The Solution of the Initial Mixed

Trends in Mathematics, 349–355 c

⃝2014 Springer International Publishing AG

The Solution of the Initial Mixed

Boundary Value Problem for Hyperbolic Equations by Monte Carlo and

Probability Difference Methods

Kanat Shakenov

Abstract. The initial mixed boundary value problem for equations of hyper- bolic type is considered. It is solved by algorithms “random walk on spheres”,

“random walk on balls” and “random walk on lattices” of Monte Carlo meth- ods and by probability difference methods.

Mathematics Subject Classification (2010).65C05.

Keywords.Hyperbolic equation,random walk,Monte Carlo,probability,dif- ference,method,approximation,expectation.

1. Introduction

We consider in a bounded closed domain Ω∈ℝ^𝑛, (𝑛= 2,3) with boundary ∂Ω and for𝑡∈(0, 𝑇) the initial mixed boundary value problem

∂_𝑡²𝑢(𝑡, 𝑥)−Δ_𝑥𝑢(𝑡, 𝑥) +𝛾²𝑢(𝑡, 𝑥) =𝑓(𝑡, 𝑥), (𝑡, 𝑥)∈Ω×(0, 𝑇), (1.1)

𝑢(0, 𝑥) =𝜑(𝑥), 𝑥∈Ω, (1.2)

∂𝑡𝑢(0, 𝑥) =𝜓(𝑥), 𝑥∈Ω, (1.3)

𝛼(𝑡, 𝑥)𝑢(𝑡, 𝑥) +𝛽(𝑡, 𝑥)∂𝑢(𝑡, 𝑥)

∂n =𝑔(𝑡, 𝑥), (𝑡, 𝑥)∈∂Ω×(0, 𝑇), (1.4) where𝛾 is a parameter,𝑓(𝑡, 𝑥), 𝜑(𝑥), 𝜓(𝑥), 𝛼(𝑡, 𝑥), 𝛽(𝑡, 𝑥), 𝑔(𝑡, 𝑥) are given func- tions, andnis the normal to∂Ω. Problems of this type in different settings have been considered in [1], [2], [3]. In monograph [1] (especially pages 7–10) the initial value problem for equation (1.1) is considered. In the AIP Conference Preceedings [2] the mixed problem for elliptic equation by Monte Carlo and by probability difference methods is solved. In article [3] the initial Neumann boundary value problem for parabolic type equation by the algorithm “random walk” of Monte Carlo methods is solved.

The initial mixed boundary value problem (1.1)–(1.4), after discretization in ⇐= ok?

the time variable 𝑡 only, for𝑛= 3, with approximation error𝑂(𝜏²) for equation (1.1), takes the form

𝑢^𝑖+1(𝑥)−2𝑢^𝑖(𝑥) +𝑢^𝑖−1(𝑥)

𝜏² −Δ𝑢^𝑖+1(𝑥) +𝛾²𝑢^𝑖+1(𝑥) =𝑓^𝑖(𝑥),

𝑖= 1,2, . . . , 𝑀−1, 𝜏 =𝑇/𝑀, 𝑥∈Ω, (1.5)

𝑢⁰(𝑥) =𝜑(𝑥), 𝑥∈Ω, (1.6)

𝑢¹(𝑥)−𝑢⁰(𝑥)

𝜏 =𝜓(𝑥), 𝑥∈Ω, (1.7)

𝛼^𝑖+1(𝑥)𝑢^𝑖+1(𝑥) +𝛽^𝑖+1(𝑥)∂𝑢^𝑖+1(𝑥)

∂n =𝑔^𝑖+1(𝑥), 𝑖= 1,2, . . . , 𝑀−1, 𝑥∈Ω.

(1.8) We can also write

L𝑢^𝑖+1(𝑥) =𝐹^𝑖(

𝑓^𝑖(𝑥), 𝑢^𝑖−1(𝑥), 𝑢^𝑖(𝑥)) ,

𝑖= 1,2, . . . , 𝑀−1, 𝑥∈Ω, (1.9) 𝛼^𝑖+1(𝑥)𝑢^𝑖+1(𝑥) +𝛽^𝑖+1(𝑥)∂𝑢^𝑖+1(𝑥)

∂n =𝑔^𝑖+1(𝑥),

𝑖= 1,2, . . . , 𝑀 −1, 𝑥∈Ω, (1.10) whereL≡(

Δ−𝜏²𝛾²)

is the elliptic (Helmholtz) operator.

2. Monte Carlo methods

The main idea of the Monte Carlo methods: we construct the probability value or the probability process in such a way that the mean value is the solution of the given problem. Then, as a rule, the variance is the precision of the solution. From

problems (1.9), (1.10) there exists an integral equation of the second type ⇐= ok?

𝑢(𝑥) =

∫

Ω

𝑘(𝑦, 𝑥)𝑢(𝑦)𝑑𝑦+𝑣(𝑥), 𝑥, 𝑦∈Ω, (2.1)

where𝑣(𝑥) is a given function and𝑘(𝑦, 𝑥) is the kernel of this given function. The ⇐= ok?

integral equation (2.1) can be is solved by Monte Carlo methods if the integral operator𝐾 of this equation satisfies the condition

∥𝐾∥_𝐿¹_(Ω)<1. (2.2)

Ifcondition (1.10) holds then the integral equation (2.1) can be solved by

“random walk on spheres” and “random walk on balls” algorithms of the Monte Carlo methods. It is also possible to construct the𝜀-displaced estimations for𝑢(𝑥), cf. [4], [2], [5], [6], [9], [10].

