Properties of a New Fractional Derivative without Singular Kernel

Transcription

1 Progr. Fract. Differ. Appl. 1, No. 2, Progress in Fractional Differentiation and Applications An International Journal Properties of a New Fractional Derivative without Singular Kernel Jorge Losada 1 and Juan J. Nieto 1,2, 1 Departamento de Análise Matemática, Universidade de Santiago de Compostela, Santiago de Compostela, Spain 2 Faculty of Science, King Abdulaziz University, P.O. Box 823, 21589, Jeddah, Saudi Arabia Received: 28 Jan. 215, Revised: 13 Feb. 215, Accepted: 15 Feb. 215 Published online: 1 Apr. 215 Abstract: We introduce the fractional integral corresponding to the new concept of fractional derivative recently introduced by Caputo and Fabrizio and we study some related fractional differential equations. Keywords: Fractional calculus, fractional derivative, fractional integral, Caputo. 1 Introduction Let us recall the well known definition of Caputo fractional derivative [1]. Given b>, f H 1,b and <α < 1, the Caputo fractional derivative of f of order α is given by C D α f t= 1 t t s α f sds, t >. Γ1 α Fractional calculus and, in particular, Caputo fractional derivative, finds numerous applications in different areas of science [2, 3, 4, 5]. By changing the kernel t s α by the function exp αt s/1 α and 1/Γ1 α by 1/ 2π1 α 2, one obtains the new Caputo-Fabrizio fractional derivative of order <α < 1, which has been recently introduced by Caputo and Fabrizio in [6]. That is, CF D α f t= 2 αmα exp α t s f sds, t, 21 α 1 α where Mα is a normalization constant depending on α. According to the new definition, it is clear that if f is a constant function, then CF D α f = as in the usual Caputo derivative. The main difference between old and new definition is that, contrary to the old definition, the new kernel has no singularity for t = s. It is well known that Laplace Transform plays an important role in the study of ordinary differential equations. In the case of this new fractional definition, it is also known see [6] that, for <α < 1, L [ CF D α f t ] s= 2 αmα 2 s+α1 s sl [ ft]s f, s>. 1 where L [gt] denotes the Laplace Transform of function g. So, it is clear that if we work with Caputo-Fabrizio derivative, Laplace Transform will also be a very useful tool. Corresponding author juanjose.nieto.roig@usc.es c 215 NSP

2 88 J. Losada, J.J. Nieto: Properties of a New Fractional Derivative 2 The associated fractional integral After the notion of fractional derivative of order < α < 1, that of fractional integral of order < α < 1 becomes a natural requirement. In this section we obtain the fractional integral associated to the Caputo-Fabrizio fractional derivative previously introduced. Let < α < 1. Consider now the following fractional differential equation, using Laplace transform, we obtain: CF D α f t=ut, t. 2 L [ CF D α f t ] s=l [ut]s, s>. That is, using 1, we have that or equivalently, 2 αmα 2 s+α1 s sl [ ft]s f = L [ut]s, s>, L [ ft]s= 1 s f+ 2α 21 α L [ut]s+ s2 αmα 2 αmα L [ut]s, s>. Hence, using now well known properties of inverse Laplace transform, we deduce that In other words, the function defined as ft= 21 α 2 αmα ut+ 2α 2 αmα ft= 21 α 2 αmα ut+ 2α 2 αmα where c Ris a constant, is also a solution of 2. We can also rewrite fractional differential equation 2 as 2 αmα t exp α t s 21 α 1 α or equivalently, usds+ f, t. 3 usds+c, t, f sds=ut, t, α exp 1 α s f sds= 21 α α 2 αmα exp 1 α t ut, t. Differentiating both sides of the latter equation, we obtain that, f t= 21 α 2 αmα Hence, integrating now from to t, we deduce as in 3, that u t+ α 1 α ut ft= 21 α 2 αmα [ut u]+ 2α 2 αmα, t. usds+ f, t. Thus, as consequence, we expect that the fractional integral of Caputo-Fabrizio type must be defined as follows. Definition 1. Let < α < 1. The fractional integral of order α of a function f is defined by, CF I α f t= 21 α 2 αmα ut+ 2α 2 αmα usds, t. c 215 NSP

