An Efficient Numerical Scheme for Solving Fractional Optimal Control Problems. 1 Introduction

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1 ISS (print), (online) International Journal of onlinear Science Vol.14(1) o.3,pp An Efficient umerical Scheme for Solving Fractional Optimal Control Problems M. M. Khader, A. S. Hendy Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt (Received 31 May 1, accepted 3 August 1) Abstract: This paper presents an accurate numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. In this technique, we approximate FOCPs and end up with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The fractional derivative approximated using the proposed formula here, along with Clenshaw and Curtis procedure for the numerical integration of a non-singular functions and the Rayleigh-Ritz method for the constrained extremum are considered. By the proposed method the given FOCP is reduced to a problem for solving a system of algebraic equations and by solving it, we obtain the solution of FOCP. The error upper bound of the obtained approximate formula is proved. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Keywords: Fractional optimal control problems; Caputo fractional derivatives; Clenshaw and Curtis formula; Rayleigh-Ritz method; Chebyshev FDM; convergence analysis. 1 Introduction This paper presents a general fractional Chebyshev finite difference method (FCheb-FDM) for solving a class of fractional optimal control problems. Fractional derivatives have recently played a significant role in many areas of sciences, engineering, fluid mechanics, biology, physics and economies [1]. Some numerical methods for solving fractional differential equations (FDEs) were appeared in ([]-[14]). Many real-world physical systems display fractional order dynamics, that is their behavior is governed by fractional order differential equations [15]. For example, it has been illustrated that materials with memory and hereditary effects, and dynamical processes, including gas diffusion and heat conduction, in fractal porous media can be more adequately modeled by fractional order models than integer order models [16]. The general definition of an optimal control problem ([17]-[19]) requires the minimization of a criterion function of the states and control inputs of the system over a set of admissible control functions. The system is subject to constrained dynamics and control variables. Additional constraints such as final time constraints can be considered. This paper introduces an original formulation and a general numerical scheme for a potentially almost unlimited class of FOCPs. FOCP is an optimal control problem in which the criterion and/or the differential equations governing the dynamics of the system contain at least one fractional derivative operator. A general formulation and a solution scheme for FOCPs were first introduced in [] where fractional derivatives were introduced in the Riemann-Liouville sense, and FOCP formulation was expressed using the fractional variational principle and the Lagrange multiplier technique. The state and the control variables were given as a linear combination of test functions, and a virtual work type approach was used to obtain solutions. In ([1], []), the FOCPs are formulated using the definition of Caputo fractional derivatives, the FDEs are substituted into Volterra-type integral equations and a direct linear solver helps calculating the solution of the obtained algebraic equations. The Grünwald and Letnikov formula is used as an approximation and the resulting equations are solved using a direct scheme. Frederico and Torres ([3], [4]), used similar definitions of the FOCPs, formulated a oether-type theorem in the general context of the fractional optimal control in the sense of Caputo and studied fractional conservation laws in FOCPs. In [5] a numerical technique in terms of Caputo operators for solving FOCPs based on the Legendre orthonormal polynomial basis is introduced. Corresponding author. address: mohamedmbd@yahoo.com Copyright c World Academic Press, World Academic Union IJS /667

