Probabilistic solutions of fractional dierential and partial dierential equations and their Monte Carlo simulations Tamer Oraby Harrinson Arrubla Erwin Suazo

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Probabilistic solutions of fractional diﬀerential and partial diﬀerential

equations and their Monte Carlo simulations

Tamer Oraby, Harrinson Arrubla, Erwin Suazo∗

School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley,

1201 W. University Dr., Edinburg, Texas, USA,

∗Corresponding author: erwin.suazo@utrgv.edu

Abstract

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional

ordinary diﬀerential equations as an expected value of a random time process. Using the latter, we present

an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional

ordinary and partial diﬀerential equations. Thirdly, we show that this approach allows us to ﬁnd the

fundamental solutions for fractional partial diﬀerential equations (PDEs), in which the fractional deriva-

tive in time is in the Caputo sense and the fractional in space one is in the Riesz-Feller sense. Lastly,

using Riccati equation, we study families of fractional PDEs with variable coeﬃcients which allow explicit

solutions. Those solutions connect Lie symmetries to fractional PDEs.

Keywords: Caputo fractional derivative, Riesz-Feller fractional derivative, Riccati equation, Lie sym-

metries, Green functions, Monte Carlo Integration, Mittag-Leﬄer function.

1. Introduction

Although it was started in the second half of the eighteenth century by Leibniz, Newton and l’Hˆopital,

[1], fractional calculus has received great attention in the last two decades. Many physical, biological and

epidemiological models have found that fractional order models could perform at least as good as well as

their integer counterparts, [2,3]. Integer order models are also appearing as special cases of the fractional

order. That makes the dynamical behavior of those models richer and in some cases ﬂexible. As advances

are made in fractional calculus and fractional modeling, understanding of the physical interpretation of

fractional derivatives is becoming clearer. A memory kernel with algebraic decay is the most common

interpretation for the change in the dynamical behavior of the system [4,5,6]. Today fractional calculus

is widely used in physical modeling. Examples include the nonexponential relaxation in dielectrics and

ferromagnets [7], [8], the diﬀusion processes [9], [10] and the Hamiltonian Chaos [11] and [12].

Anomalous diﬀusion could be modeled using fractional order stochastic processes and their Fokker-

Planck equations [13,14,15]. Continuous-time random walk (CTRW) is one approach used to model

anomalous diﬀusion [16]. In particular, a CTRW with inﬁnite-mean time to jump exhibits sub-diﬀusion

behavior. The time to jump could be modeled by a heavy tail distribution with index βsuch that 0 < β < 1

leading to a mean square displacement that is of order tβdepicting the short-range jump and so thus sub-

diﬀusion. A long-range jump leads to super-diﬀusion, i.e., β > 1, [17].

A subordinator is a non-decreasing L´evy process, i.e. a non-negative process with independent and

stationary increments. It is used as a random time, or operational time, in deﬁning time-changed-processes.

Its density decays algebraically as 1/tα+1 as t→ ∞, with α∈(0,1). A diﬀerent type of time-change can

be made by using the inverse subordinator (i.e the hitting time of a subordinator) which sometimes leads

to sud-diﬀusion processes [18,19].

arXiv:2210.02955v2 [math.DS] 6 Nov 2022

The Riccati equation has played an important role in ﬁnding explicit solutions for Fisher and Burgers

equations, (see [20] and [21] and references therein). Also, similarity transformations and the solutions

of Riccati and Ermakov systems have been extensively applied thanks to Lie groups and Lie algebras

[22], [23], [24], [25] and [26]. In this work, we show that this approach allows us to ﬁnd the fundamental

solutions for fractional PDEs; the fractional derivative in time is in the Caputo sense and the fractional

derivative in space one is in the Riesz-Feller sense. We establish a relationship between the coeﬃcients

through the Riccati equation; we study families of fractional PDEs with variable coeﬃcients which allow

explicit solutions and ﬁnd those explicit solutions. Those solutions connect Lie symmetries to fractional

PDEs.

Formulating solutions of fractional diﬀerential equations and partial diﬀerential equations as expected

values with respect to heavy-tail or power law distributions could be enabled using Monte-Carlo integration

methods and Sampling Importance Integration to evaluate them. The mean problem is in the number of

simulations or random number generations that need to be done to guarantee convergence and small

standard error.

