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 Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

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 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.

scholarly journals FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

2020 ◽  
Vol 4 (3) ◽  
pp. 313-322
Author(s):  
Sunday Obomeviekome Imoni ◽  
D. I. Lanlege ◽  
E. M. Atteh ◽  
J. O. Ogbondeminu
Keyword(s):  
Basis Function ◽  
Initial Value Problems ◽  
Hermite Polynomials ◽  
Multistep Method ◽  
Linear Multistep Method ◽  
Initial Value ◽  
Numerical Examples ◽  
First Order ◽  
Block Scheme ◽  
Multi Step Methods

ABSTRACT In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

scholarly journals Fractional order differential inclusions on an unbounded domain with infinite delay

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 167-178
Author(s):  
Mohamed Helal
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Fréchet Spaces ◽  
Initial Value ◽  
Nonlinear Alternative ◽  
Hyperbolic Differential Inclusions

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

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