[EN] This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener¿Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge¿Kutta second-order stochastic scheme as an illustrative example.This work was partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, TRA2007-68006-C02-02,
DPI2010-20891-C02-01 as well as the Universidad Politécnica de Valencia grant PAID-06-09 (ref. 2588).Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD.; Santamaría Navarro, C. (2011). Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method. Computers and Mathematics with Applications. 61(8):1946-1950. https://doi.org/10.1016/j.camwa.2010.07.057S1946195061

C. Santamaría

Cameron

El-Tawil

Ghanem

J.-C. Cortés

J.-V. Romero

Kloeden

Linz

M.-D. Roselló

English

AbstractThis paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener–Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge–Kutta second-order stochastic scheme as an illustrative example

Cortés, J.-C.

Romero, J.-V.

Roselló, M.-D.

Santamaría, C.

Elsevier - Publisher Connector 

Solving random diffusion models with nonlinear perturbations by the Wiener–Hermite expansion method 

[EN] This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener¿Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge¿Kutta second-order stochastic scheme as an illustrative example.This work was partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, TRA2007-68006-C02-02,
DPI2010-20891-C02-01 as well as the Universidad Politécnica de Valencia grant PAID-06-09 (ref. 2588).Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD.; Santamaría Navarro, C. (2011). Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method. Computers and Mathematics with Applications. 61(8):1946-1950. https://doi.org/10.1016/j.camwa.2010.07.0571946195061

Cortés López, Juan Carlos

Romero Bauset, José Vicente

Roselló Ferragud, María Dolores

Santamaría Navarro, Cristina

RiuNet

Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method

Crossref

Solving random diffusion models with nonlinear perturbations by the Wiener–Hermite expansion method

[EN] This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener¿Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge¿Kutta second-order stochastic scheme as an illustrative example.This work was partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, TRA2007-68006-C02-02,
DPI2010-20891-C02-01 as well as the Universidad Politécnica de Valencia grant PAID-06-09 (ref. 2588).Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD.; Santamaria Navarro, C. (2011). Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method. Computers and Mathematics with Applications. 61(8):1946-1950. doi:10.1016/j.camwa.2010.07.057S1946195061

