numerical solutions for stochastic partial differential equations via

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IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY NUMERICAL SOLUTIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS VIA ACCELERATED GENETIC ALGORITHM Dr. Eman A. Hussain*, Dr. Yaseen M. Alrajhi * Al-Mustansiriyah University, College of Science , Department of Mathematics, Iraq Al-muthanna University, College of Science , Department of Mathematics, Iraq

ABSTRACT This paper introduced a new accelerated Genetic Algorithms (GAs) method to find a numerical solutions of stochastic Partial differential equations driven by space-time white nose wiener process . The numerical scheme is based on a representation of the solution of the equation involving a stochastic part arising from the noise and a deterministic partial differential equation . By using Doss-Sussmann transformation that enables us to work with a partial differential equation instead of the stochastic partial differential equation. Then compare these solutions obtained by our method with saul'yev method and deterministic solution. KEYWORDS: SPDS, Accelerated Genetic Algorithm method, Numerical solution of stochastic partial differential equations.

INTRODUCTION In this paper we want to take a quicker look at the numerical solutions for stochastic partial differential equations (SPDEs). Working on the numerical solutions for SPDEs we face many difficulties. On the one hand we have to consider problems known from numerically solving deterministic partial differential equations. On the other hand we are faced with problems triggered by numerically solving stochastic ordinary differential equations (SODEs). And additionally new issues arise resulting from the infinite dimensional nature of the underlying noise processes ,[1]. Stochastic partial differential equations (SPDEs) are used as a model in many applications. This area of mathematics is especially motivated by the need to describe random phenomena studied in natural sciences like physics, chemistry, biology, and in control theory, [2]. So, we can define SPDEs, by combine deterministic partial differential equations with some kind of noise. Consider the SPDE with space-time white noise ,[4]. 𝑑𝑢(𝑡, 𝑥) = 𝑢𝑥𝑥 (𝑡, 𝑥)𝑑𝑡 + 𝑔(𝑢(𝑡, 𝑥))𝑑𝑊(𝑡, 𝑥) (1) with 0 < 𝑥 < 1 , and 𝑢(0, 𝑥) = 𝑢0 , 𝑢(𝑡, 0) = 𝑢(𝑡, 1) = 0 , 𝑎𝑛𝑑 𝑢𝑡 (𝑡, 0) = 𝑢𝑡 (0,1) = 0 Then , two different ways of writing this equation (1) are : 𝜕𝑢 𝜕 2 𝑢 𝜕2𝑊 = 2 + 𝑔(𝑢) (2) 𝜕𝑡 𝜕𝑥 𝜕𝑡𝜕𝑥 or ∞

𝑑𝑢 = 𝑢𝑥𝑥 𝑑𝑡 + ∑ 𝑔(𝑢)ℎ𝑘 (𝑥)𝑑𝑊𝑘 (𝑡)

(3)

𝑘=1

Such that , three kinds of space-time white noise as in [4] are :  

Brownian Sheet – 𝑊(𝑡, 𝑥) = 𝜇([0, 𝑇] × [0, 𝑥]) Cylindrical Brownian motion – family of Gaussian random variables http: // www.ijesrt.com

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IC™ Value: 3.00 𝐵𝑡 = 𝐵𝑡 (ℎ) , ℎ ∈ 𝐻 a Hilbert space, s.t. 𝐸[𝐵𝑡 (ℎ)] = 0, 𝐸[𝐵𝑡 (ℎ)𝐵𝑠 (𝑔)] = 〈ℎ, 𝑔〉𝐻 (𝑡 ∧ 𝑠) 𝜕2 𝑊



𝑘 𝑘 Space-time white noise 𝑑𝑊(𝑡, 𝑥) = = ∑∞ 𝑘=1 ℎ (𝑥)𝑑𝑊𝑘 (𝑡) , where {ℎ } is assumed a Basis of the 𝜕𝑡𝜕𝑥 Hilbert space we’re in , if {ℎ𝑘 , 𝑘 > 1} is a complete orthonormal system, then {𝐵𝑡 (ℎ𝑘 ), 𝑘 > 1} independent standard Brownian motion. The connection between the three kinds: If 𝐻 = 𝐿2 (ℝ) 𝑜𝑟 𝐻 = 𝐿2 (0, 1), then 𝜕𝑊 𝐵𝑡 (ℎ) = ∫ ℎ(𝑥)𝑑𝑥 𝜕𝑥 and ∞

