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.

On generalized fractional operators and a gronwall type inequality with applications

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5457-5473 ◽  
Author(s):  
Yassine Adjabi ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Type Inequality ◽  
Weighted Spaces ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Derivatives

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

scholarly journals Iterative Bernstein splines technique applied to fractional order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zoltan Satmari
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Spline Approximation ◽  
Initial Value ◽  
Numerical Examples ◽  
Fractional Order Differential Equations ◽  
Banach’S Fixed Point Theorem

<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

scholarly journals Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 614 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Integral Representations ◽  
Initial Value ◽  
Impulsive Conditions ◽  
Fractional Differential ◽  
Basic Concepts

In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case.

On fractional order derivatives and Darboux problem for implicit differential equations

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Aleksandr Vityuk
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Darboux Problem ◽  
Hyperbolic Differential Equations

AbstractIn this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of functional hyperbolic differential equations by using some fixed point theorems.

Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Satwinder Jit Singh ◽  
Anindya Chatterjee
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Fractional Order Systems ◽  
Order Operator ◽  
Differential Algebraic Equation ◽  
Initial Value ◽  
Numerical Examples ◽  
Excellent Accuracy

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy.

scholarly journals A Novel Interval Arithmetic Approach for Solving Differential-Algebraic Equations with ValEncIA-IVP

2009 ◽  
Vol 19 (3) ◽  
pp. 381-397 ◽  
Author(s):  
Andreas Rauh ◽  
Michael Brill ◽  
Clemens Günther
Keyword(s):  
Dynamical Systems ◽  
Interval Arithmetic ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Theoretical Background ◽  
Future Research ◽  
Algebraic Equations ◽  
Initial Value ◽  
Simulation Techniques

A Novel Interval Arithmetic Approach for Solving Differential-Algebraic Equations with ValEncIA-IVPThe theoretical background and the implementation of a new interval arithmetic approach for solving sets of differential-algebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on ValEncIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations. For the application to DAEs, ValEncIA-IVP has been extended by an interval Newton technique to solve nonlinear algebraic equations in a guaranteed way. In addition to verified simulation of initial value problems for DAE systems, the developed approach is applicable to the verified solution of the so-called inverse control problems. In this case, guaranteed enclosures for valid input signals of dynamical systems are determined such that their corresponding outputs are consistent with prescribed time-dependent functions. Simulation results demonstrating the potential of ValEncIA-IVP for solving DAEs in technical applications conclude this paper. The selected application scenarios point out relations to other existing verified simulation techniques for dynamical systems as well as directions for future research.

scholarly journals Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

2020 ◽  
Author(s):  
Andreas Rauh ◽  
Julia Kersten
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Conditions ◽  
Infinite Horizon ◽  
Robustness Analysis ◽  
Interval Uncertainty ◽  
Ongoing Research ◽  
Limit Values ◽  
Simulation Techniques ◽  
Past Data

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical <em>linear</em> ordinary differential equations, <em>linear</em> fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of &plusmn;20 dB per frequency decade and, respectively, deviate from integer multiples of &plusmn;&pi;/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.

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