Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations

Transcription

1 Progr. Fract. Differ. Appl., No., 99- (8) 99 Progress in Fractional Differentiation and Applications An International Journal Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations Kolade M. Owolabi,, Department of Mathematical Sciences, Federal University of Technology, PMB 7, Akure, Ondo State, Nigeria Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 93, South Africa Received: 7 April 7, Revised: 9 July 7, Accepted: 3 Oct. 7 Published online: Apr. 8 Abstract: In this work, the stability analysis and numerical treatment of chaotic time-fractional differential equations are considered. The classical system of ordinary differential equations with initial conditions is generalized by replacing the first-order time derivative with the Riemann-Liouville fractional derivative of order, for <. In the numerical experiments, we observed that analysis of pattern formation in time-fractional coupled differential equations at some parameter value is almost similar to a classical process. A range of chaotic systems with current and recurrent interests which have many applications in biology, physics and engineering are taken to address any points and queries that may naturally occur. Keywords: Chaotic systems, fractional calculus, numerical simulations, Riemann-Liouville derivative. Introduction Fractional calculus has a long history [ 3]. It is seen as the generalization of ordinary differentiation and integration to non-integer orders. Over the years, fractional calculus was found to play an important role in the modeling of a considerable number of real-life or physical phenomena; for instance, the predator-prey dynamic [, ], the modeling of memory-dependent and complex media such as porous media [ 9] and references therein. It has emerged as an essential tool for the study of dynamical systems where standard methods are not effective with strong limitations. A lot of research findings have revealed that maority of models that are based only on the integer (classical) order derivatives do not provide enough information to describe the complexity of such phenomena, on the basis of their mathematical and physical considerations. Some system of equations present delays which may be finite, infinite, or state-dependent. Others are subect to an impulsive effect. This paper aims to capture a wide reader of specialists such as biologist, economist, engineers, mathematicians as well as physicists on the application of fractional calculus to address various nonlinear problems. A general time-fractional differential equation of order is given by ϕ n D n t + + ϕ D t +ϕ D t () where ϕ δ R, δ,,,...,n are the coefficients of the differential equations. Without loss of generality, we let n > n > n > >. Following [3], we present the analytical solution of () in the general form ϕ n k n ( ϕi i ( ) k k! (k;δ,δ,δ,...,δ n ) δ +δ +δ + +δ n k ) ( δi χ k ϕ n t, ϕ n ϕ n ; n n, n + Corresponding author mkowolax@yahoo.com n ( n )δ + ), ()

2 K. M. Owolabi: Riemann-Liouville fractional derivative and application... where (k;δ,δ,δ,...,δ n ) are multinomial coefficients and the term χ δ (t,λ ;ξ,τ) denotes the Mittag-Leffler function type, bear in mind that δ ;δ ;...,δ n, as suggested by Podlubny [3, ]. This function is given by ( χ δ (t,λ ;ξ,τ)t ξ δ+τ E (δ) λt ξ), for δ,,,... ξ,τ where E (δ) (ν) is the δ th derivative of two parameters Mittag-Leffler function [] ξ,τ ξ,τ (ν) (i+δ)!ν i, for δ,,,... i Γ(ξ i+ξ δ + τ) i! E (δ) The Laplace transform of function χ δ (t,±λ ;,η) is defined by L{χ δ (t,±λ ;,η)} δ!y η (y ± λ) δ+. (3) for y> λ /. Readers are referred to [3, ] for a list of some useful Laplace transforms and their inverse functions. The non-local nature of fractional derivatives can be used to simulate accurately many natural phenomena containing long memory, for example, the groundwater and geo-hydrology models [, 7, 3]. In recent years, there has been considerable interest in formulating different methods to numerically solve various types of differential and integral equations. One of the outstanding techniques is the use of spectral methods [ ], due to their flexibility of implementation over finite and infinite intervals. A good survey of many appropriate numerical approximations can be found in [, 3]. Precisely speaking, we consider the fractional ordinary differential system of the form D,t f(t,), t (,T], T >, () where f C([,T] R,R), (,), and n>is an integer not less than. Here, we denote the term D,t is the Riemann-Liouville th-order derivative of function defined by d n t RL D,t Γ(n ) dt n (t ξ) n u (n) dξ, n < < n, n Z +. () The rest of this paper is structured as follows. There are several definitions of fractional derivatives, a quick tour based on the most useful and widely applied cases, such as the Riemann-Liouville, the Caputo, and the Grünwald-Letnikov types are briefly discussed in Section. Numerical method of approximation is given in Section 3. Modelling of three chaotic problems arising in biology, physics and financial economics is considered in Section. Finally, the main conclusion is outlined in Section. Preliminary Background In this section, we briefly introduced some of the useful preliminaries regarding fractional calculus theory [, 3, ]. The Riemann-Liouville fractional integral of order for a function C ([,b],r n ); b> is given by RL I t (t ξ) u(ξ)dξ, < <, () Γ() where Γ( ) denotes the Euler s gamma function. The left Riemann-Liouville derivative and the right Riemann-Liouville derivative with order > of the given function C ([,b],r n ) are respectively given as RL D,t dn [ D n dt n a,t ] and [ RL Dt,b dn ( )n dt n Dt,b n d n t Γ(n ) dt n ] ( )n d n Γ(n ) dt n where n is an integer which satisfies n < n. a (t ξ) n u(ξ)dξ (7) b t (ξ t) n u(ξ)dξ, (8)

