Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations

2010 ◽  
Vol 60 (8) ◽  
pp. 788-797 ◽  
Author(s):  
A.M.A. El-Sayed ◽  
I.L. El-Kalla ◽  
E.A.A. Ziada
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Analytical And Numerical Solutions ◽  
Fractional Orders

scholarly journals Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiyong Li ◽  
Siqing Gan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Analytical And Numerical Solutions ◽  
Mean Square Stability ◽  
Delay Differential

This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsizeΔt=τ/mwhen1/2≤θ≤1, and they are exponentially mean-square stable if the stepsizeΔt∈(0,Δt0)when0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.

scholarly journals Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
X. Liu ◽  
Y. M. Zeng
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Analytical And Numerical Solutions ◽  
Impulsive Delay Differential Equations ◽  
Nonlinear Delay Differential Equations ◽  
Impulsive Delay ◽  
Delay Differential ◽  
Nonlinear Delay

A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

scholarly journals Vector SDE Based Stochastic Analysis of Transformer

2021 ◽  
Vol 15 (1) ◽  
pp. 82-107
Author(s):  
Rawid Banchuin ◽  
Roungsan Chaisricharoen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Analysis ◽  
Numerical Solutions ◽  
Low Frequency ◽  
Equation Approach ◽  
Analytical And Numerical Solutions

In this research, the stochastic behaviours oftransformer have been analysed by using the stochasticdifferential equation approach where both noise in thevoltage source applied to the transformer and the randomvariations in elements and parameters of transformers havebeen considered. The resulting vector stochasticdifferential equations of the transformer have been bothanalytically and numerically solved in the Ito sense wherethe Euler-Maruyama scheme has been adopted fordetermining the numerical solutions which have been theirsample means have been used for verification. With theobtained analytical and numerical solutions, the stochasticproperties of the transformer’s electrical quantities havebeen studied and the influences of noise in the voltagesource and random variations in elements and parametersof transformers to those electrical quantities have beenanalysed. The causes of high and low frequency stochasticvariations of such electrical quantities in both transient andsteady state have been pointed out. Moreover, furtherextension of our stochastic differential equations and therelated mathematical formulations has also been given.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

2019 ◽  
Vol 28 (14) ◽  
pp. 1950247 ◽  
Author(s):  
Sadiye Nergis Tural-Polat
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Riccati Differential Equations ◽  
The Third ◽  
Chebyshev Wavelet ◽  
Fractional Orders

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

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