𝑥

𝐵𝑡 (𝑥) = 𝐵𝑡 (𝑋[0,𝑋] ) = ∑ ∫ (ℎ𝑘 (𝑦)𝑑𝑦𝑊𝑘 (𝑡)) = 𝑊(𝑡, 𝑥) 𝑘=1 0

where we assume that ℎ𝑘 (𝑥) = √2 sin(𝑘𝜋𝑥) Then we get equations of the form :

(4) ∞

𝑑𝑈(𝑡, 𝑥) = [𝒜𝑈(𝑡, 𝑥) + 𝑐(𝑥)𝑈(𝑡, 𝑥)]𝑑𝑡 + ∑ ℎ𝑙 (𝑡)𝑈(𝑡, 𝑥)𝑑𝐵𝑡𝑙

(5)

𝑙=1

Or in integral form 𝑡

𝑡

𝑡

𝑛

𝑈(𝑡, 𝑥) = 𝑈0 (𝑥) + ∫ 𝒜𝑈(𝑠, 𝑥)𝑑𝑠 + ∫ 𝑐(𝑥)𝑈(𝑠, 𝑥)𝑑𝑠 + ∑ ∫ ℎ𝑙 (𝑠)𝑈(𝑠, 𝑥)𝑑𝐵𝑠𝑙 (6) 0

𝑙=1 0

0

For 0 ≤ 𝑡 ≤ 𝑇 . The process 𝐵 = (𝐵𝑡𝑙 , … , 𝐵𝑡𝑛 )0≤𝑡≤𝑇 is an n-dimensional Brownian motion . The operator 𝒜 is defined as : 𝑑

𝒜𝑢 =

𝑑

1 𝜕2𝑢 𝜕𝑢 ∑ 𝑎𝑖𝑗 + ∑ 𝑏𝑖 , 2 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥𝑖 𝑖,𝑗=1

𝑖=1

𝑚

𝑤𝑖𝑡ℎ 𝑎𝑖𝑗 (𝑥) = ∑ 𝜎𝑖𝑘 𝜎𝑗𝑘

(7)

𝑘=1

Where the diffusion matrix 𝜎: ℝ𝒅 → ℝ𝒅×𝒎 and the drift coefficient 𝑏: ℝ𝒅 → ℝ𝒅 . the initial condition 𝑈0 , the functions 𝑐 , 𝜎 𝑎𝑛𝑑 𝑏 are suppose to be smooth functions of the space variable , (ℎ𝑙 (𝑡))1≤𝑙≤𝑛 are bounded and holder continuous of order 1/2. Thus the equation (5) has a unique regular strong solution. In this paper, we focus on the stochastic heat equation. Thus, we simplify the above equation to : 𝑑𝑈(𝑡, 𝑥) = 𝒜𝑈(𝑡, 𝑥)𝑑𝑡 + 𝒮(𝑈(𝑡, 𝑥))𝑑𝑊(𝑡, 𝑥) (8) where 𝒮 is a multiplication operator of the form (𝒮(𝑣)𝑢)(𝑥) = 𝑏(𝑥, 𝑣(𝑥)). 𝑢(𝑥) Taking a closer look at the noise in this equation we see that we can split it into two types, additive and multiplicative noise. We speak of additive noise if the operator 𝒮 is a constant operator and of multiplicative noise if 𝒮 is not constant. The objective of our work is to develop a numerical scheme for the random field 𝑈. The problem of numerical solutions of (5) has been studied by many authors with different approaches. The ideas that lead us to propose a new scheme are twofold.  On the one hand we wish to propose a numerical scheme that separates the noise 𝒮 from the second order operator 𝒜. This idea has been used in [2] ,[5] in a filtering context in which the authors’ scheme first performs off-line a wide number of solutions of partial differential equations by the finite element method. The stochastic part of the simulation is done after this first step.  On the other hand we want to use the accelerated genetic algorithm method to find numerical solution for the partial differential equations that may appear in our scheme. In order to implement the above ideas, we need to introduce the d-dimensional Markov process X = (Xt )0≤t≤T whose infinitesimal generator is given by the second order operator in equation (7) . The Markov process 𝑋 is governed by infinitesimal generator 𝒜 of the stochastic differential equation is : http: // www.ijesrt.com