3 Progr. Fract. Differ. Appl., No., 99- (8) / RL D The Riemann-Liouville fractional derivative of order (,] for a function C ([,b],r n ); b> is given by dn dtni d n Γ( ) dt n t (t ξ) n u(ξ)dξ, (9) for all t [,b] and n < < n, where n> is an integer. The left Caputo derivative and the right Caputo derivative with order > of the given function, for t (,b) are respectively given as C D,t D n,t [ u (n) (t) ] t (t ξ) n u (n) (ξ)dξ, () Γ(n ) and [ C Dt,b ( )n D,t n u (n) (t) ] ( )n b (ξ t) n u (n) (ξ)dξ () Γ(n ) t where n is an integer which satisfies n < < n. The Caputo fractional derivative of order (,] for a function C ([,b],r n ); b> is given by C D,t t (t ξ) n u n (ξ)dξ, () Γ(n ) for all t [,b]. The Grunwald-Letnikov fractional derivative of order > of a function is given by GL D,t lim h t [ ( h ( ) k k where h is the time-step. The one-parameter Mittag-Leffler function E (z) is defined by E (z) h ] k k ω k ) u(t kh), (3) Γ(k+. () This function arises in the solution of fractional integral equations and interpolates between the power law and exponential law (when, we obtain E (z)e z ) scenarios for phenomena modelled by classical and fractional ordinary equations [7 3]. 3 Numerical Approximation and Stability Analysis of Time-Fractional Differential System with Riemann-Liouville Derivative In this section, we first present the numerical approximation of the Riemann-Liouville time-fractional derivative with the power law kernel [3], and later discuss the stability analysis of time-fractional differential equations. 3. Riemann-Liouville Approximation We shall present directly the numerical approximation, by definition, we have Γ( ) t RL D,t { f(t)} d dt, (t ξ) f(ξ)dξ, d dt u(t + u(t )), t ()

4 K. M. Owolabi: Riemann-Liouville fractional derivative and application... where and u(t + ) u(t ) Our numerical approximation is presented as follows: u(t + ) similarly, u(t ) Γ( ) Γ( ) Γ( ) Γ( ) Γ( ) Γ( ) Thus t + s s s t s (t + ξ) f(ξ)dξ, ts+ RL D,t { f(t)} tγ( ) where s +E,, (t + ξ) f(t s+)+ f(t s ) t s t + (t + ξ) f(ξ)dξ Γ( ) t (t ξ) f(ξ)dξ Γ( ) dξ, f(t s+ )+ f(t s ) ts+ (t + ξ) dξ, t s f(t s+ )+ f(t s )[ (t + t s+ ) (t + t s ) ], (t ξ) f(ξ)dξ, f(t s )+ f(t s )[ (t t s ) (t t s ) ] +O( t). E, { s f(t s+ )+ f(t s )[ (t + t s+ ) (t + t s ) ] f(t s )+ f(t s )[ (t t s ) (t t s ) ]} (8) tγ( ) { ts+ f(u) f(t s+ ) s t s (t + u) du s } f(u) f(t s+ ) t s (t u) du Theorem 3(Atangana and Gómez-Aguilar [3]) Let f denotes a function not necessarily differentiable on [a, T], then the fractional derivative of f of order in the Riemann-Liouville sense is defined by { RL D,t {} f(t s+ )+ f(t s )[ tγ( ) (t + t s+ ) (t + t s ) ] s where s +E,, f(t s )+ f(t s )[ (t t s ) (t t s ) ]} ( ) E, C t + t. Proof.Readers are referred to Atangana and Gómez-Aguilar [3] for details. ts+. () (7)

