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On Tempered and Substantial Fractional Calculus
1,2 1 2,∗
Jianxiong Cao , Changpin Li and YangQuan Chen
Abstract—In this paper, we discuss the differences between the Momoniat[11] compare the numerical solutions of three kinds
tempered fractional calculus and substantial fractional operators of fractional Black-Merton-Scholes equations with tempered
in anomalous diffusion modelling, so that people can better fractional derivatives. Recently, high order numerical scheme
understand the two fractional operators. We Ņrst introduce for tempered diffusion equation is presented in [12]. However,
the deŅnitions of tempered and substantial fractional operators, numerical algorithms for solving these problems are limited.
and then analyze the properties of two deŅnitions. At last, we
prove that the tempered fractional derivative and substantial As an extension of the concept of CTRWs to phase space,
derivative are equivalent under some conditions. A diffusion Friedrich et al. derived a new fractional Kramers–Fokker–
problem deŅned by using tempered derivative is also given to Planck equation [13], which involved a fractional substantial
illustrate the slow convergence of an anomalous diffusion process. derivative, it has important nonlocal couplings in both time
Keywords—Fractional calculus, Tempered fractional calculus, and space. In 2011, based on the CTRW models with coupling
Substantial fractional calculus, Anomalous diffusion PDFs, Carmi and Barkai obtained a deterministic equation by
using fractional substantial derivative [14]. The properties and
I. INTRODUCTION numerical discretizations of the fractional substantial operators
The process of anomalous diffusion is one of common are recently discussed in [15]. To our best knowledge ,
phenomena in nature, and the continuous time random walks whether the tempered fractional operators [3] or the fractional
(CTRWs) framework [1] is a useful tool to describe this substantial operators [13] is originated from the tempered
phenomenon. The CTRWs are often governed by the waiting function space. Motivated by this, we try to let people know
time probability density function (PDF) and jump length PDF. the relationship of these two fractional operators.
When the PDFs are power law, the anomalous transport The work is organized as follows. In Section II, we intro-
process can be depicted by fractional diffusion equations. duce three common used deŅnitions of fractional integrals and
While the PDFs are exponentially tempered power law, then derivatives. Two classes of fractional operators called tempered
tempered anomalous diffusion models are derived in [2–8]. and substantial operators are introduced in Section III. Nu-
As Meerschaert [9] pointed out, tempered stable processes merical experiment is carried out to show the effectiveness of
are the limits of random walk models where the power law tempered model in describing exponentially tempered power-
probability of long jumps is tempered by an exponential factor. law behavior. Finally, we conclude the paper in the last section.
These random walks converge to tempered stable stochastic
process limits, whose probability densities solve tempered
fractional diffusion equations. Tempered power law waiting II. PRELIMINARIES
times lead to tempered fractional time derivatives, which have
proven useful in geophysics. Meerschaert et al. proposed a In this section, we give some preliminaries about fractional
tempered diffusion model to capture the slow convergence of calculus. There are sveral different deŅnitions of fractional
subdiffusion [6]. derivatives, but the most frequently used are the following three
Baeumer and Meerschaert studied tempered stable Levy´ deŅnitions, i.e. Grunw¨ ald–Letnikov derivative, the Riemann–
motion in [2], they proposed Ņnite difference and particle Liouville derivative and the Caputo derivative [16–21]. We
tracking methods to solve the tempered fractional diffusion introduce the deŅnitions in the following way.
equation with drift. In view of the efŅciency of tempered DeŅnition II.1. The fractional integral of order α>0 for a
fractional calculus in describing exponentially tempered power function f(t) is deŅned by
law behavior and its variants, it has attracted many researchers
to study numerical methods to solve these problems. Baeumera 1 t
−α α−1
andMeerschaert[2]derivedŅnite differenceandparticle track- aDt f(t)=Γ(α) a (t−s) f(s)ds, (1)
ing methods. Cartea et al. [10] presented a general Ņnite dif-
ference scheme to numerically solve a Black-Merton-Scholes where Γ(·) is the Euler’s function.
model with tempered fractional derivatives. Momoniat and DeŅnition II.2. The left and right Grunwald–Letnikov deriva-
¨
1Department of Mathematics, Shanghai University, Shanghai 200444, China tives of order α>0 of f(t) are deŅned as
(lcp@shu.edu.cn)
2 N
School of Engineering, University of California, Merced, CA 95343, USA α −α j α f(t−jh),
(yqchen@ieee.org, yangquan.chen@ucmerced.edu) GLDa,tf(t) = lim h (−1) (2)
∗ Corresponding author. Tel. 1(209)2284672; Fax: 1(209)2284047 h→0 j=0 j
Nh=t−a
978-1-4799-2280-2/14/$31.00 ©2014 IEEE
and 2) The right Riemann-Liouville tempered fractional integral of
N order α for f(t) is deŅned as
α −α j α
GLDt,bf(t) = lim h (−1) f(t+jh), (3) 1 b
h→0 j=0 j −α,λ −λ(τ−t) α−1
Nh=b−t RLDb,t f(t)=Γ(α) t e (τ −t) f(τ)dτ.
