On Sensitivity Analysis of Parameters for Fractional Differential Equations with Caputo Derivatives

In this paper, we discuss the effect of parameter variations on the performance of fractional differential equations and give the concept of fractional sensitivity functions and fractional sensitivity equations. Meanwhile, by employing Laplace transform and the inverse Laplace transform, some main results on fractional differential equations are proposed. Finally, two simple examples with numerical simulations are provided to show the validity and feasibility of the proposed theorem.


Introduction
Fractional differential equations (FDEs) have become one of the most attractive topics in the last few years [4,10,13,17,19,23,26,27].The reason for this is that FDEs could well reflect the long-memory and non-local properties of many dynamical models, such as fractional oscillator equation in the coulomb damping vibration [25], fractional Schrödinger equation in quantum mechanics [21], fractional Langevin equation in the anomalous diffusion [3], fractional reaction-diffusion equation in biochemistry [11], fractional Lotka-Volterra equation in the historical biological systems [6], fractional Cattaneo equation in the laser short-pulse heating process [24], and so on.Some fundamental theories of the solution of FDEs have been published [2,7,8,14,17,18,20,22,23,[28][29][30], like existence, uniqueness, continuous dependence on the order α and on the initial values, etc.These results are not introduced particularly in this paper.As we all know, the performance of many dynamical systems can be closely related to parameter variations.Actually, it is very important that there should be described the effect of small parameter variations on solutions before the dynamical analysis of systems.However, how to characterize this relation is not yet well established in the published literature.Motivated by the previous works, the paper studies the differentiability and continuous dependence of the solution of FDEs with respect to parameters.And this is to derive the concept of fractional sensitivity functions and fractional sensitivity equations that depict the relationship between the performance of systems and parameter variations.Meanwhile, by employing Laplace transform and the inverse Laplace transform, some main results on FDEs are proposed.Finally, two simple examples with numerical simulations are given to show the effectiveness of our theoretical results.
Throughout the paper, R + and Z + are the sets of positive real and integer numbers, respectively; while R and C denotes separately the sets of real and complex numbers.R n represents the n-dimensional Euclidean space.For a vector x = [x 1 , x 2 , . . ., x n ] T ∈ R n , we use x ∞ = max i=1,2,...,n |x i | to denote respectively the 1-norm, 2-norm and infinity norm of vector x, while x represents an arbitrary norm of vector x.C D and 0 I α t denote the Caputo fractional derivative and Riemann-Liouville fractional integral of order α on [0, t], respectively.

Preliminaries
In this section, we recall some basic definitions and properties related to fractional calculus which will be needed later.More detailed information on fractional calculus can be found in the literatures [17,23].In fractional calculus, the fractional integrals and derivatives usually employ the Riemann-Liouville definition and Caputo definition, respectively.Besides, without loss of generality, the lower limit of all fractional integrals and derivatives is supposed to be zero throughout the paper.We list their representation that will be used in our proofs.
Definition 2.2.The Caputo fractional derivative with order α ∈ R + of function x(t) is defined as follows where n is an integer such that 0 3) The Laplace transform (LT) of the Caputo fractional derivative is given by where X (s) denotes the Laplace transform of x(t), t and s are the variable in the time domain and complex-frequency domain, respectively.Furthermore, the Laplace transform of 0 I α t x(t) takes the particularly simple form In order to study the solution of FDEs, the Mittag-Leffler function is employed frequently.To this end, the Mittag-Leffler function with two parameters is defined by The Laplace transform of the Mittag-Leffler function in two parameters is where λ ∈ R, Re(s) denotes the real part of s and LT{•} stands for the Laplace transform.
Property 2.4.Let x(t) be a continuously differentiable function defined on an interval [0, b] of the real axis R, then where n is an integer such that 0

Nonlinear fractional differential equations
Applying the Caputo derivative, a fractional-order differential equation with nonzero initial value can be defined by where α ∈ (0, 1) is a real constant.By using the property of Riemann-Liouville fractional integral, one gets ).From Property 2.4 and Definition 2.1, we have where α ∈ (0, 1) is a real constant.
From the above processes, it is recognized that this solution x(t) depends continuously on the order α, the initial condition x 0 , and the right-hand function f (t, x).The corresponding theorem of such problems has been found in [7,8].Here, a new criterion is devoted to the estimates of solutions of fractional-order equations.