In Section 1, we will review fundamental deﬁnitions and classical results needed from the classical theory

of fractional diﬀerential equations. In Section 2, we present the ﬁrst main result of this work, Lemma 1,

which allow us to see a fractional functions and their fractional derivatives as Wright type transformations

of some functions and their derivatives. They could be also interpreted as expected values of functions

in a random time process. Also, in Section 2 we present Theorem 2.1 which allow us to solve fractional

ordinary diﬀerential equations through solutions of regular ordinary diﬀerential equations. In Section 3, we

derive fractional green functions for some important fractional partial diﬀerential equations, like diﬀusion,

Schr¨odinger, and wave equation. In Section 4, we derive green functions of fractional partial diﬀerential

equations with variable coeﬃcients with application to Fokker-Planck equations. In Section 5, we use the

integral transform or the expected value interpretation of the solutions of fractional equations to carry out

Monte Carlo simulations of their solution.

1.1. Preliminaries

In this section we give the required background of Caputo and Riesz-Feller fractional diﬀerentiation.

Caputo Derivative.

Let Dnbe the Leibniz integer-order diﬀerential operator given by

Dnf=dnf

dtn=f(n),

and let Jnbe an integration operator of integer order given by

Jnf(t) = 1

n−1! Zt

(t−τ)n−1f(τ)dτ, (1.1)

where n∈Z+. Let us use D=D1for the ﬁrst derivative. For fraction-order integrals, we use

Jn−βf(t) = 1

Γ(n−β)Zt

(t−τ)n−β−1f(τ)dτ, (1.2)

where n−1< β ≤n. Now, deﬁne the Caputo fractional diﬀerential operator Dβ

Cto be

Dβ

Cf(t) = Jn−βDnf(t),

where n−1< β ≤n, for n∈N.

Riemann-Liouville Derivative.

The Riemann-Liouville fractional diﬀerential operator Dβ

RL is deﬁned to be

Dβ

RLf(t) = DnJn−βf(t),

where n−1< β ≤n, for n∈N. We will use ∂β

tF:= ∂βF

∂tβand use ∂tF:= ∂F

∂t .

The Riemann-Liouville fractional is related to the Caputo fractional derivative through [27]:

Dβ

RLf(t) = Dβ

Cf(t) +

n−1

k=0

f(k)(0)tk

k!.

While we will not discuss the Riemann-Liouville fractional derivatives in the paper, the results presented

in this paper are valid for Riemann-Liouville fractional derivatives, when f(k)(0) = 0 for k= 0,1, . . . , n−1.

We will consider n= 1 in this work; that is 0 < β ≤1. Some of the results be extended through

the remark that for 0 < β ≤1, Dn+β−1

Cf(t) = Dβ

Cf(n−1)(t) for n≥1. Note also that when 0 < β ≤1,

Dβ

RLf(t) = Dβ

Cf(t) + f(0).

Riesz-Feller Derivative

The Riesz-Feller fractional diﬀerential operator Dα,θ

RF is deﬁned to be [28]

Dα,θ

RF f(x) = Γ(1 + α)

πsin((α+θ)π

2)Z∞

f(x+y)−f(x)

y1+αdy

+Γ(1 + α)

πsin((α−θ)π

2)Z∞

f(x−y)−f(x)

y1+αdy

for fractional order 0 < α ≤2, and the skewness parameter θ≤min(α, 2−α). The symmetric Riesz-Feller

diﬀerential operator is deﬁned at θ= 0 and is simply denoted by Dα

RF .

Transformations.

The Laplace transform of a function f(t) is deﬁned as

L(f)(s) = e

f(s) = Z∞

e−stf(t)dt.

The inverse Laplace transform is deﬁned by

L−1e

f(t) = 1

2πi ZC

est e

f(s)ds

where Cis a contour parallel to the imaginary axis and to the right of the singularities of e

f. The Laplace

transform of the Caputo fractional derivative of a function is given by

LDβ

Cf(s) = sβe

f(s)−

n−1

k=0

sβ−1−kf(k)(0).(1.3)

The Fourier transform of a function f(x) is deﬁned as

F(f)(y) = b

f(y) = Z∞

−∞

eixyf(x)dx.