2 88 International Journal of onlinear Science, Vol.14(1), o.3, pp Clenshaw and Curtis [6] introduced a procedure for the numerical integration of a non-singular function y(x) by expanding the function in a series of Chebyshev polynomials and integration term by term. Elbarbary introduced Chebyshev finite difference approximation for the boundary value problems of integer derivatives ([7]-[9]). The fractional derivatives of the function y(x) at the point x k, k are expanded as linear combination from the values of the function y(x) at the shifted Gauss-Lobatto points x k = L L πk cos( ), k =, 1,..., associated with the interval [, L]. In addition to the procedure for the numerical integration introduced by Clenshaw and Curtis to approximate the integral by a finite sum. Finally, we approximate the variational problem and end up with a finite dimensional problem. The main characteristic of this new technique is that it gives a straight-forward algorithm in converting FOCPs to a system of algebraic equations. The suggested method is more accurate in comparison to the finite difference and finite element methods as the approximation of the fractional derivatives is defined over the whole domain. The main aim of the presented paper is concerned with an extension the previous work on fractional differential equations and derive some general approximate formulae of fractional Chebyshev-FDM and then we applied this approach to obtain the numerical solution of FOCPs. Also, we present the study of the convergence analysis of the proposed method. The structure of this paper is arranged in the following way: In section, we introduce some basic definitions about Caputo fractional derivatives and properties the shifted Chebyshev polynomials. In section 3, we give the basic formulation of the proposed operational matrix method using FCheb-FDM. In section 4, we prove an error upper bound of the approximated fractional derivatives. In section 5, we introduce the numerical scheme for solving FOCPs. In section 6, we give numerical examples to solve FOCPs and show the accuracy of the presented method. Finally, in section 7, the paper ends with a brief conclusion and some remarks. Preliminaries and notations In this section, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory required for our subsequent development..1 The Caputo fractional derivatives Definition 1 The Caputo fractional derivative operator D (ν) of order ν is defined in the following form: D (ν) f(x) = where m 1 < ν m, m. 1 Γ(m ν) x f (m) (ξ) dξ, ν >, x >, (x ξ) ν m+1 Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation where λ and µ are constants. For the Caputo s derivative we have D (ν) (λf(x) + µg(x)) = λ D (ν) f(x) + µ D (ν) g(x), (1) D (ν) C =, C is a constant, () {, for n D (ν) x n and n < ν ; = Γ(n+1) (3) Γ(n+1 ν) xn ν, for n and n ν. We use the ceiling function ν to denote the smallest integer greater than or equal to ν and = {, 1,,...}. Recall that for ν, the Caputo differential operator coincides with the usual differential operator of integer order. For more details on fractional derivatives definitions and its properties see ([3], [31]).. The definition and properties of the shifted Chebyshev polynomials The well known Chebyshev polynomials are defined on the interval [ 1, 1] and can be determined with the aid of the following recurrence formula IJS for contribution: editor@nonlinearscience.org.uk

3 M. M. Khader, A.S. Hendy: An Efficient umerical Scheme for Solving Fractional Optimal Control Problems 89 T n+1 (z) = z T n (z) T n 1 (z), T (z) = 1, T 1 (z) = z, n = 1,,.... It is well known that T n ( 1) = ( 1) n, T n (1) = 1. The analytic form of the Chebyshev polynomials T n (z) of degree n is given by T n (z) = n/ where n/ denotes the integer part of n/. The orthogonality condition is 1 1 ( 1) i n i 1 n (n i 1)! (i)! (n i)! zn i, (4) i= T i (z) T j (z) 1 z π, for i = j = ; π dz =, for i = j ;, for i j. In order to use these polynomials on the interval [, L] we define the so called shifted Chebyshev polynomials by introducing the change of variable z = x L 1. The shifted Chebyshev polynomials are defined as (5) T n(x) = T n ( x L 1), where T (x) = 1, T 1 (x) = x L 1. The analytic form of the shifted Chebyshev polynomial T n(x) of degree n is given by T n(x) = n ( 1) n k where, T n() = ( 1) n, T n(l) = 1. The orthogonality condition of these polynomials is L k= (n + k 1)!k (n k)!