Computers and Mathematics with Applications 61 (2011) 1946–1950
Contents lists available at ScienceDirect
Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Solving random diffusion models with nonlinear perturbations by the
Wiener–Hermite expansion method
J.-C. Cortés ∗, J.-V. Romero, M.-D. Roselló, C. Santamaría
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camino de Vera s/n, Edificio 8G, 2°, 46022 Valencia, Spain
a r t i c l e i n f o
Keywords:
Random differential equation
Wiener–Hermite expansion
Perturbation method
a b s t r a c t
This paper deals with the construction of approximate series solutions of random nonlin-
ear diffusion equations where nonlinearity is considered by means of a frank small pa-
rameter and uncertainty is introduced through white noise in the forcing term. For the
simpler but important case in which the diffusion coefficient is time independent, we pro-
vide a Gaussian approximation of the solution stochastic process by taking advantage of
the Wiener–Hermite expansion together with the perturbation method. In addition, ap-
proximations of the main statistical functions associated with a solution, such as the mean
and variance, are computed. Numerical values of these functions are compared with re-
spect to those obtained by applying the Runge–Kutta second-order stochastic scheme as
an illustrative example.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Deterministic differential equations of the form x˙(t) = a(t)x(t) constitute the basic form of so-called diffusion or
transport problems which appear in relevant models such as: the growth population geometric (or Malthusian) model in
biology, where a(t) represents the per capita growth rate; the neutron and gamma ray transport model in physics, where
coefficient a(t) involves the geometry of the cross-sections of the medium; the continuous composed interest rate models
for studying the evolution of an investment under time-variable interest rate r(t) in which case a(t) = 1+ r(t); etc. Despite
the usefulness of these basicmodels, they do not often cover all possible situations observed from a practical point of view. In
fact, as a simple but illustrative example, if a(t) = a > 0, the Malthus model predicts unlimited growth of a species despite
the fact that resources are always limited. Then, the logistic (or Verhulst) model introduces a nonlinear term in order to
overcome this drawback by considering the differential equation x˙(t) = ax(t)− b(x(t))2, a, b > 0, where the nonlinearity
intensity is given by parameter b. In many practical situations it is appropriate to assume that the nonlinear term affecting
the phenomena under study is small enough; then its intensity is controlled by means of a frank small parameter, say ϵ.
Relevant examples in this sense appear for instance in epidemiology, where the so-called SIS models become nonlinear
differential equations where the nonlinear term coefficient denoting the contagious rate can be assumed to be a frank small
parameter in many situations [1]. In addition to these considerations, diffusion models with nonlinear perturbations can
also consider the introduction of a forcing term in order to model external aspects which can become very complex, such
as: the environment in biology; unexpected material changes in the surrounding medium in physics; and foreign political
events that can affect the markets where an investment has been ordered in finance. Stochastic differential equations based
on the white noise process provide a powerful tool for dynamically modeling these complex and uncertain aspects. Over
∗ Corresponding author. Tel.: +34 963879144.
E-mail addresses: jccortes@imm.upv.es, jccortes@mat.upv.es (J.-C. Cortés), jvromero@imm.upv.es (J.-V. Romero), drosello@imm.upv.es
(M.-D. Roselló), crisanna@imm.upv.es (C. Santamaría).
0898-1221/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2010.07.057
J.-C. Cortés et al. / Computers and Mathematics with Applications 61 (2011) 1946–1950 1947
the last few years, new and relevant methods for finding the exact solutions of such equations have been developed. They
include the homotopy perturbation method [2–4] and the Exp-function method [5,6].
This paper deals with the solution of stochastic differential models of the form
x˙(t) = a(t)x(t)− ϵ(x(t))2 + λn(t), t > 0, x(0) = x0, (1)
where the diffusion coefficient a(t) and initial condition x0 are deterministic, ϵ is a small parameter and n(t) = n(t)(ω) is
the white noise process, whose intensity is given by parameter λ, and ω is a random outcome for a triple probability space
(Ω,A, P)whereΩ is a sample space,A is a σ -algebra associated withΩ and P is a probability measure.
The paper is organized as follows. Section 2 summarizes the main results of the Wiener–Hermite expansion (WHE)
providing a powerful technique for representing any stochastic process in terms of certain deterministic kernels to be
determined as well as the so-called Wiener–Hermite (WH) polynomials. In Section 3, the WHE is applied in order to obtain
two initial integro-differential equations that are satisfied by these kernels. By taking advantage of the perturbationmethod
the solutions of these equations are obtained in Section 4. Previous development is illustrated for the simpler but important
casewhere the diffusion coefficient is autonomous. In addition,we compute approximations for itsmain statisticalmoments
such as the mean and variance. A comparison of the results obtained with respect to the Runge–Kutta second-order
stochastic scheme for solving stochastic differential equations is also provided. Conclusions are shown in Section 5.
2. The Wiener–Hermite expansion (WHE)
For the sake of clarity in the presentation, we summarize the main ideas of the Wiener–Hermite expansion (WHE) in
this section on the basis of the Wiener–Hermite (WH) polynomials. For further details we recommend [7,8,4]. The WH
polynomials constitute a complete set of statistically orthogonal stochastic processes, say H(i) = H(i)(t1, . . . , ti), defined in
terms of white noise n(t) and the Dirac delta function δ(·) through the following recurrence relations:
H(i)(t1, . . . , ti) = H(i−1)(t1, . . . , ti−1)H(1)(ti)−
i−1
j=1
H(i−2)(ti1 , . . . , tii−2)δ(ti−j − ti), i ≥ 2, (2)
starting fromH(0) = 1 andH(1)(t1) = n(t1). Taking into account the following statistical properties ofwhite noise processes:
E [n(t)] = 0, E [n(t1)n(t2)] = δ(t1 − t2), (3)
where E [·] denotes the expectation operator, one can establish thatWH polynomials are centered with respect to the origin
(except E

H(0)
 = 1) and they are statistically orthogonal:
E

H(i)
 = 0, ∀i ≥ 1; E H(i)H(j) = 0, ∀i ≠ j. (4)
As a consequence of the completeness of theWH set [8], any arbitrary stochastic process, say x(t) = x(t;ω), ω ∈ Ω , can be
expanded in terms of a WH polynomial set and this expansion converges to the original stochastic process, i.e.,
x(t) = x(0)(t)+
∫
R
x(1)(t; t1)H(1)(t1) dt1 +
∫
R2
x(2)(t; t1, t2)H(2)(t1, t2) dt1dt2 + · · · , (5)
where x(0) = x(0)(t), x(i) = x(i)(t; t1, . . . , ti), i ≥ 1, are called the (deterministic) kernels of the WHE of x(t). The first two
terms of the right-hand side define the Gaussian representation of x(t) (the zeroth-order termbeing just itsmean or average,
i.e., E [x(t)] = x(0)(t)), while the second-order and higher terms correspond to the non-Gaussian part. The variance of x(t)
can be expressed as follows:
Var [x(t)] =
∫
R