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IC™ Value: 3.00 𝑡

𝑡

𝑋𝑡 = 𝑥0 + ∫ 𝑏(𝑋𝑠 )𝑑𝑠 + ∫ 𝜎(𝑋𝑠 ) 𝑑𝑊𝑠 , 0 ≤ 𝑡 ≤ 𝑇 0

(10)

0

where the initial condition 𝑥0 in ℝ𝑑 .

TYPES OF SOLUTIONS OF SPDES Stochastic partial differential equations of the form 𝑑𝑋𝑡 (𝑥) = [𝒜𝑋𝑡 + 𝐹(𝑋𝑡 )]𝑑𝑡 + 𝒮(𝑋𝑡 )𝑑𝑊𝑡 (𝑥) have different notions of solutions. As in [1] we find :

, 𝑋(0) = 𝜉

(11)

Definition 1: 𝐷(𝒜)-valued predictable process 𝑋(𝑡) , 𝑡 ∈ [0 , 𝑇] is called an analytical strong solution of the problem (6) if 𝑡

𝑡

𝑋(𝑡) = ∫[𝒜𝑋𝑠 + 𝐹(𝑋𝑠 )]𝑑𝑠 + ∫ 𝒮(𝑋𝑠 )𝑑𝑊𝑠 0

, 𝑃 − 𝑎. 𝑠.

(12)

0

In particular ,the integral in the right-hand side have to be well-define ,[1],[4]. Definition 2: H-valued predictable process 𝑋(𝑡) , 𝑡 ∈ [0 , 𝑇] is called an analytical weak solution of the problem (6) if 𝑡

𝑡 ∗

< 𝑋(𝑡), 𝜁 >= ∫[< 𝑋(𝑠), 𝒜 𝜁 > +< 𝐹(𝑋𝑠 ), 𝜁 >]𝑑𝑠 + ∫ < 𝜁, 𝒮(𝑋𝑠 )𝑑𝑊𝑠 > , 𝑃 − 𝑎. 𝑠. (13) 0

0

For each 𝜁𝜖𝐷(𝒜 ∗ ) , in particular ,the integral in the right-hand side have to be well-define. Definition 3: H-valued predictable process 𝑋(𝑡) , 𝑡 ∈ [0 , 𝑇] is called an mild solution of the problem (6) if 𝑡

𝑋(𝑡) = ∫[𝑒

𝑡 𝒜(𝑡−𝑠)

𝐹(𝑋𝑠 )]𝑑𝑠 + ∫ 𝑒 𝒜(𝑡−𝑠) 𝒮(𝑋𝑠 )𝑑𝑊𝑠

0

, 𝑃 − 𝑎. 𝑠.

(14)

0

In particular ,the integral in the right-hand side have to be well-define,[1],[4].

STOCHASTIC INTEGRAL WITH RESPECT TO CYLINDRICAL WIENER PROCESS We denote by 𝐿20 = 𝐿2 (𝑈0 , 𝑌) the space of Hilbert-Schmidt operators acting from 𝑈0 into Y , and by 𝐿 = 𝐿(𝑈, 𝑌) , we denote the space of linear bounded operators from U into Y ,[1],[4]. Let us consider the norm of the operator 𝜓 ∈ 𝐿02 : ∞

‖𝜓‖2𝐿2 0

= ∑
2𝑌

ℎ,𝑘=1

1

2

= ∑ 𝜆ℎ < 𝜓𝑒ℎ , 𝑓𝑘 >2𝑌 = ‖𝜓𝒬 2 ‖

𝐻𝑆

ℎ,𝑘=1

= 𝑡𝑟(𝜓𝒬𝜓 ∗ ) (15)