respectively. DeŅnition III.2. [3, 12] Let f(t) be (n − 1)-times continu-
DeŅnition II.3. Suppose that f(t) be (n − 1)-times continu- ously differentiable on (a,∞), and its n-times derivatives be
ously differentiable on (a,∞), and its n-times derivatives be integrable on any subinterval [a,∞). Then the left tempered
integrable on any subinterval [a,∞). Then the left Riemann- fractional derivative of order α>0 for a given function f(t)
Liouville derivative of order α>0 of f(t) is deŅned by is deŅned as
n α,λ −λt α λt
α d −(n−α) RLD f(t)=(e RLD e )f(t)
RLDa,tf(t)=dtn aDt f(t) a,t a,t
−λt n t
n t (4) e d n−α−1 λτ
1 d n−α−1 = n (t − τ) e f(τ)dτ,
= Γ(n−α)dtn a (t−s) f(s)ds, Γ(n−α)dt a (8)
andtheright Riemann-Liouvillefractionalderivative is deŅned and the right tempered fractional derivative is deŅned as
as Dα,λf(t)=(eλt Dα e−λt)f(t)
RL t,b RL t,b
n n b n λt n b
α (−1) d n−α−1 (−1) e d n−α−1 −λτ
RLDt,bf(t)=Γ(n−α)dtn t (s−t) f(s)ds, = Γ(n−α)dtn (τ −t) e f(τ)dτ,
(5) t (9)
respectively, where n is a nonnegative integer and n − 1 ≤ respectively, where n is a nonnegative integer and n − 1 ≤
α0 for f(t) is deŅned as II.3.
α −(n−α) (n) Remark III.2. The variants of the left and right Riemann-
CDa,tf(t)=aDt f (t)
t (6) Liouville tempered fractional derivatives are deŅned as [2, 12,
1 n−α−1 (n) 22]
= Γ(n−α) a (t−s) f (s)ds,
α,λ α
α,λ RLDa,t f(t)−λ f(t), 0 <α<1,
and the right Caputo derivative is deŅned by RLD f(t)=
a,t α,λ α−1 α
n b RLDa,t f(t)−αλ ∂tf(t) −λ f(t),1 <α<2,
α (−1) n−α−1 (n) (7) (10)
CDt,bf(t)=Γ(n−α) t (s−t) f (s)ds, and
α,λ α
respectively, where n is a nonnegative integer and n − 1 < α,λ RLDt,b f(t)−λ f(t), 0 <α<1,
RLD f(t)=
α0 for f(t) is
deŅned by
DeŅnition III.1. [3, 12] Suppose that f(t) is piecewise 1 t
continuous on [a,∞) and integrable on any Ņnite subinterval −α −λ(t−τ) α−1
of [a,∞), α>0,λ≥0.Then Ds f(t)=Γ(α) a e (t − τ) f(τ)dτ,
1) The left Riemann-Liouville tempered fractional integral of where λ can be a constant or a function not related to t.
order α of function f(t) is deŅned by DeŅnition III.4. [13, 15] Suppose that α>0, f(t) be (n−1)-
−α,λ 1 t −λ(t−τ) α−1 times continuously differentiable on (a,∞), and its n-times
RLDa,t f(t)=Γ(α) a e (t − τ) f(τ)dτ. derivatives be integrable on any subinterval [a,∞). Then the
substantial fractional derivative of order α>0 for f(t) is UMERICALSIMULATION
IV. N
deŅned by In this section, based on the discussion of tempered and
α n −(n−α) substantial derivatives, we use Ņnite difference method to solve
Dsf(t)=Ds Ds f(t) , a tempered diffusion problem.
n Example IV.1. Solve the following tempered fractional diffu-
where Dn = d +λ . sion equation
s dt
Remark III.3. If λ ≥ 0, it is clear that DeŅnition III.1 is ∂u(x,t) = RLD0.5,λu(x,t)+f(x,t), 0 0 of a function f(t) with TheoremIV.1. Thelocaltruncationerror ofdifferencescheme
respect to another function z(t) and weight w(t) are deŅned (19) is O(τ + h).