Corollary 3.3.
Let Ω ⊂ R n is a domain that contains x = 0. Assume that x(t) ∈ Ω is a solution of Cauchy type problem (3.1), for all t ≥ 0; and there exists a constant l > 0 such that f (t, x(t)) ≤ l x(t) on [0, ∞) × Ω.Then, for any α ∈ (0, 1) (ii) x 0 (E α (−2lt α )) According to the foregoing processes, one gets For the left-hand side of (3.3), there exists a non-negative function The Laplace transform (LT) of Equation (3.4) yields Taking the inverse Laplace transform of (3.5), one obtains where * denotes the convolution operator, and since The proof of the right-hand side of (3.3) is similar to the above procedure, so we omit it here.Therefore, we have Taking the square root yields This completes the proof.
In the next section, we give the closeness of solutions for FDEs involving Caputo derivative on a finite interval of the real axis in spaces of continuous functions.
Proof.The solutions u(t) and v(t) are given as, respectively.
. Then there exists a nonnegative function M(t) satisfying Taking Laplace transform (LT) of (3.6), we obtain where s α − l and the inverse Laplace transform using the formula (2.4) gives the formula where * denotes the convolution operator.And since This completes the proof.
Remark 3.5.Theorem 3.4 here is an extension of Theorem 3.4 of the Ref. [16] about the closeness of solutions of integer-order to fractional-order differential equations.
Remark 3.6.This bound is useful only on a finite time interval [0, b], since the right side of the inequality will be unbounded as b is large enough (i.e. the Mittag-Leffler function grows unbound as t → ∞).
Corollary 3.7.Let x 1 (t) and x 2 (t) be differentiable functions on an interval [0, b] such that x 1 (0) It follows from the Property 2.4 that Consequently, Similar to the proof of Theorem 3.4, one gets This completes the proof.

Fractional differential equations with perturbation
Let us consider the fractional differential equation with perturbation where α ∈ (0, 1) is a real constant, µ ∈ R P could represent physical parameters of the equation, and the study of these parameters explains the change of modeling errors, aging, or uncertainties and disturbances, which exist in any realistic problem.
Proof.In order to prove this theorem, we need structure an invariant set D such that the solution of fractional-order equation starts in D at some time, and stays in D for all future time.This method is similar to the proof of Theorem 3.5 in [16].Since φ(t, µ 0 ) is continuous with respect to t, then φ(t, µ 0 ) is bounded on the closed interval [0, b].Consequently, there exists a small enough ε to ensure that the set Apparently, D is compact set, then f (t, x, µ) satisfies the Lipschitz condition with respect to x on D with Lipschitz constant l > 0. By the continuity of f (t, x, µ) concerning µ, for any ζ > 0, there exists η > 0 (with η < r), such that Taking ζ < ε and φ 0 − ϕ 0 < ζ, by the existence and uniqueness theorem, then there exists a unique solution ϕ(t, µ) of C 0 D α t x(t) = f (t, x(t), µ) defined on [0, b], with ϕ(0, µ) = ϕ 0 .We will prove that, by electing a small enough ζ, the solution stays in D for all t ∈ [0, b].That is, Based on Theorem 3.4, it follows that where δ = min(ζ, η).From Definition 3.9, the solution of fractional-order equation is continuously dependent on parameter µ.This completes the proof.Corollary 3.12.Let f (t, x) is continuous in (t, x) and has continuous first partial derivative with respect to x, for all (t, x) ∈ [0, b] × R n .Let ψ(t, t 0 , x t 0 ) be the solution of equation C t 0 D α t x(t) = f (t, x(t)) that starts at x(t 0 ) = x t 0 for all t 0 ≥ 0; Further let ψ(t, t 0 , x t 0 ) is fully determined by x(t 0 ) = x t 0 in the usual sense and ψ(t, t 0 , x t 0 ) is well-defined on [t 0 , b], while does not consider knowledge of the infinite time interval x(τ) (−∞ < τ ≤ t 0 ).Then, for any real constant α (0 < α < 1), (i) ψ(t, t 0 , x t 0 ) is continuously differentiable with respect to t 0 and x t 0 .
(iii) Under the assumptions of (ii), if the Jacobian matrix Proof.(i) By using the property of Riemann-Liouville fractional integral, one has From Property 2.4 and Definition 2.1, one gets Apparently, ψ(t, t 0 , x t 0 ) is continuously differentiable with respect to t 0 and x t 0 since f and its partial derivative with respect to ψ are continuous in (t, ψ).(ii) From (i), it yields that