5 Progr. Fract. Differ. Appl., No., 99- (8) / Stability Analysis of Fractional Differential System in Riemann-Liouville Sense Here, we shall consider the stability of fractional differential system in the sense of the Riemann-Liouville fractional derivative [, 33] RL D,t F, < <, (9) where [u (t),u (t),...,u n (t)] T R n, F ( f i ) n n R n n, with initial condition taken the form RL D,t t u [u,u,...,u n ] T. () In what follows we analyze equation (9) for stability when F has zeros and non-zero eigenvalues. Theorem 3 If all the eigenvalues of F satisfies then the zero solution of fractional differential system (9) is asymptotically stable. arg(λ(f)) > π, () Proof.By taking the Laplace transform on both sides of equation (9), subect to the initial condition (), we get u(s)s u FU(s), which means that the solution of fractional differential system (9) can be written as u t E, (Ft ). By following [, 3], we assume that F is diagonalizable, such that where A denotes an invertible matrix. Then, Λ A FAdiag(λ,λ,...,λ n ). E, (Ft )AE, (Λt )A A diag[e, (λ t ),E, (λ t ),E, (λ 3 t ),...,E, (λ n t )]A. By using the two-parameter Mittag-Leffler function and (), we have So that, E, (λ t ) E, (z) k s k s z s Γ( s) +O( z k ), () (λ t ) s Γ( s) +O( λ t k ), t +, n. E, (Λt ) diag[e, (λ t ),E, (λ t ),E, (λ 3 t ),...,E, (λ n t )]. Again, we recall that any matrix F R n n can be written in its Jordan canonical form where J denotes the diagonal block matrix Jdiag(J,J,J 3,...,J M ), and λ. J λ..,,,...,m. F AJA (3)... λ n n

6 K. M. Owolabi: Riemann-Liouville fractional derivative and application... Clearly, and E, (J t ) s s E, (Ft )Adiag[E, (J t ),E, (J t ),...,E, (J M t )]A, (J t ) s Γ(s+) (t ) s s Γ(s+) Js λ s Ks λ s K n (t ) s λ s.... Γ(s+)... K s λ s s λ s n + ) n E, (λ t )! λ E, (λ t ) (n )! λ E, (λ t ) E, (λ t. )......! λ E, (λ t ) E, (λ t ) λ s ( () The term Ks, i n i denotes the binomial coefficients. It is noticeable that if arg(λ (F)) > π, M and t, with further calculations, we have ( ) E, (λ t i ), E,(λ t ) i! λ, for i n, M. In fact, it is obvious that E, (λ t ) k s (λ t ) s Γ( s) +O( λ t k ), which means that E, (λ t ) as t, and ( ) i E,(λ t ) ( ) { } i k (λ t i! λ i! λ ) s Γ( s) +O( λ t k ) k s s ( ) i (s+i )...(s+)sλ i s t s +O( λ k i t k ) i!γ( s) k ( ) i (s+i )!λ i m t s +O( λ k i t k ), s i!(s )!Γ( s) ( ) i which results to i! λ E, (λ t ), i n as t. It follows that u t E, (Ft ) as t, for any non-zero initial value u. This completes the proof. Theorem 33 Let < <. For internal stability, we consider the fractional autonomous system at a, that is with [,,..., n ] T which has n dimensional representation RL D,t u (t) f u (t)+ f u (t)+ + f n u n (t), RL D,t u (t) f u (t)+ f u (t)+ + f n u n (t), RL D,t F, (). RL D n,t u n(t) f n u (t)+ f n u (t)+ + f nn u n (t), ()

7 Progr. Fract. Differ. Appl., No., 99- (8) / where i s are fractional numbers between and. Suppose ϕ is the least common multiple of the denominators x i s of i s, where i y i /x i, x i Z + for i,,...,n and we set β /κ. We define λ κ f f f n f λ κ f f n det... (7). f n f n λ κ n f nn If all i s are fractional number, the characteristic polynomial equation (7) can be transformed to integer order polynomial. Then the zero solution of n-dimensional fractional differential system () is globally asymptotically stable provided all roots λ i s of the characteristic equation (7) satisfy arg(λ i ) >β π, i. By putting λ h β in equation (7), the characteristic equation can be written in the form det( h β I F). Proof.See Deng et al. [3] for a similar proof. Theorem 3 Assume all the eigenvalues of matrix F satisfy (), that is, arg(λ(f)) π, and the critical eigenvalues found satisfying arg(λ(f)) π have the same geometric and algebraic multiplicities, then we can say that the zero solution of fractional differential system (9) is not asymptotically stable but rather stable. Proof.Let us assume there is a critical eigenvalue λ ρ which satisfies arg(λ ρ ) π with both geometric and algebraic multiplicity equal to unity. Then, from the above, we present the solution of (9) as u t E, (Ft ) u t A diag[e, (J t ),...,E, (J ρ t ),E, (λ ρ t ),E, (J ρ+ t ),...,E, (J ρ t )]A where J q s are Jordan block matrices with order q, arg(λ q(f)) > π, and ρ q n q+ s qρ+ n q+ n,q,...,ρ,ρ+,...,ω. We obtain from above Assume E, (λ ρ t ) (λ ρt ) ( )/ exp((λ ρ t ) / ) ( λ ρ s cos π s q π ) + isin, (8) (λ ρ t ) q Γ( q) +O( (λ ρt ) s ). (9) recall that i, and ω denotes the modulus of λ ρ. Then, E, (λ ρ t ) [ ωt ( cos π π )] ( ( )/ [ + isin exp ωt ( cos π π )] ) ( )/ + isin [ s ωt ( cos π )] π q [ + isin ωt ( +O cos π π ) ] s + isin q Γ( q) ( ω ( )/ t ( sin π π )) ( ) + icos exp iω / t [ s ω q t q( cos qπ isin qπ )] q +O ( (ωt ) s), (3) Γ( q) q similarly, we have t E, (λ ρ t ) ( ω ( )/( sin π π )) + icos [ ω q t q( cos qπ isin qπ Γ( q) t s q ( exp ) iω / t )] q +O ( ωt s ). (3)

