Fractional Differential-algebraic Equations Research Articles

Mitigation of numerical issues appearing in transient analyses when applying fractional derivative approximations

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup’s method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup’s method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda’s method and Charef’s method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used.

Excitation system voltage regulator modeling with the use of fractional calculus

The article presents a new and measurement-verified approach in the modeling of a voltage regulator, which is used in the power industry and applied for the purpose of regulation of generating units. The voltage regulator extended model, proposed in this paper, involves nonlinearities and the application of fractional calculus. The addition of these features has not only significantly increased the number of parameters determining the model response, it has also established a challenge of developing a solver capable of computing the resulting complex system of fractional differential–algebraic equations (FDAE). The mathematical formulation of a single simulation of a transient process for the model has been described. The application of the Oustaloup Recursive Approximation (ORA) of the fractional derivative has been considered in order to increase the speed of the model evaluation. Generalized equations leading to an approximating system of Differential–Algebraic Equations (DAE) have been presented in detail. In order to obtain the parameters for the fractional nonlinear model, a parameter estimation procedure has been applied, which consists of implementations of several algorithms that are executed sequentially. The usefulness and drawbacks of the application of the ORA has also been discussed.

Analytical solutions for fractional partial delay differential-algebraic equations with Dirichlet boundary conditions defined on a finite domain

In this paper, we investigate the solution of multi-term time-space fractional partial delay differential-algebraic equations (MTS-FPDDAEs) with Dirichlet boundary conditions defined on a finite domain. We use Laplace transform method to give the solutions of multi-term time fractional delay differential-algebraic equations (MTS-FDDAEs). Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the MTS-FPDDAEs into the MTS-FDDAEs. By applying our obtained solutions to the resulting MTS-FDDAEs, the desired analytical solutions of the MTS-FPDDAEs are obtained. Finally, we give the solutions of some special cases.

Application of Said Ball Curve for Solving Fractional Differential-Algebraic Equations

The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is proved. Some examples are presented to show the efficiency and accuracy of the method. Based on the obtained results, the SBC is more accurate than the Bezier curve method.

Numerical solution of the time-fractional Navier–Stokes equations for incompressible flow in a lid-driven cavity

This paper addresses a computational technique for solving 2D unsteady Navier–Stokes equations (NSEs) with time-fractional order in the Caputo sense in the formulation of stream function-vorticity. The finite difference-based method of lines is used to discretize the time-fractional NSEs on a collocated grid that construct a fractional differential algebraic equations system. After solving the discretized complementary Poisson’s equation, this system is reduced to a system of fractional differential equations (FDEs). The resulting FDEs are solved by fractional backward differentiation formulas. The flow in a square lid-driven cavity is considered as the model problem.

A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

Piecewise Linear Approximate Solution of Fractional Order Non-Stiff and Stiff Differential-Algebraic Equations by Orthogonal Hybrid Functions Based Numerical Method

Piecewise Linear Approximate Solution of Fractional Order Non-Stiff and Stiff Differential-Algebraic Equations by Orthogonal Hybrid Functions Based Numerical Method

System of fractional differential algebraic equations with applications

Abstract One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo’s definition, Caputo–Fabrizio’s definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method.

Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations

In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate.

Nonnegativity of solutions of nonlinear fractional differential-algebraic equations

Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.

A new approach for solving fractional differential-algebraic equations

In this paper, the Bezier curves method is implemented to give approximate solutions for fractional differential-algebraic equations (FDAEs). This methods in applied mathematics can be used as approximated method for obtaining approximate solutions for different types of fractional differential equations. An illustrative example is included to demonstrate the validity and applicability of the suggested approach.

Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.

An efficient scheme for solving a system of time- fractional order differential-algebraic equations by using fractional Laplace iteration method

In this article, we propose an efficient algorithm for solving system of time- fractional differential-algebraic equations by using a fractional Laplace iteration method. The scheme is tested for some examples and the results demonstrate reliability and accuracy of this method.

New efficiency algorithm for solving fractional order differential-algebraic system

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System

The existence and uniqueness of the solution of a new kind of system—linear fractional differential-algebraic equations (LFDAE)—are investigated. Fractional derivatives involved are under the Caputo definition. By using the tool of matrix pair, the LFDAE in which coefficients matrices are both square matrices have unique solution under the condition that coefficients matrices make up a regular matrix pair. With the help of equivalent transformation and Kronecker canonical form of the coefficients matrices, the sufficient condition for existence and uniqueness of the solution of the LFDAE in which coefficients matrices are both not square matrices is proposed later. Two examples are given to justify the obtained theorems in the end.

An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives

In this paper, we propose an algorithm to obtain approximate solutions of Fractional Differential-Algebraic Equations with Caputo-type derivatives. The method, Iterative Decomposition Method presents solutions as rapidly convergent infinite series of easily computable terms. Numerical examples are considered to highlight the significant features of the IDM, as well as illustrate the efficiency and accuracy of the method, when compared with known methods.

An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives

In 1990s the government adopted D-by-D policy with a very good intention of reforming its local governments and let them enjoy the decision making powers promised by the D-by-D policy. One of the crucial of elements of decentralization was fiscal decentralization which gives the local jurisdictions to exercise adequate authority over important decisions with regard to setting development priorities, financing local services, revenue mobilization and expenditure authority continue. This article explored the perception of council on exercise of their fiscal authority in selected local authorities in Tanzania. The findings show that council exercise reasonable authority over revenues mobilization, the setting of the local tax rates and full expenditure authority over the locally generated revenue and also the discretionary grants.

Waveform relaxation method for fractional differential-algebraic equations

Abstract The waveform relaxation method has been successfully applied into solving fractional ordinary differential equations and fractional functional differential equations [11, 5]. In this paper, the waveform relaxation method is further used to solve fractional differential-algebraic equations, which often arise in integrated circuits with new memory materials. We give the iteration scheme of the waveform relaxation method and analyze the convergence of the method under linear and nonlinear conditions for the right-hand of the equations. Numerical examples illustrate the feasibility and efficiency of the method.

Numerical comparison of methods for solving fractional differential–algebraic equations (FDAEs)

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are implemented to give approximate solutions for fractional differential–algebraic equations (FDAEs). Both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. This paper presents a numerical comparison between these two methods and the homotopy analysis method (HAM) for solving FDAEs. Numerical results reveal that the VIM and the ADM are quite accurate and applicable.