x(1)(t; t1)
2
dt1 + 2
∫
R2

x(2)(t; t1, t2)
2
dt1dt2 + · · · . (6)
3. Application of the WHE to approximate the solution of the nonlinear problem
In this section we will apply the WHE in order to analyze the response of the nonlinear model (1) to the Gaussian
stochastic process n(t) with intensity λ. The procedure can be described as follows: first, from the original governing
equation (1), we expand the unknown x(t) by means of the WHE given by (5); then, integral–differential deterministic
equations are derived for the dynamics of the unknown kernel functions x(i) of the WHE of the response. For that, we take
advantage of the stochastic orthogonality properties of WH polynomials.
In practice, the WHE series for the response must be truncated after a few terms. Henceforth, we are only interested
in obtaining the Gaussian approximation of the response x(t) of problem (1); then two integral–differential equations
for x(0)(t) and x(1)(t; t1) must be established. For the first one, we just follow the previous procedure: we substitute the
WHE (5) of x(t) in model (1); next we take the expectation operator over the resulting expression and, finally, we take
advantage of properties (4) as well as the fact that E

H(1)(t1)H(1)(t2)
 = δ(t1 − t2) and E H(2)(t1, t2)H(2)(t3, t4) =
δ(t1 − t3)δ(t2 − t4)+ δ(t1 − t4)δ(t2 − t3). This leads to
1948 J.-C. Cortés et al. / Computers and Mathematics with Applications 61 (2011) 1946–1950
x˙(0)(t) = a(t)x(0)(t)− ϵ

x(0)(t)
2 + ∫
R

x(1)(t; t1)
2
dt1

, x(0)(0) = x0, (7)
where the initial condition has been derived by setting t = 0 in (5), next applying the expectation operator, and again
taking advantage of the first property given by (4). In order to establish another (deterministic) differential equation for
x(1)(t; t1), firstly we multiply WHE (5) of x(t) by H(1)(t5), next we take the expectation operator, and then we again
apply the above properties together with E

H(1)(t1)H(1)(t2)H(1)(t3)
 = 0, E H(1)(t1)H(2)(t2, t3)H(2)(t4, t5) = 0 and
E

H(1)(t1)H(1)(t2)H(2)(t3, t4)
 = δ(t1 − t3)δ(t2 − t4)+ δ(t1 − t4)δ(t2 − t3). In this way, one gets
x˙(1)(t; t1) = a(t)x(1)(t; t1)− 2ϵx(0)(t)x(1)(t; t1)+ λδ(t − t1), x(1)(0; t1) = 0, ∀t1. (8)
In this case, the initial condition has been derived by multiplying the WHE (5) by H(1)(t2), next setting t = 0, then taking
the expectation operator and, finally, applying the first property of (4) as well as E

H(1)(t1)H(1)(t2)
 = δ(t1 − t2).
4. The application of the perturbation method. An illustrative example
In order to compute the Gaussian part of the stochastic process solution of problem (1), we need to solve the nonlinear
coupled deterministic problems (7)–(8). Note that both problems depend on the small parameter ϵ > 0. Then a reliable
technique for solving them is the so-called perturbation method under which the deterministic kernels can be represented
in a first approximation as
x(0)(t) = x(0)0 (t)+ ϵx(0)1 (t), x(1)(t; t1) = x(1)0 (t; t1)+ ϵx(1)1 (t; t1). (9)
Substituting these representations in Eqs. (7)–(8) and neglecting these powers of ϵ whose exponents are greater than 1, one
obtains the following initial value problems:
x˙(0)0 (t) = a(t)x(0)0 (t), x(0)0 (0) = x0, (10)
x˙(0)1 (t) = a(t)x(0)1 (t)−

x(0)0 (t)
2 − ∫ ∞
0

x(1)0 (t; t1)
2
dt1, x
(0)
1 (0) = 0, (11)
x˙(1)0 (t; t1) = a(t)x(1)0 (t; t1)+ λδ(t − t1), x(1)0 (0; t1) = 0, ∀t1 ≥ 0, (12)
x˙(1)1 (t; t1) = a(t)x(1)1 (t; t1)− 2x(0)0 (t)x(1)0 (t; t1), x(1)1 (0; t1) = 0, ∀t1 ≥ 0. (13)
Example 1. Let us consider the frequently encountered situation where the diffusion coefficient does not depend on time,
i.e., a(t) = a. In this case, we first compute directly the solutions of problems (10) and (12), and after that we solve problems
(11) and (13). The results obtained are
x(0)0 (t) = x0eat , x(0)1 (t) = −