8 K. M. Owolabi: Riemann-Liouville fractional derivative and application... From the above, it is obvious that the absolute value of the first term is ω( )/, but the remainder terms tend to zero as t +. We can see from the result presented in Theorem 3 that E, (J q t ) as t +, for q,...,ρ,ρ+,...,ω, which show that the solution of fractional differential system (9) is stable but not asymptotically stable. This completes the proof. Numerical Experiments In this section, we present some numerical experiments of the chaotic time-fractional differential equations arising from the fields of mathematical biology, physics and finance in which the time classical derivatives are replaced with the Riemann-Liouville fractional derivatives of order (, ). Numerical approximation is done with the forward difference scheme which we execute with the Matlab Ra package.. Lotka-Volterra System Let us consider the time-fractional order Lotka-Volterra System [] RL D,t a+εu (t) β τu (t) RL D,t c+d RL D,t ρ+τu (t) (3) where >, >, > are the two-predator and one-prey densities. The parameters a,β,c,d,ε,ρ,τ are Fig. : Simulation result for fractional Lotka-Volterra system (3). The upper and lower rows correspond to species evolution at.73 and.9 respectively. Other parameters are given in (33). Simulation runs for t. assumed positive. Whenever ρ, τ, we recall the fractional order one-predator and one-prey model discussed in [3]. The Lotka-Volterra system (3) displays chaotic attractors in Figure at different values of (as given in the caption) for the parameter values a, β, c, d, ε, ρ 3, τ.7, u(), v()., w(). (33)

9 Progr. Fract. Differ. Appl., No., 99- (8) / 7. Hyperchaotic System Here, we consider the time-fractional order problem which describe a novel hyperchaotic system given by the fourdimensional dynamics [3] RL D,t γ ( ) + + z(t) RL D,t β + z(t) D,t RL D,t z(t) RL (3) τ ρ ( + ) z(t) z(t) where,, and z(t) denotes the states, and γ, β, τ, ρ are positive (constants) parameters. The D system (3) as z(t) Fig. : Numerical results at.9 showing the 3D (upper-row) and D (lower-row) proections of the novel hyperchaotic system (3) with initial conditions u()., v()., w(). and z().3. Other parameters are given in (3). Simulation runs for t. t t 3 u v w z u,v,w,z u,v,w,z 3 u v w z.. time 8 time Fig. 3: Time series result of the novel hyperchaotic system (3) at.87 with different instances of simulation time t,. Other parameters are given in Figure. presented in Figures and 3, displays a strange hyperchaotic attractors for the parameter values γ 8, β, τ, ρ. (3) c 8 NSP

10 8 K. M. Owolabi: Riemann-Liouville fractional derivative and application....3 Halvorsen Circulant Chaotic System Lastly, we consider the Halvorsen circulant chaotic system, described by the 3D dynamics [37] Fig. : Strange chaotic attractors of the Halvorsen system (3) inr andr 3 obtained at.93 with upper-row (γ.8,β 3.8) and lower-row (γ.7, β ). Simulation runs for time t. t u v w t u v w t u v w u,v,w u,v,w u,v,w 8 3 time 8 time time Fig. : Time series result of the Halvorsen circulant chaotic system (3) showing chaotic oscillations at.8 for some instances of simulation time t,, with γ.7 and β. RL D,t γ β(+) v (t) RL D,t γ β(+) w (t) (3) RL D,t γ β(+) u (t) which is symmetric with respect to cyclic interchanges of components u, v and w. Halvorsen have shown that the circulant chaotic system described in (3) is chaotic when γ.7 and β. In the numerical experiment, we simulate with the initial conditions u()., v(). and w().. Figure illustrates the strange attractor of system (3) in R 3. Various D proections on (u,v),(u,w) and (v,w) coordinate are shown. It should be mentioned that, apart from the patterns obtained for the examples considered here, other chaotic and more-complex structures are obtainable, depending on the choices of initial conditions and other parameter values.

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