Note that
Hence Consequently, denote that y(t) ψ t 0 (t) + ψ x t 0 (t) Namely, Taking Caputo type fractional derivative of both sides of (3.9), it is clearly seen that y(t) = ψ t 0 (t) + ψ x t 0 (t) f (t 0 , x t 0 ) t 0 I α t (t − t 0 ) −1 satisfies the fractional-order equation This completes the proof.Remark 3.13.If the fractional-order differential equations are linear and autonomous (i.e.f (t, x(t)) = f (x(t)), and f (x(t)) is noted for the linear combination of the vector x(t)), then the result (iii) holds certainly.

Sensitivity equations of Caputo fractional derivative
Let the nonlinear function f (t, x, µ) and its partial derivative with respect to x and µ be continuous in (t, x, µ) for all (t, x, µ) ∈ [0, b] × R n × R p .Before existing the perturbation of parameter µ, let us assume the nominal equation has a unique solution x(t, µ 0 ) on a finite time interval [0, b], where α ∈ (0, 1) is a real constant and µ 0 is a nominal value of µ.When there is the perturbation of parameter µ, the nominal equation can be written the following form If µ − µ 0 is chosen as sufficiently small constant, then it follows from Theorem 3.11 that the equation (3.11) has a unique solution x(t, µ) on a finite time interval [0, b] and it is close to the nominal solution x(t, µ 0 ).Therefore, the equation (3.11) is equivalent to a Volterra integral equation (3.12) By the continuous differentiability of f (t, x, µ) with respect to x and µ, one obtains that the solution x(t, µ) is differentiable with respect to µ on some neighborhood of µ 0 .That is, By setting x µ (t, µ) = ∂x(t,µ) ∂µ , we can write the equation (3.13) as At µ = µ 0 , x µ (t, µ 0 ) depends only on some time t and describes the time evolution of the sensitivity of parameter µ.Taking Caputo type fractional derivative of both sides of (3.13), it yields that ∂µ , as µ is sufficiently close to µ 0 , P(t, µ) and Q(t, µ) are well-defined on [0, b].Furthermore, the evolution of equation (3.14) is fully determined by knowledge of the vector x µ (t, µ) at a time t = 0 in the usual sense, and does not depend on information of the infinite time interval x µ (τ, µ) (−∞ < τ ≤ 0).When µ = µ 0 , the equation (3.14) is described by the follows form where S(t) = x µ (t, µ 0 ) is called the fractional-order sensitivity function, and formula (3.15) is call the fraction-order sensitivity equation.Apparently, S(t) is also well-defined on on a finite time interval [0, b], which is the unique solution of the equation (3.15).The matrices P(t, µ 0 ) and Q(t, µ 0 ) are given as follows.
Remark 3.14.The sensitivity function and sensitivity equation of Caputo fractional derivative is an extension of sensitivity analysis of parameters from the integer-order to the fractionalorder.The corresponding theorem has been found in [16].
In the next section, two simple examples are given to illustrate the solution process of the sensitivity function.