Where 𝑔𝑖 = √𝜆𝑖 𝑒𝑖 and {𝜆𝑖 }, {𝑒𝑖 } are eigenvalues and eigenfunctions of the operator 𝒬 , {𝑔𝑖 }{𝑒𝑖 } and {𝑓𝑖 } are orthonormal bases of spaces 𝑈0 , 𝑈 and 𝑌 , respectively. The space 𝐿02 is a separable Hilbert space with the norm ‖𝜓‖2𝐿2 = 𝑡𝑟(𝜓𝒬𝜓 ∗ ) (16) 0 In particular 1.When 𝒬 = 𝐼 then 𝑈0 = 𝑈 and the space 𝐿20 becomes 𝐿2 (𝑈, 𝑌) . 2. When 𝒬 is a nuclear operator, that is 𝑡𝑟𝒬 < +∞, then 𝐿(𝑈, 𝑌 ) ⊂ 𝐿2 (𝑈0 , 𝑌). For, assume that 𝐾 ∈ 𝐿(𝑈, 𝑌), that is 𝐾 is linear bounded operator from the space 𝑈 into 𝑌. 1

Let us consider the operator 𝜓 = 𝐾|𝑈0 , that is the restriction of operator 𝐾 to the space 𝑈0 , where 𝑈0 = 𝒬 2 (𝑈) 1

. Because 𝒬 is nuclear operator, then 𝒬 2 is Hilbert- Schmidt operator.

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IC™ Value: 3.00 Proposition.1 the formula

∞

𝑊𝑐 (𝑡) = ∑ 𝑔𝑗 𝛽𝑗 (𝑡)

,𝑡 ≥ 0

(17)

𝑗=1

defines Wiener process in 𝑈1 with covariance operator 𝒬1 such that 𝑡𝑟𝒬1 < +∞ . Proposition.2 For any 𝑎 ∈ 𝑈 the process

∞

< 𝑎, 𝑊𝑐 (𝑡) >𝑈 = ∑ < 𝑎, 𝑔𝑗 >𝑈 𝛽𝑗 (𝑡)

(18)

𝑗=1

is real-valued Wiener process and 𝐸 < 𝑎, 𝑊𝑐 (𝑡) >𝑈 < 𝑏, 𝑊𝑐 (𝑡) >𝑈 = (𝑡 ∧ 𝑠) < 𝒬𝑎, 𝑏 >𝑈 , 𝑎, 𝑏 ∈ 𝑈 1

−

1

Additionally, 𝐼𝑚𝒬12 = 𝑈0 and ‖𝑢‖𝑈0 = ‖𝒬1 2 𝑢‖ . 1 2

𝑈1

In the case when 𝒬 is nuclear operator, 𝒬 is Hilbert-Schmidt operator. Taking 𝑈1 = 𝑈, the process 𝑊𝑐 (𝑡) , 𝑡 ≥ 0, defined by (17) is the classical Wiener process introduced. Definition.4 The process 𝑊𝑐 (𝑡) , 𝑡 ≥ 0 , defined in (17), is called cylindrical Wiener process in 𝑈 when 𝑡𝑟𝒬1 < +∞. As shown in Fig.1 below .

Fig.1 Cylindrical White noise with its distribution and spectral density

The stochastic integral with respect to cylindrical Wiener process is defined as follows. As we have already written above, the process 𝑊𝑐 (𝑡) , 𝑡 ≥ 0 defined by (10) is a Wiener process in the space 𝑈1 with the covariance operator 𝒬1 such that 𝑡𝑟𝒬1 < +∞. Then the stochastic integral , 𝑡

∫ 𝑔(𝑠)𝑑𝑊𝑐 (𝑠) ∈ 𝑌

(19)