3 Progr. Fract. Differ. Appl. 1, No. 2, / 89 Remark. Note that, according to the previous definition, the fractional integral of Caputo-Fabrizio type of a function of order < α < 1 is an average between function f and its integral of order one. Imposing we obtain an explicit formula for Mα, 21 α 2 αmα + 2α 2 αmα = 1, Mα= 2, α 1. 2 α Due to this, we propose the following definition of fractional derivative of order < α < 1. Definition 2. Let < α < 1. The fractional Caputo-Fabrizio derivative of order α of a function f is given by, CF D α f t= 1 exp α t s f sds, t. 1 α 1 α 3 Some fractional differential equations In this section we study some simple but useful fractional differential equations. Lemma 1. Let <α < 1 and f be a solution of the following fractional differential equation, CF D α f t=, t. 4 Then, f is a constant function. The converse, as indicated in the Introduction, is also true. Proof. From 3, we obtain that the solution of 4 must satisfy ft= f for all t. Hence, it is clear that f must be a constant function. Proposition 1. Let <α < 1. Then, the unique solution of the following initial value problem is given by where I 1 σ denotes a primitive of σ and CF D α f t=σt, t, 5 f= f R; 6 ft= f + a α σt σ + bα I 1 σt, t, 7 a α = 21 α 2 αmα, b 2α α = 2 αmα. 8 Proof. Suppose that the initial value problem 5-6 has two solutions, f 1 and f 2. In that case, we have that CF D α f 1 t CF D α f 2 t= [ CF D α f 1 f 2 ] t= and f1 f 2 =. So, by Lemma 1, we have that f 1 f 2 =. That is f 1 t= f 2 t for all t. By 3, it is clear that the function defined by 7 is a solution of the fractional differential equation 5. Moreover, if we substitute t by in 7, we obtain f. Hence, the function defined by 7 is the unique solution of initial value problem 5-6. Remark. For α = 1, we have that the solution of 5 is the usual primitive of σ. Now, we consider the following linear fractional differential equation where λ R, λ λ = corresponds to the case previously studied. CF D α f t=λ ft+ut, t, 9 c 215 NSP

4 9 J. Losada, J.J. Nieto: Properties of a New Fractional Derivative From Proposition 1, we have that solving equation 9 is equivalent to find a function f such that [ ] ft= f + a α λ ft f + ut u + bα [λ f + u]sds, t where a α, b α are given by 8. Equivalently, we must find f such that where If λ a α = 1, we obtain: ft λ b α I 1 f t= f + a α ut u + bα I 1 ut, t. In the other case, i. e., λ a α 1, we have that: ft= a α u t b α ut, t. λ b α λ ft λ b α I 1 f t= σt, t, 1 σt= f + a α ut u + b α I 1 ut, t. The case λ = is trivial, and we obtain f = σ. If λ, we see that 1 can be rewritten as ft λ I 1 f t= σt, t, where Hence, λ = λ b α. f t= λ ft+ σt, t. Thus, we have obtained an ordinary differential equation, which has a unique solution if we consider an initial condition. In consequence, we have proved the following result. Proposition 2. Let < α < 1. Then, initial value problem given by CF D α f t=λ ft+ut, t, f= f R; has a unique solution for any λ R. 4 Nonlinear fractional differential equations Theorem 1. Let <α < 1, T > and ϕ : [,T] R R a continuous function such that there exits L> satisfying, Ifa α + b α TL<1, then the initial value problem given by has a unique solution on C[,T]. ϕt,s 1 ϕt,s 2 L s 1 s 2 for all s 1, s 2 R. CF D α f t=ϕt, ft, t [,T], 11 f= f R; 12 c 215 NSP

5 Progr. Fract. Differ. Appl. 1, No. 2, / 91 Proof. Let C[,T] be the space of all continuous functions defined on the interval[,t] endowed with the usual supremum norm, that is, f = sup t [,T] ft for all f C[,T]. We consider the operator N : C[,T] C[,T] defined by, where N f t=c+a α ϕt, ft+b α ϕs, fsds, for all f C[,T], c= a α ϕ, f + f By 3, finding a solution of in C[,T] is equivalent to finding a fixed point of the operator N. Since for all f 1, f 2 C[,T] and all t [,T] we have that N f 1 t N f 2 t = a α ϕ t, f 1 t ϕ t, f 2 t + b α ϕ s, f 1 s ds ϕ s, f 2 s ds a α ϕ t, f 1 t ϕ t, f 1 t + b α ϕ s, f 1 s ϕ s, f 2 s ds a α L f 1 t f 2 t +b α L f 1 s f 2 s ds a α + b α TL f 1 f 2, we conclude that operator N is a contraction. The statement follows now from Banach s Fixed Point Theorem. 5 Application to fractional falling body problem Consider a mass m falling due to gravity. The net force acting on the body is equal to the rate of change of the momentum of that body. For constant mass, applying the classical Newton second law, we have mv t=mg kvt, where g is the gravitational constant, and the air resistance is proportional to the velocity with proportionality constant k. If air resistance is negligible, then k= and the equation simplifies to v t=g. If we replace D 1 = v by D α we have the following fractional falling body equation CF D α vt= k m vt+g. For an initial velocity v=v then, according to Proposition 2, it has a unique solution. Acknowledgment The work of J.J. Nieto and J. Losada has been partially supported by the Ministerio de Economía y Competitividad of Spain under grants MTM and MTM P, Xunta de Galicia under grant R214/2, and co-financed by the European Community fund FEDER. The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. c 215 NSP

6 92 J. Losada, J.J. Nieto: Properties of a New Fractional Derivative References [1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 26. [2] R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol. 3, [3] M. Caputo, Linear models of dissipation whose Q is almost frequency independent: II, Geophys. J. R. Astron. Soc. 13, [4] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91, [5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [6] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1:2, [7] X.-L. Ding and J. J. Nieto, Analytical Solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain, Fract. Calc. Appl. Anal. 8, 215. To appear. c 215 NSP