Towards Solving Fractional Order Delay Variational Problems Using Euler Polynomial Operational Matrices

In this paper, we introduce an approximate method for solving fractional order delay variational problems using fractional Euler polynomials operational matrices. For this purpose, the operational matrices of fractional integrals and derivatives are designed for Euler polynomials. Furthermore, the delay term in the considered functional is also decomposed in terms of the operational matrix of the fractional Euler polynomials. It is applied and substituted together with the other matrices of the fractional integral and derivative into the suggested functional. The main equations are then reduced to a system of algebraic equations. Therefore, the desired solution to the original variational problem is obtained by solving the resulting system. Error analysis has been discussed. An illustrative example is given in order to illustrate that the proposed method is very accurate and efficient for solving such kinds of problems


Introduction ISSN: 0067-2904
Hussien and Mohammed Iraqi Journal of Science, 2023, Vol. 64, No. 9, pp: 4693-4703 4694 Fractional calculus is one of the most interdisciplinary fields of applied mathematics which deals with the derivative and integrals of any order.Nowadays, it is used to advance mathematical models of real-world phenomena in various areas of science and engineering [1][2][3].Fractional order derivatives are naturally related to the systems with memory that dominate most of the scientific system models.Models and applications containing fractional derivatives can be found in chemical physics, probability physics, astrophysics, and various fields of engineering [4][5][6].There are many definitions of a fractional derivative, The commonly known fractional derivatives are the classical Riemann-Liouville and Caputo derivative.Fractional derivatives and integrals of these Riemann-Liouville and Caputo types have a huge number of applications in many fields of science and engineering [7][8][9][10][11].
The calculus of variation has a long history of communications with other fields of mathematics such as differential equations, geometry and with physics.However, the calculus of variation has found applications in economics and some branches of engineering [12][13][14].A fractional calculus of variations problem is a problem in which either the objective functional or the constraint equations or both consist of at least one fractional derivative term.In recent years, many numerical and approximate methods have been used to solve fractional order problems such as the homotopy analysis method, variational iteration method, homotopy perturbation method, wavelet method, collocation method, spectral tau method, finite element method and other methods, see [15][16][17][18][19][20][21].Recently, many researchers used different functions and polynomials.For some orthogonal polynomials, the operational matrices of fractional integrals and derivatives have been derived such as Bernstein polynomials, the Legendre polynomials, Jacobi polynomials, Chebyshev polynomials and Laguerre polynomials [22][23][24][25][26]. Inclusion of delay in the fractional order variational problems seems to be opening new vistas, especially in the field of bioengineering [27].
In this paper, the fractional order Euler functions based on Euler polynomials are used for solving fractional order delay variational problems.The operational matrix is derived for the fractional integration.By using the operational matrix of fractional integration and the fractional order Euler functions, we convert the varational problem into a system of linear algebraic equations.Numerical solutions are obtained by solving this linear system.By comparing the exact solution with the numerical solution using the proposed method, we exhibit the precision and efficiency of the proposed technique for various values of .

Preliminaries and notations
In this section, we introduce some basic definitions of fractional calculus, namely the definition of Riemann-Liouville fractional order integral and Caputo fractional order derivative [28].

Definition 1:
The Riemann-Liouville fractional integral operator   of order α of a function  ∈   and  ≥ −1 is defined as follows: where Γ(α) is the Gamma function.

Fractional order Euler functions
In this section, we give some definitions and basic properties of the Euler polynomials that are used in this paper [29].

Euler polynomials Properties
The Euler polynomials have the following interesting properties for all where   () are the Bernoulli polynomials of order  for ,  = 0,1, … , which are defined as follows: Using the last property, the Euler polynomials can be expressed in the following matrix form: ) 2 (0)

Formulation of fractional order Euler functions
The fractional order Euler functions are constructed by replacing the variable  by   , ( > 0) in the Euler polynomials.Let the fractional order Euler functions   (  ) be the basis of degree  and denoted by    (), then by using Eq. ( 3) we get: +    () = 2  (9) and,   () =     () (10) where ) 1 (0) ] And the first fractional order Euler functions are given by: , and so on.Moreover, the fractional order Euler functions satisfy the following formula: Therefore, the fractional order Euler functions are complete basis over the interval [0,1].

Fractional order Euler functions approximation
A function () which is square integrable in [0,1] can be expanded as : .Then we can compute the matrix  by using Eq. ( 11).

Conclusion
This paper introduces an efficient technique for approximating the solution to fractional order delay variational problems using the operational matrices of the fractional Euler polynomials.The fractional derivative in the present paper is defined in the Caputo sense.The unknown function was decomposed in terms of the fractional Euler polynomials operational matrices which contain the unknown vector of coefficients.The proposed technique converted the original variational problem into a system of algebraic equations.Solving the resulting system gives us the unknown coefficients vector and then the desired solution.The numerical results approve that the proposed method is accurate and relatively simple to implement and has good accuracy.

Figure 1 : 2 :
Figure 1: Represent the Error curve between the Figure 2: Represent the approximate solution for App.solution and the exact solution. = 1 and the exact solution at same .

Table 1 :
Comparison between the approximate solution of Eq. (22) for different values of  and the exact solution