Illustrative examples
where a, b, c and w are real positive parameters of equation.If the nominal values of parameters are chosen as a 0 = 0.3, b 0 = 1, c 0 = 39 and w 0 = 1, then the nominal equation is written as Differentiating with respect to x = (x 1 , x 2 ) T and µ = (a, b, c, w) T , respectively, we see that the Jacobian matrices are described by 3  1 cos(wt) −cw sin(wt) .
where µ = (a, b, c) T are real positive parameters that represent the Prandtl number, the geometric factor and the Rayleigh number, respectively.When the nominal values of parameters are chosen as a 0 = 10, b 0 = 28 and c 0 = 8 3 , then the nominal equation is described by Similar to the procedure of Example 4.1, one obtains Let the sensitivity function     When the initial value (x 10 , x 20 , x 30 ) = (1, 1, 1), Fig. 4.3 shows the sensitivities of x 1 (t) with respect to x 4 (t), x 6 (t) and x 8 (t), that is the sensitivity of parameters (a, b, c) for x 1 (t).Also, Figs.4.4-4.5 depict the sensitivity of parameters (a, b, c) for x 2 (t) and x 3 (t), respectively.From Figs. 4.3-4.5, we can see that the solution is more sensitive for parameter c than for parameters a and b.In addition, Fig. 4.6 displays a much closer relationship between the Rayleigh number c and the spatial temperature distribution in the fluid layer under gravity (i.e. the variable x 3 (t)) than measures of the fluid velocity (i.e. the variables x 1 (t) and x 2 (t)).
Refs. [12,15] show that the performance of the system is closely related to variations in the parameter c.Therefore, it is suggested that there should be described the effect of small parameter variations on solutions before the dynamical analysis of fractional Lorenz system.

Conclusion
This letter shows the effect of parameter variations on the performance of fractional differential equations and give the concept of fractional sensitivity functions and fractional sensitivity equations.At the same time, by employing Laplace transform and the inverse Laplace transform, some main results on fractional differential equations are proposed.Finally, two simple examples with numerical simulations are provided to show the validity and feasibility of the proposed theorem.x 2 (t) x 3 (t)

Definition 2 . 1 .
Let x(t) be a continuous function on an interval [0, b].The Riemann-Liouville fractional integral of order α ∈ R + is defined as 0

Theorem 3 . 4 .
Let Ω be an open and connected set in R n and f

Remark 3 . 8 .
For the case (δ = λ 1 = λ 2 = 0), Corollary 3.7 shows that the initial value problem has only one solution as the right side of the differential equation satisfies a Lipschitz condition.The corresponding result for ordinary differential equations of integer-order has been found in[5, Theorem 5].

Theorem 3 . 11 .
t), µ) are the same thing, that is x 00 = x 01 x 0 .Let Ω be an open and connected set in R n and the nonlinear function f

Example 4 . 1 .
Let us consider the following Duffing forced-oscillation equation with Caputo fractional derivative C 0

Figure 4 . 2 :
Figure 4.2: Evolution of the sensitivities of x 2 (t) on parameters (a, b, c, w). t

.2) Remark 3.1. From
(3.2), we can clearly see that the solution of Cauchy type problem (3.1) is fully determined by the initial value x 0 and the nonlinear function f in the usual sense, while does not consider information of the infinite time interval x(τ) (−∞ < τ ≤ 0).In this case, the solution x(t) of Cauchy type problem (3.1) is well-defined on t > 0. It is thus assumed that the initial value of FDEs involving Caputo derivative is a constant function of time, and x(t) = x(0 + ) for all t ≤ 0 throughout the paper.

Lemma 3.2. Let
Ω be an open and connected set in R n , and assume that the nonlinear function f (t, x) : [0, b] × Ω → R n is piecewise continuous in t and satisfies the Lipschitz condition in x.