instabilities. They are the main novelties of the DSA in comparison with the FPA. The error analysis of the method is proved. Furthermore, the main theorem of the CESTAC method is presented. Using this theorem we can apply a new termination criterion instead of the traditional absolute error. Several examples are approximated based on the FPA and the DSA. The numerical results show the applications and advantages of the DSA than the FPA.

scholarly journals Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1031
Author(s):  
Samad Noeiaghdam ◽  
Sanda Micula
Keyword(s):  
Mathematical Model ◽  
Convergence Theorem ◽  
Malaria Infection ◽  
Absolute Error ◽  
Numerical Approximations ◽  
Cestac Method ◽  
Cadna Library ◽  
Numerical Instabilities ◽  
Better Than

This study focuses on solving the nonlinear bio-mathematical model of malaria infection. For this aim, the HATM is applied since it performs better than other methods. The convergence theorem is proven to show the capabilities of this method. Instead of applying the FPA, the CESTAC method and the CADNA library are used, which are based on the DSA. Applying this method, we will be able to control the accuracy of the results obtained from the HATM. Also the optimal results and the numerical instabilities of the HATM can be obtained. In the CESTAC method, instead of applying the traditional absolute error to show the accuracy, we use a novel condition and the CESTAC main theorem allows us to do that. Plotting several ℏ-curves the regions of convergence are demonstrated. The numerical approximations are obtained based on both arithmetics.

scholarly journals A Valid Dynamical Control on the Reverse Osmosis System Using the CESTAC Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 48
Author(s):  
Samad Noeiaghdam ◽  
Denis Sidorov ◽  
Alyona Zamyshlyaeva ◽  
Aleksandr Tynda ◽  
Aliona Dreglea
Keyword(s):  
Reverse Osmosis ◽  
Numerical Procedure ◽  
Optimal Solution ◽  
Absolute Error ◽  
Optimal Approximation ◽  
Successive Approximations ◽  
Mathematical Methods ◽  
Optimal Error ◽  
Integration Rule ◽  
Cestac Method

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Moreover, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library instead of the mathematical methods based on the floating point arithmetic (FPA). Applying this technique, we would be able to find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorems are proved to support the method analytically. Based on these theorems, we can apply a new stopping condition in the numerical procedure instead of the traditional absolute error. These theorems show that the number of common significant digits (NCSDs) of exact and approximate solutions are almost equal to the NCSDs of two successive approximations. The numerical results are obtained for both SE and DE Sinc integration rules based on the FPA and the SA. Moreover, the number of iterations for various ε are computed in the FPA. Clearly, the DE case is more accurate and faster than the SE for finding the optimal approximation, the optimal error and the optimal iteration of the RO system.

scholarly journals Numerical validation in quadruple precision using stochastic arithmetic

10.29007/5c91 ◽  
2018 ◽  
Author(s):  
Stef Graillat ◽  
Fabienne Jézéquel ◽  
Romain Picot ◽  
François Févotte ◽  
Bruno Lathuilière
Keyword(s):  
Double Precision ◽  
Rounding Errors ◽  
Numerical Validation ◽  
Stochastic Arithmetic ◽  
Quadruple Precision ◽  
Cadna Library ◽  
Numerical Instabilities ◽  
Significant Performance ◽  
Arbitrary Precision

Discrete Stochastic Arithmetic (DSA) enables one to estimate rounding errors and to detect numerical instabilities in simulation programs. DSA is implemented in the CADNA library that can analyze the numerical quality of single and double precision programs. In this article, we show how the CADNA library has been improved to enable the estimation of rounding errors in programs using quadruple precision floating-point variables, i.e. having 113-bit mantissa length. Although an implementation of DSA called SAM exists for arbitrary precision programs, a significant performance improvement has been obtained with CADNA compared to SAM for the numerical validation of programs with 113-bit mantissa length variables. This new version of CADNA has been successfully used for the control of accuracy in quadruple precision applications, such as a chaotic sequence and the computation of multiple roots of polynomials. We also describe a new version of the PROMISE tool, based on CADNA, that aimed at reducing in numerical programs the number of double precision variable declarations in favor of single precision ones, taking into account a requested accuracy of the results. The new version of PROMISE can now provide type declarations mixing single, double and quadruple precision.

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

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 260
Author(s):  
Samad Noeiaghdam ◽  
Denis Sidorov ◽  
Abdul-Majid Wazwaz ◽  
Nikolai Sidorov ◽  
Valery Sizikov
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Absolute Error ◽  
Floating Point ◽  
Cestac Method ◽  
Stochastic Arithmetic ◽  
Floating Point Arithmetic ◽  
Adomian Decomposition ◽  
Discontinuous Kernels ◽  
Point Arithmetic

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.

A Valid Scheme to Evaluate Fuzzy Definite Integrals by Applying the CADNA Library

2017 ◽  
Vol 6 (4) ◽  
pp. 1-20 ◽  
Author(s):  
Mohammad Ali Fariborzi Araghi ◽  
Samad Noeiaghdam
Keyword(s):  
Optimal Number ◽  
Fuzzy Integral ◽  
Step Size ◽  
Integration Rule ◽  
Stochastic Arithmetic ◽  
Definite Integrals ◽  
Cadna Library ◽  
Optimal Step Size ◽  
Termination Criterion ◽  
Romberg Integration

The aim of this paper is to estimate the value of a fuzzy integral and to find the optimal step size and nodes via the stochastic arithmetic. For this purpose, the fuzzy Romberg integration rule is considered as an integration rule, then the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied which is a method to describe the discrete stochastic arithmetic. Also, in order to implement this method, the CADNA (Control of Accuracy and Debugging for Numerical Applications) is applied which is a library to perform the CESTAC method automatically. A theorem is proved to show the accuracy of the results by means of the concept of common significant digits. Then, an algorithm is given to perform the proposed idea on sample fuzzy integrals by computing the Hausdorff distance between two fuzzy sequential results which is considered to be an informatical zero in the termination criterion. Three sample fuzzy integrals are evaluated based on the proposed algorithm to find the optimal number of points and validate the results.

scholarly journals Mathematical Modelling of  RLC Circuit  Through Mixed Quadrature Rule

2018 ◽  
Author(s):  
Saumya jena ◽  
Damayanti Nayak
Keyword(s):  
Integral Equation ◽  
Mathematical Modelling ◽  
Electromotive Force ◽  
Quadrature Rule ◽  
Absolute Error ◽  
Definite Integral ◽  
Numerical Result ◽  
Rlc Circuit ◽  
Instantaneous Current

In this study, a mixed rule of degree of precision nine has been developed and implemented in the field of electrical sciences to obtain the instantaneous current in the RLC- circuit for particular value .The linearity has been performed with the Volterra’s integral equation of second kind with particular kernel . Then the definite integral has been evaluated through the mixed quadrature to obtain the numerical result which is very effective. A polynomial has been used to evaluate Volterra’s integral equation in the place of unknown functions. The accuracy of the proposed method has been tested taking different electromotive force in the circuit and absolute error has been estimated.

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