The inverse Fourier transform is deﬁned by

F−1b

f(x) = 1

2πZ∞

−∞

e−ixy b

f(y)dy.

The Fourier transform of the Riesz-Feller fractional derivative of a function is given by

FDα,θ

RF f(y) = −ψθ

α(y)b

f(y),(1.4)

where ψθ

α(y) = |y|αeisign(y)θπ

Mittag-Leﬄer Function

The Mittag-Leﬄer function, which generalizes the exponential function, can be written as follows,

Eβ(z) =

∞

k=0

Γ(βk + 1),β∈R+, z ∈C,(1.5)

and the more general Mittag-Leﬄer function with two-parameters is deﬁned to be

Eβ,α(z) =

∞

k=0

Γ(βk +α),β, α ∈R+, z ∈C.(1.6)

Wright function.

The Wright function is another special function of importance to fractional calculus and is deﬁned by

[29],

Wβ,α(z) =

∞

k=0

k!Γ(βk +α),β > −1, α ∈C, z ∈C.(1.7)

The following Wright type function will be fundamental in the rest of this work

gβ(x;t) = 1

tβW−β,1−β−x

tβ,(1.8)

which is a probability density function of the random time process Tβ(t) for all t > 0 [30,31]. Tβ(t) can

be seen as the inverse of a βstable subordinator with density gβ(x, t), see [34,35,36]. It has a Laplace

transform

L(gβ(·;t)) (s) = Z∞

e−sxgβ(x;t)dx =Eβ−stβ(1.9)

for <(s)>0 and moments E(Tβ(t))k= Γ(k+ 1) tkβ

Γ(kβ + 1) for k≥1 [37,38]. At the same time

Z∞

e−stgβ(x;t)dt =sβ−1e−xsβ.(1.10)

It is shown in [30] that for 0 < β, α < 1,

gβα(x;t) = Z∞

gβ(x;s)gα(s;t)ds (1.11)

for x > 0. For more details about Wright function see [30,31]. From (1.9) and (1.11), we get

Z∞

Eβ−stβgα(t;r)dt =Z∞

0Z∞

e−sxgβ(x;t)gα(t;r)dxdt =Eβα −srβα,

and from (1.10) and (1.11), we get

Z∞

Eα(−txα)gβ(x;t)dt =Z∞

sβ−1e−xsβgα(s;x)ds.

L´evy α-stable distribution.

The L´evy α-stable distribution with stability index 0 < α ≤2, Lθ

α(x) has a Fourier transform given by

FLθ

α(·)(y) := b

Lθ

α(y) = e−ψθ

α(y).(1.12)

The density Lθ

α(x) has a fat tail proportional to |x|−(1+α).

Deﬁne Lθ

α(y;x) = 1

Lθ

αy

αfor y∈Rand x > 0 which is a probability density function of the

α-stable random process with asymmetry parameter θ, denoted by Lθ

α(x) for x > 0, [39].

Z∞

−∞

eisyLθ

α(y;x)dy =e−ψθ

α(s)x.(1.13)

For 0 < α < 1, and t, x > 0,

L−α

α(t;x) = xα

tgα(x;t).(1.14)

See [40] for more details. Also, for 0 < α ≤1

Lθα

βα(x;t) = Z∞

Lθ

β(x;s)L−α

α(s;t)ds. (1.15)

2. Fractional Derivative as Expected Value with Respect to a L´evy Distribution

In this Section we establish the ﬁrst main result of this paper.

2.1. Caputo Fractional Derivative

The following lemma is one of the main results of this work for its wide applicability.

Lemma 1.

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摘要：

ProbabilisticsolutionsoffractionaldierentialandpartialdierentialequationsandtheirMonteCarlosimulationsTamerOraby,HarrinsonArrubla,ErwinSuazoSchoolofMathematicalandStatisticalSciences,TheUniversityofTexasRioGrandeValley,1201W.UniversityDr.,Edinburg,Texas,USA,Correspondingauthor:erwin.suazo@utrgv....

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Probabilistic solutions of fractional dierential and partial dierential equations and their Monte Carlo simulations Tamer Oraby Harrinson Arrubla Erwin Suazo

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