System Models, Differential Equations, and Initial-Value Problems

2007 ◽  
pp. 1-45
Author(s):  
Panos J. Antsaklis ◽  
Anthony N. Michel
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Initial Value ◽  
System Models

scholarly journals Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations

2021 ◽  
Vol 5 (1) ◽  
pp. 17
Author(s):  
Andreas Rauh ◽  
Luc Jaulin
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Order System ◽  
Interval Methods ◽  
Initial Value ◽  
System Models ◽  
Simulation Techniques ◽  
Software Implementations

Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants.

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

scholarly journals Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations

2020 ◽  
Vol 6 (1) ◽  
pp. 7
Author(s):  
Fadugba Sunday Emmanuel ◽  
Adebayo Kayode James ◽  
Ogunyebi Segun Nathaniel ◽  
Okunlola Joseph Temitayo
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Initial Value
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