0

where 𝑔(𝑠) ∈ 𝐿(𝑈1 , 𝑌), with respect to the Wiener process 𝑊𝑐 (𝑡) is well defined on 𝑈1 . We denote by 𝑁(𝑌 ) the space of all stochastic processes ∅: [0, 𝑇] × Ω → 𝐿2 [𝑈0 , 𝑌] Such that 𝑇

𝐸 (∫ ‖∅(𝑡)‖2𝐿2[𝑈0,𝑌] 𝑑𝑡) < +∞

(20)

0

and for all 𝑢 ∈ 𝑈0 , ∅(𝑡)𝑢 is a Y-valued stochastic process measurable with respect to the filtration ℱ𝑡 . The stochastic integral 𝑡

∫ ∅(𝑠)𝑑𝑊𝑐 (𝑠) ∈ 𝑌 0

with respect to cylindrical Wiener process, given by (10) for any process ∅ ∈ 𝑁(𝑌) can be defined as the limit

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𝑚

𝑡

𝑡

∫ ∅(𝑠)𝑑𝑊𝑐 (𝑠) = lim ∑ ∫ ∅(𝑠)𝑔𝑗 𝑑𝛽𝑗 (𝑠) 𝑚→∞

0

𝑖𝑛 𝑌

(21)

𝑗=1 0

In 𝐿2 (Ω) sense .

MATHEMATICAL SETTING AND ASSUMPTIONS ,[1] Let 𝑇 > 0 and let (Ω, ℱ, Ρ) be a probability space with a normal filtration ℱ𝑡 , 𝑡 ∈ [0, 𝑇] . in addition let (𝐻, ) be a separable Hilbert space with norm denoted by |. | .We will interpret the SPDE (1) in such a space 𝐻 . The objects 𝒜 , 𝑥0 , 𝐹, 𝑊𝑡 , here are specified through the following assumptions. Assumption 1: linear operator 𝒜. There exist sequence of real eigenvalues 0 ≤ 𝜆1 ≤ 𝜆2 ≤ ⋯ and eigenfunctions {𝑒𝑛 }𝑛≥1 of 𝒜 such that the linear operator 𝐴: 𝐷(𝒜) ⊂ 𝐻 → 𝐻 is given by : ∞

𝒜𝑣 = ∑ −𝜆𝑛 < 𝑒𝑛 , 𝑣)𝑒𝑛 ,

(22)

𝑛=1

For all

∞

𝑣 ∈ 𝐷(𝐴)𝑤𝑖𝑡ℎ 𝐷(𝒜) = {𝑣 ∈ 𝐻| ∑ |𝜆𝑛 |2 | < 𝑒𝑛 , 𝑣 > |2 < ∞} 𝑛=1

Let 𝐷((−𝒜)𝑟 ) 𝑤𝑖𝑡ℎ 𝑟 ∈ ℝ denote the interpolation space of the operator (−𝒜) ,[8]. Assumption 2: Cylindrical Brownian motion 𝑊𝑡 . there exist a sequence of 𝑞𝑛 ≥ 0 , 𝑛 ≥ 1 , of positive real numbers 𝛾 ∈ (0,1) such that ∞

∑(𝜆𝑛 )2𝛾−1 𝑞𝑛 < ∞

(23)

𝑛=1

And independent real valued ℱ𝑡 − 𝐵𝑟𝑜𝑤𝑛𝑖𝑎𝑛 𝑚𝑜𝑡𝑖𝑜𝑛 𝛽𝑡𝑛 , 𝑡 ≥ 0, 𝑛 ≥ 1 , i.e. each 𝛽𝑡𝑛 is ℱ𝑡 -adapted and the 𝑛 increments 𝛽𝑡+ℎ − 𝛽𝑡𝑛 , ℎ > 0 , are independent of ℱ𝑡 . Then the cylindrical Brownian motion 𝑊𝑡 is given by : ∞

𝑊𝑡 (𝑥) = ∑ √𝑞𝑛 𝑒𝑛 (𝑥)𝛽𝑡𝑛

(24)