(k)!l k xk, (6) T j (x)t k (x)w(x)dx = δ jk h k, (7) 1 where, the weight function w(x) =, h Lx x k = b k π, with b =, b k = 1, k 1. The function y(x) which belongs to the space of square integrable in [, L], may be expressed in terms of shifted Chebyshev polynomials as y(x) = c n Tn(x), where the coefficients c n are given by c n = 1 h n L y(x)t n(x)w(x)dx, n =, 1,,.... (8) 3 Basic formulation of the operational matrix method using FCheb-FDM The well known shifted Chebyshev polynomials of the first kind of degree n are defined on the interval [, L] as in Eq.(6). We choose the grid (interpolation) points to be the Chebyshev- Gauss Lobatto points associated with the interval [, L], x r = L L πr cos( ), r =, 1,...,. These grids can be written as L = x < x 1 <... < x 1 < x =. Clenshaw and Curtis [6] introduced an approximation of the function y(x), we reformulate it to be used on the shifted Chebyshev polynomials as follows, y (x) = a n Tn(x), a n = y(x r ) T n(x r ). (9) The summation symbol with double primes denotes a sum with both first and last terms halved. The fractional derivative of the function y(x) at the point x s is given in the following theorem. r= IJS homepage:

4 9 International Journal of onlinear Science, Vol.14(1), o.3, pp Theorem 1 The fractional derivative of order ν in the Caputo sense for the function y(x) at the point x s is given by such that d (ν) s,r = 4θ r n= ν j= k= ν D (ν) y (x s ) = r= d (ν) s,r y(x r ), ν >, (1) nθ n b j ( 1) n k (n + k 1)! Γ(k ν + 1 ) L ν Γ(k + 1 ) (n k)! Γ(k ν j + 1) Γ(k ν + j + 1) T n(x r ) T j (x s ), where, s, r =, 1,,..., with θ = θ = 1, θ i = 1 i = 1,,..., 1. Proof. The fractional derivative of the approximate formula for the function y(x) in Eq.(9) is given by Employing Eqs.() and (3) in Eq. (6) we have then, D (ν) y (x) = a n D (ν) Tn(x). (11) D (ν) T n(x) =, n =, 1,..., ν 1, D (ν) y (x) = a n D (ν) Tn(x). (1) n= ν Therefore, for n = ν, ν + 1,..., and by using Eqs.()-(3), in for formula (6) we get D (ν) T n(x) = n n k (n + k 1)! k ( 1) (n k)! (k)! L k D(ν) x k = n k= ( 1) n k (n + k 1)! k k! (n k)! (k)! L k Γ(k ν + 1) xk ν. k= ν ow, x k ν can be expressed approximately in terms of shifted Chebyshev series, so we have (13) x k ν = c kj Tj (x), (14) j= where, c kj is obtained from (8) with y(x) = x k ν [3]. If only the first (+1)-terms from shifted Chebyshev polynomials in Eq.(9) are considered, the approximate formula for the fractional derivative of the shifted Chebyshev polynomials introduced by Doha [3] as follows D (ν) T n(x) = From Eqs.(1) and (15), we have D (ν) y (x) = 4 j= k= ν n= ν r= ( 1) n k n (n + k 1)! Γ(k ν + 1 ) b j L ν Γ(k + 1 ) (n k)! Γ(k ν j + 1) Γ(k ν + j + 1) T j (x). (15) j= k= ν ( 1) n k n(n + k 1)!Γ(k ν + 1 )y(x r)tn(x r )Tj (x) b j L ν Γ(k + 1 (16) ) (n k)!γ(k ν j + 1)Γ(k ν + j + 1). From Eq.(16), the fractional derivative of order ν for the function y(x) at the point x s leads to the desired result. The coefficients d (ν) s,r which are defined in Theorem 1 are the elements of the s-th row of the matrix D ν which is defined in the following relation [y (ν) ] = D ν [y], where, D ν is a square matrix of order (+1) and the column matrices [y (ν) ] and [y] are given by y (ν) r = y (ν) (x r ) and y r = y(x r ), respectively. IJS for contribution: editor@nonlinearscience.org.uk

5 M. M. Khader, A.S. Hendy: An Efficient umerical Scheme for Solving Fractional Optimal Control Problems 91 4 Error bound for the approximate fractional derivatives In this section, we will derive an error upper bound for the approximate formula of the fractional derivative D (ν) of the function y(x). To acheive this aim, we state and prove the following two theorems. Theorem [33] Suppose that H is a Hilbert space and Y is a closed subspace of H such that dimy < and y 1, y,..., y n is a basis for Y. Let x be an arbitrary element in H and y be the unique best approximation to x out of Y. Then x y = G(x; y 1, y,..., y n ) G(y 1, y,..., y n ), where G(x; y 1, y,..., y n ) = x, x x, y 1 x, y n y 1, x y 1, y 1 y 1, y n y n, x y n, y 1 y n, y n Theorem 3 The error upper bound for the approximated fractional derivative D (ν) of the function y(x) is defined in the following manner ( G(x D (ν) y(x) D (ν) y (x) k ν ; T,..., T a n Ω ) ) 1 n G(T,..., T ), (17) where Ω n = k= ν ( 1) n k n (n + k 1)! Γ(k ν + 1 ) b j L ν Γ(k + 1 (18) ) (n k)! Γ(k ν j + 1) Γ(k ν + j + 1). Proof. Consider the following statement in which we call e ν D the error vector of the operational matrix e ν D := D (ν) y(x) D (ν) y (x), e ν D = [ e ν D e ν D 1 e ν D ] T. From the approximate relation (14) and according to Theorem 4.1, we have x k ν ( G(x c kj Tj k ν ; T,..., T (x) = ) ) 1 G(T,..., T ), (19) j= also, according to Eqs.