eat − 1 eat(2a(x0)2 + λ2)− λ2
2a2
, (14)
x(1)0 (t; t1) =

λea(t−t1) if t ≥ t1,
0 if t < t1,
(15)
x(1)1 (t; t1) =
−2e
a(t−t1) eat − 1 λx0
a
if t ≥ t1,
0 if t < t1.
(16)
Taking into account that E [x(t)] = x(0)(t) = x(0)0 (t)+ ϵx(0)1 (t), one gets the following approximation of the mean of x(t):
E [x(t)] = x0eat − ϵ

eat − 1 eat(2a(x0)2 + λ2)− λ2
2a2
. (17)
As regards the variance approximation, note that by the perturbation method and (6) one gets
Var [x(t)] =
∫ ∞
−∞

x(1)0 (t; t1)
2 + 2ϵx(1)0 (t; t1)x(1)1 (t; t1)+ ϵ2 x(1)1 (t; t1)2 dt1, (18)
which leads in our case to
Var [x(t)] = λ
2
2a

e2at − 1− 2ϵ λ2x0
a2

eat − 12 eat + 1+ 2ϵ2 λ2(x0)2
a3

eat − 13 eat + 1 . (19)
J.-C. Cortés et al. / Computers and Mathematics with Applications 61 (2011) 1946–1950 1949
Fig. 1. Comparison of the expectation (left) and the variance (right) obtained by using the Wiener–Hermite expansion technique for problem (1) with
a(t) = 1/2, λ = 1, ϵ = 10−2 and x0 = 0.5 on the interval [0, 2] and a Runge–Kutta stochastic scheme considering m = 100 000 simulations and taking
as the step h = 10−4 .
Table 1
Numerical values of the expectation and variance as well as the relative errors obtained by using the Wiener–Hermite expansion technique for problem
(1) with a(t) = 1/2, λ = 1, ϵ = 10−2 and x0 = 0.5 on the interval [0, 2] and a Runge–Kutta stochastic scheme consideringm = 100 000 simulations and
taking as the step h = 10−4 .
t E

xWHE(t)

RelErrE Var

xWHE(t)

RelErrVar
0.00 0.5 0 0 0
0.25 0.565465 0.001495642 0.282515 0.004461187
0.50 0.638576 0.000480526 0.641372 0.004179709
0.75 0.720045 0.0009411 1.09676 0.00440265
1.00 0.810596 0.00063616 1.67398 0.00537721
1.25 0.9110935 0.00100017 2.4046 0.00656068
1.50 1.02172 0.00034244 3.32786 0.01095191
1.75 1.14352 0.0006563 4.49228 0.01562586
2.00 1.27674 0.0006662 5.95747 0.01874725
Columns E

xWHE(t)

and Var

xWHE(t)

of Table 1 show the results for the average and variance obtained from (17) and (19),
respectively, for λ = 1, ϵ = 10−2, a = 1/2, x0 = 0.5. In Fig. 1, we compare these results with respect to those obtained
by using a Runge–Kutta second-order stochastic scheme [9], where the Brownian motion involved has been simulated
taking m = 100 000 simulations and step h = 10−4. In addition, the third and fifth columns of Table 1 show the relative
errors for the average (RelErrE) and variance (RelErrVar) with respect to the Runge–Kutta scheme results. Note that the
approximations obtained from the two approaches agree.
5. Conclusions
This paper shows that the WHE technique constitutes a powerful tool for constructing approximate solutions for the
stochastic process for random diffusion models with nonlinear perturbations where uncertainty is considered by means of
an additive term defined by white noise. As has been highlighted, suchmodels appear in frequent applications in fields such
as physics and epidemiology. Although the success of the WHE method depends heavily on the complexity encountered
in dealing with integro-differential equations, a large number of deterministic techniques for solving them are available,
includingmathematical software [10]. Besides computing theGaussian approximation of the solution,wehave also provided
approximations of its average and variance. As we have shown, these results agree with those obtained by applying other
stochastic numerical methods. Finally, we remark that in the near future, we will report on the corresponding results for
the non-Gaussian approximation.
Acknowledgements
This work was partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, TRA2007-68006-C02-02,
DPI2010-20891-C02-01 as well as the Universidad Politécnica de Valencia grant PAID-06-09 (ref. 2588).
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Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method

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