in the following way
α [w(t)]−1 t w(τ)z′(τ)f(τ) Proof: According to (16), (17) and (18), we deŅne the
I f (t)= dτ, (14) local truncation error Rk of difference scheme (19) as below:
a,+;[z;w] Γ(α) [z(t) − z(τ)]1−α i
a
i+1
u(x ,t ) −u(x ,t )
and k i k i k−1 −α 1,α
R = −Kh g u(x ,t )
i t m i−m+1 k
b ′ m=0
α [w(t)] w(τ)z (τ)f(τ) −f(x ,t )
I f (t)= dτ, (15) i k
b,−;[z;w] Γ(α) t [z(τ) −z(t)]1−α ∂u(x ,t ) u(x ,t ) −u(x ,t )
= i k − i k i k−1
respectively. ∂t t
i+1
Remark III.4. If we take z(t)=t, w(t)=eλt, then the left +K(Dαu(x ,t )−h−α g1,αu(x ,t ))
and right generalized integrals reduce to the left and right s i k m i−m+1 k
m=0
tempered fractional integrals. =O(τ)+KO(h)=O(τ+h).
The proof ends. Journal of Physics A: Mathematical and Theoretical,vol.
45, no. 25, pp. 255101, 2012.
[6] Mark M. Meerschaert, Yong Zhang, and Boris Baeumer,
Let λ =0, 0.5, 1.0, the analytical and numerical solutions “Tempered anomalous diffusion in heterogeneous sys-
are displayed in Fig. 1. It can be seen that the numerical tems,” Geophysical Research Letters, vol. 35, no. 17,
solutions Ņt the analytical solutions very well. When λ =0, 2008.
the equation (16) reduces to the Riemann–Liouville diffusion [7] Arijit Chakrabarty and Mark M. Meerschaert, “Tempered
equation, Fig. 1 (a) and (b) show that solution peak is high. For stable laws as random walk limits,” Statistics & Proba-
λ=0.5 and λ =1.0, the solution are plotted in Fig. 1(c), (d) bility Letters, vol. 81, no. 8, pp. 989–997, 2011.
and Fig. 1(e), (f), respectively. From Fig. 1, we can see that the [8] Dumitru Baleanu, Fractional Calculus: Models and
peak of the solutions of tempered diffusion equation becomes Numerical Methods, vol. 3, World ScientiŅc, 2012.
more and more smooth as exponential factor λ increases. [9] Farzad Sabzikar, Mark M Meerschaert, and Jinghua
Chen, “Tempered fractional calculus,” Journal of Com-
V. CONCLUSION putational Physics, 2014.
[10] Alvaro Cartea and Diego del Castillo-Negrete, “Frac-
In this paper, we introduce two classes of fractional tional diffusion models of option prices in markets with
operators for anomalous diffusion, and further discuss the jumps,” Physica A: Statistical Mechanics and its Appli-
properties of tempered and substantial derivatives. We obtain a cations, vol. 374, no. 2, pp. 749–763, 2007.
theorem on two deŅnitions under some conditions. It is easy to [11] O. Marom and E. Momoniat, “A comparison of numer-
conclude that tempered and substantial fractional calculus are ical solutions of fractional diffusion models in Ņnance,”
the generalization of fractional calculus, and both of them are Nonlinear Analysis: Real World Applications, vol. 10, no.
special cases of generalized fractional calculus. Although sub- 6, pp. 3435–3442, 2009.
stantial derivative is equivalent to tempered derivative when the [12] Can Li and Weihua Deng, “High order schemes for the
parameter λ ≥ 0, they are introduced from different physical tempered fractional diffusion equations,” arXiv preprint
backgrounds. Mathematically the fractional substantial calcu- arXiv:1402.0064, 2014.
lus is time-space coupled operator but the tempered fractional [13] R. Friedrich, F. Jenko, A. Baule, and S. Eule, “Anomalous
calculus is not. However, the tempered fractional operators are diffusion of inertial, weakly damped particles,” Physical
the more commonly used in truncated exponential power law review letters, vol. 96, no. 23, pp. 230601, 2006.
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ACKNOWLEDGMENT E, vol. 84, no. 6, pp. 061104, 2011.
The work was partially supported by the Natural Science [15] Minghua Chen and Weihua Deng, “Discretized fractional
Foundation of China under Grant No. 11372170, the Key substantial calculus,” arXiv preprint arXiv:1310.3086,
Program of Shanghai Municipal Education Commission under 2013.
Grant No. 12ZZ084 and China Scholarship Council. [16] Keith B. Oldham and Jerome Spanier, The fractional
calculus: theory and applications of differentiation and
integration to arbitrary order, vol. 111, Academic press
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