𝑛=1

Remark 1. The above series may not converge in 𝐻, but in some space 𝑈1 into which 𝐻 can be embedded, ([7] and [8]). In our example with the Laplace operator in one dimension, we will have 𝜆𝑛 = −𝜋 2 𝑛2 and 𝑞𝑛 ≡ 1 , 𝑓𝑜𝑟 𝑛 ≥ 1 . This is the important case of space–time white noise. Assumption 3: nonlinearity 𝑓. The nonlinearity 𝑓: 𝐻 → 𝐻 is two times continuously differentiable, it and its derivatives satisfy |𝑓 ′ (𝑥) − 𝑓 ′ (𝑦)| ≤ 𝐿|𝑥 − 𝑦|, |(−𝒜)(−𝑟) 𝑓 ′ (𝑥)(−𝒜)𝑟 𝑣| ≤ 𝐿|𝑣| 1 For all 𝑥, 𝑦 ∈ 𝐻 , 𝑣 ∈ 𝐷(−𝒜)𝑟 𝑎𝑛𝑑 𝑟 = 0, , 1 and they satisfy 2

1

1

|𝒜 −1 𝑓 ′′ (𝑥)(𝑣, 𝑤)| ≤ 𝐿 |(−𝒜)−(2) 𝑣| |(−𝒜)−(2) 𝑤| , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣, 𝑤, 𝑥 ∈ 𝐻, 𝐿 > 0 Remark 2. The function 𝑓 is usually given as a real-valued function of a real variable, but in the SPDE (1) it is considered as a function defined on 𝐻 and taking values in some function space such as a subspace of 𝐻. Assumption 4: initial value 𝑋0 . The initial value 𝑋0 is a 𝐷((−𝒜)𝑟 ) valued random variable, which satisfies 1

𝐸|(−𝒜)−(2) 𝑋0 |4 < ∞ where 𝛾 > 0 is given in assumption 2.2. With the above assumptions we get by [JK11] that

(25)

𝑑𝑋𝑡 = [𝑘∆𝑋𝑡 + 𝑓(𝑥, 𝑥𝑡 )]𝑑𝑡 + 𝑏(𝑥, 𝑋𝑡 )𝑑𝑊𝑡 (𝑥)

(26)

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IC™ Value: 3.00 𝑋0 (𝑥) = 0 𝑎𝑛𝑑 𝑋𝑡 (0) = 𝑋𝑡 (1) = 0 𝑓𝑜𝑟 𝑥 ∈ (0,1), 𝑡 ∈ [0, 𝑇) has unique mild solution 𝑋: [0, 𝑇] × Ω → 𝐻𝛽+1

(27 )

(28) .

2

A RELATED PARTIAL DIFFERENTIAL EQUATION We present in this section a transformation that enables us to work with a partial differential equation instead of the stochastic partial differential equation (5). This method is classical and it is known as the Doss-Sussmann transform when one applies it to stochastic differential equation ([7] and [8]). It is a useful trick that permits to rewrite a large class of one dimensional stochastic dynamic as a one dimensional random ordinary dynamic (by stochastic dynamic we mean stochastic differential equation or stochastic partial differential equation). It has been successfully used in [8] in which the authors have estimated the probability of finite-time blowup of positive solutions of stochastic partial differential equations with Dirichlet boundary condition. Doss-Susmann transform The particular form of (5) will allow us to use a Doss-Susmann transform. We may write that ,[5]. 𝑡

𝑛

𝑈(𝑡, 𝑥) = exp (∑ ∫ ℎ𝑙 (𝑠)𝑑𝐵𝑠𝑙 ) × 𝑣(𝑡, 𝑥)

(29)

𝑙=1 0

with 𝑣 that solves the partial differential equation

𝑛

1 2 𝑑𝑣(𝑡, 𝑥) = 𝒜𝑣(𝑡, 𝑥)𝑑𝑡 + (𝑐(𝑥) − ∑(ℎ𝑙 (𝑡)) ) 𝑣(𝑡, 𝑥)𝑑𝑡 2

(30)

𝑙=1

As regard to the expression of the function 𝑣, it is clear that 𝑣 can be simulated off-line. Indeed the coefficients in the above partial differential equation are 𝑐, (ℎ𝑙 )1