(15) and (19) we have so, we obtain e ν Dj = D (ν) T n(x) ( G(x Ω n Tj k ν ; T,..., T (x) Ω ) ) 1 n G(T,..., T ), () j= D (ν) y(x) D (ν) y (x) = A combination from Eqs.() and (1) leads to the desired result. [ ] a n D (ν) Tn(x) Ω n Tj (x). (1) 5 Fractional Chebyshev finite difference formulation for FOCPs In this section, we implement the proposed algorithm for solving FOCPs. Our problem formulation as follows subject to the system dynamics minimum J(u) = 1 L j= [ p(t)x (t) + q(t)u (t) ] dt, < t < L, () IJS homepage:

6 9 International Journal of onlinear Science, Vol.14(1), o.3, pp D (ν) x(t) = a(t)x(t) + b(t)u(t), (3) and subject to the initial condition x() = x. Where α represents the order of fractional derivatives, p(t), q(t) >, b(t) and a(t) are given functions. Here u(t) is the control variable and x(t) is the state variable. For more details about this model and its existence and uniqueness of the solution of such problem see ([], [5], [34]). The procedure of the implementation is given by the following steps: 1. Substitute by Eq.(3) into Eq.() gives minimum J(u) = 1 L [ p(t)x (t) + q(t) (b(t)) (D(ν) x(t) a(t)x(t)) ] dt. (4). Approximate the function x(t) using Clenshaw and Curtis formula (9) and its Caputo fractional derivative D (ν) x(t) using the proposed formula (1), then Eq.(4) transformed to the following approximated form J = 1 L where d (ν) r,s is defined in (1). [ p(t)( a n Tn(t)) + q(t) (b(t)) ( d (ν) r,s x(t r ) a(t) a n Tn(t)) ] dt, (5) 3. Use the transformation t = L (η + 1) to change Eq.(5) to the following form J = r= [ p(η)( a n Tn(η)) + q(η) (b(η)) ( d (ν) r,s x(η r ) a(η) a n Tn(η)) ] dη. (6) 4. Use the formula [6] developed by Clenshaw and Curtis 1 F ( η ) dη = m m θ s F (η s )[ T 1 m i + 1 s (η i ) Ts (η i+ ) ], (7) where s= i= θ = θ m = 1, θ s = 1 s = 1,,..., m 1, η i = cos[(πi)/m] i < m, η i = 1 i > m, to approximate the definite integral (6) as a finite sum of shifted Chebyshev polynomials in the following approximated formula J(x(t ),..., x(t )) = 1 m m θ s ( p(ηs )( a n T m i + 1 n(η s )) (8) s= i= r= r= + q(η s) (b(η s )) ( d (ν) r,s x(η r ) a(η s ) a n Tn(η s )) )[ Ts (η i ) Ts (η i+ ) ]. 5. The extremal values of functionals of the general form () or (8), according to Rayleigh-Ritz method gives J x(t 1 ) =, J x(t ) =,..., J x(t ) =, so, after using the initial condition, we obtain a system of algebraic equations. 6. Solve the resulting algebraic system by using the ewton iteration method to obtain x(t 1 ), x(t ),..., x(t ), then the state function x(t) which extremes FOCPs and the control variable u(t) have the following form (after using the inverse transform) x(t) = x(t r ) T n(t r ) Tn(t), (9) r= u(x) = 1 ( D (ν) x(t) a(t)x(t) ). (3) b(t) IJS for contribution: editor@nonlinearscience.org.uk

7 M. M. Khader, A.S. Hendy: An Efficient umerical Scheme for Solving Fractional Optimal Control Problems 93 6 Applications and numerical results In this section, we demonstrate the capability of the introduced approach. For this aim, we solve two widely used examples from the literature. The introduced problems are stated in the traditional FOCP framework and then reformulated via our introduced methodology. Problem 1: (Linear time-invariant problem) Consider the following linear time invariant fractional optimal control problem ([], [5]) minimum J(u) = 1 1 [ x (t) + u (t) ] dt, < t < 1, (31) subject to the system dynamics D (ν) x(t) = x(t) + u(t), < ν 1, (3) and subject to the initial condition x() = 1. Our aim is to find the control u(t) which minimizes the performance index J. For this problem we have the exact solution in the case of ν = 1 as follows x(t) = cosh( t) + β sinh( t), u(t) = (1 + β) cosh( t) + ( + β) sinh( t), where β = cosh( ) + sinh( ) cosh( ) + sinh( ).98. The behavior of the numerical solutions of this problem with different values of and ν are given in figures 1-3. Where in figures 1-, the numerical solutions x(t) and u(t), are plotted, respectively, at = 3 for different values of ν and the exact solution at ν = 1. In figure 3, the numerical solutions x(t) at ν =.8 for different values of ( = 3, 4, 5) are plotted. From these figures, we can conclude that the numerical results obtained by using the proposed method are in excellent agreement with the exact solution and the numerical results obtained using the numerical techniques ([], [5]). Problem : (Linear time-variant problem) In this example, we consider the linear time varying fractional optimal control problem ([], [5]) minimum J(u) = 1 1 [ x (t) + u (t) ] dt, < t < 1, (33) subject to the system dynamics D (ν) x(t) = tx(t) + u(t), < ν 1, (34) and subject to the initial condition x() = 1. Our aim is to find the control u(t) which minimizes the performance index J. The behavior of the numerical solutions of this problem with different values of and ν are given in figures 4-5. Where in figure 4, the numerical solutions x(t) at = 3 for different values of ν and the exact solution at ν = 1 are plotted. In figure 5, the numerical solutions x(t) at ν =.8 for different values of ( = 3, 4, 5) are plotted. From these figures, we can conclude that the numerical results obtained by using the proposed method are in excellent agreement with the exact solution and the numerical results obtained using the numerical techniques ([], [5]). IJS homepage:

8 94 International Journal of onlinear Science, Vol.14(1), o.3, pp Figure 1: The behavior of x(t) for problem 1 at = 3 with different values of ν. Figure : The behavior of u(t) for problem 1 at = 3 with different values of ν. Figure 3: The behavior of x(t) for problem 1 at ν =.8 with different values of. Figure 4: The behavior of x(t) for problem at = 3 with different values of ν. Figure 5: The behavior of x(t) for problem at ν =.8 with with different values of. IJS for contribution: editor@nonlinearscience.org.uk

9 M. M. Khader, A.S. Hendy: An Efficient umerical Scheme for Solving Fractional Optimal Control Problems 95 7 Conclusions In this article, we introduced an accurate numerical scheme for solving a wide class of fractional optimal control problems. In the proposed method, the FOCP is transformed to a system of algebraic equations. The error upper bound of the obtained approximate formula of the fractional derivative is stated and proved. The numerical results show that the algorithm converges as the number of terms is increased. The solution is expressed as a truncated Chebyshev series and so it can be easily evaluated for arbitrary values of t using any computer program without any computational effort. From illustrative examples, it can be seen that the proposed approach can obtain very accurate and satisfactory results. For all examples, the solution for the integer order case of the problem is also obtained for the purpose of comparison. In the end, from our numerical results using the proposed method, we can conclude that, the solutions are in excellent agreement with the exact solution and with the numerical results obtained using Lnite element method and the Legendre approximation method. All computational results are made using Matlab program 8. Acknowledgement The authors are very grateful to the editor and referees for carefully reading the paper and for their comments and suggestions which have improved the paper. References [1] R. L. Bagley and P. J. Torvik. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech., 51(1984): [] M. M. Khader. On the numerical solutions for the fractional diffusion equation. Communications in onlinear Science and umerical Simulation, 16(11): [3] M. M. Khader. Introducing an efficient modification of the variational iteration method by using Chebyshev polynomials. Application and Applied Mathematics: An International Journal, 7(1)(1): [4] M. M. Khader and A. S. Hendy. The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method. International Journal of Pure and Applied Mathematics, 74(1)(3): [5] M. M. Khader,. H. Sweilam and A. M. S. Mahdy. An efficient numerical method for solving the fractional diffusion equation. J. of Applied Math. and Bioinformatics, 1(11):1 1. [6]. H. Sweilam and M. M. Khader. A Chebyshev pseudo-spectral method for solving fractional order integrodifferential equations. AZIAM, 51(1): [7]. H. Sweilam, M. M. Khader and A. M. agy. umerical solution of two-sided space-fractional wave equation using finite difference method. Journal of Computional and Applied Mathematics, 35(11): [8]. H. Sweilam, M. M. Khader and A. M. S. Mahdy. Crank-icolson finite difference method for solving timefractional diffusion equation, Journal of Fractional Calculus and Applications, (1):()1 9. [9]. H. Sweilam and M. M. Khader. On the convergence of VIM for nonlinear coupled system of partial differential equations. Int. J. of Computer Maths., 87(1):(5) [1]. H. Sweilam, M. M. Khader and F. T. Mohamed. On the numerical solutions of two dimensional Maxwell s equations. Studies in onlinear Sciences, 1(1):(3)8 88. [11]. H. Sweilam, M. M. Khader and A. M. S. Mahdy. umerical studies for fractional-order Logistic differential equation with two different delays. Accepted in Journal of Applied Mathematics, (1):1-14. [1] G. D. Smith. umerical Solution of Partial Differential Equations. Oxford University Press, [13] M. A. Snyder. Chebyshev Methods in umerical Approximation. Prentice- Hall, Inc. Englewood Cliffs,. J [14] I. Podlubny. Fractional Differential Equations. Academic Press, ew York, [15] A. Oustaloup, F. Levron and B. Mathieu. Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Transactions on Circuits and Systems, 47():5 39. [16] M. Zamani, M. Karimi-Ghartemani and. Sadati. FOPID controller design for robust performance using particle swarm optimization. Journal of Fractional Calculus and Applied Analysis (FCAA), 1(7):() [17] C. Tricaud, Y. Q. Chen. An approximation method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59(1): IJS homepage:

10 96 International Journal of onlinear Science, Vol.14(1), o.3, pp [18] O. P. Agrawal, D. Baleanu. A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13(7): [19] O. P. Agrawal. On a general formulation for the numerical solution of optimal control problems, onlinear Dynam. 5(1989)(): [] O. P. Agrawal. A general formulation and solution scheme for fractional optimal control problems. onlinear Dynamics, 38(4)(1): [1] O. P. Agrawal. A quadratic numerical scheme for fractional optimal control problems, ASME Journal of Dynamic Systems, 13(8)(1): [] O. P. Agrawal. Fractional optimal control of a distributed system using eigenfunctions. ASME Journal of Computational and onlinear Dynamics, 3(8)(): [3] G. Frederico and D. Torres. Fractional conservation laws in optimal control theory. onlinear Dynamics, 53(8)(3):15. [4] G. Frederico and D. Torres. Fractional optimal control in the sense of Caputo and the fractional oethers theorem. International Mathematical Forum, 3(8)(1): [5] A. Lotfi, M. Dehghanb and S. A. Yousefi. A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications, 6(11): [6] C. Clenshaw and A. Curtis. A method for numerical integration of an automatic computer. umer. Math., (196): [7] E. M. E. Elbarbary and M. El-Kady. Chebyshev finite difference approximation for the boundary value problems. Appl. Math. Comput., 139(3): [8] E. M. E. Elbarbary and. S. Elgazery. Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder with variable properties and suction (numerical study). Acta Mechanica, 176(5):13 9. [9] A. K. Khalifa, E. M. E. Elbarbary and M. A. Abd-Elrazek. Chebyshev expansion method for solving second and fourth-order elliptic equations. Applied Mathematics and Computation, 135(3): [3] K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wily and Sons, Inc. ew York, [31] S. Samko, A. Kilbas and O. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London, [3] E. H. Doha, A. H. Bahrawy and S. S. Ezz-Eldien. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Modeling, 35(11): [33] E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley and sons, Inc., [34] E. R. Pinch. Optimal Control and the Calculus of Variations, Oxford University Press, IJS for contribution: editor@nonlinearscience.org.uk