It can also be solved by algorithms “random walk on spheres” and “random walk on lattices”, cf. [2], [5], [6], of the Monte Carlo and probability difference methods, cf. [11].

Let ∂Ω be a Lyapunov surface, and let the surface Ω be convex. Then the norm of the integral operator acting in𝐶(Ω) is less than 1. Hence, it is possible to apply the Neumann–Ulam scheme to equation (1.9), cf. [9].

The integral equation (2.1) can also be solved by “random walk on spheres”

and by “random walk on balls” algorithms of the Monte Carlo methods, cf. [10].

By reaching the𝜀-boundary the Markov chain is reflected with probability 𝑝_∂Ω𝜀 = ∣𝛽^𝑖+1(𝑥)∣

∣𝛼^𝑖+1(𝑥)∣+∣𝛽^𝑖+1(𝑥)∣

and adsorbed with probability

𝑞∂Ω𝜀= ∣𝛼^𝑖+1(𝑥)∣

∣𝛼^𝑖+1(𝑥)∣+∣𝛽^𝑖+1(𝑥)∣.

At transition from one condition to the following condition, the “weight” of node, which is defined by the recurrence relation

𝑄₀= 1, 𝑄_𝑖+1=𝑄_𝑖 𝑘(𝑥_𝑖, 𝑥_𝑖+1)

𝑝Ω(𝑥𝑖, 𝑥𝑖+1), 𝑖= 0,1, . . . ,

is taken into account. On a border, the “weight” of border, proportional to 𝑄∂Ω= 𝑔^𝑖+1(𝑥)

∣𝛼^𝑖+1(𝑥)∣+∣𝛽^𝑖+1(𝑥)∣, is taken into account.

Let us denote by ℎ a step of the difference scheme in each coordinate di- rection and by 𝑒𝑖 the coordinate unit vector in the 𝑖th coordinate direction. We approximate the domain Ω and the operator L by the finite difference method, 𝑝^ℎ(𝑥, 𝑥±𝑒𝑖ℎ),𝑝^ℎ(𝑥, 𝑥+𝑒𝑖ℎ±𝑒𝑗ℎ),𝑝^ℎ(𝑥, 𝑥−𝑒𝑖ℎ±𝑒𝑗ℎ) and𝑝^ℎ(𝑥, 𝑦) = 0, for the others 𝑥, 𝑦 ∈Ω∈𝑅^𝑛_ℎ. Function 𝑝^ℎ(𝑥, 𝑦) is nonnegative, the sum in 𝑦 is equal to 1 for each 𝑥. This means that 𝑝^ℎ(𝑥, 𝑦) is the probability of transitions of some Markov chain that we denote by{

𝜉_𝑛^ℎ}

⋅𝑝^ℎ(𝑥, 𝑦), and these will be the coefficients ⇐= ok?

in finite difference approximation.

3. Probability difference method

We divide a discrete border into the reflecting part∂Ω^ℎ_𝑅 and the absorbing part

∂Ω^ℎ_𝐴. Then it is possible to construct the𝜀-displaced approximation of the unique decision in the point𝑥. For example, it is possible to construct a Markov chain by

“random walk on lattices” and to define { 𝜉^ℎ_𝑛}

along this chain.

Let the set∂Ω^ℎ_𝑅 approximate∂Ω “from within”. That is, either 𝑥∈Ω∩ 𝑅³_ℎ or 𝑥 ∈ ∂Ω or a straight line connecting 𝑥 with one of the nearest nodes 𝑥𝑖 ± 𝑒𝑖ℎ, 𝑥𝑖±𝑒𝑖ℎ±𝑒𝑗ℎor𝑥𝑖±𝑒𝑖ℎ∓𝑒𝑗ℎtouches∂Ω. The set is determined in Ω∩

𝑅³_ℎ. Let us define a digitization Ω_ℎ= Ω∩

𝑅³_ℎ−∂Ω^ℎ_𝑅 of interior Ω and a digitization of a stopping set∂Ω^ℎ_𝐴=𝑅³_ℎ−Ωℎ−∂Ω^ℎ_𝑅. Then𝑝^ℎ gives the transition probabilities of the approximating chain {

𝜉^ℎ_𝑖}

in Ω_ℎ. The chain breaks at the first contact

with ∂Ω^ℎ_𝐴. We notice that E𝑥

{𝜉_𝑛+1^ℎ −𝜉_𝑛^ℎ∣𝜉_𝑛^ℎ =𝑦𝑖 ∈∂Ω^ℎ_𝑅}

=𝜐(𝑦)ℎ/∣𝜐(𝑦)∣. It is coordinated so that the reflection from the point ∂Ω^ℎ_𝑅 happens along direction 𝜐(𝑦). Here 𝜐(𝑦) is the direction of hit into the interior node, see [2], [5], [6], [11].

After an approximation of domains Ω, ∂Ω and (1.9), (1.10), we obtain the problem by the finite difference method

L^ℎ𝑢^𝑖+1_ℎ =𝐹_ℎ^𝑖(

𝑓_ℎ^𝑖, 𝑢^𝑖−1_ℎ , 𝑢^𝑖_ℎ)

, (3.1)

𝛼^𝑖+1_ℎ 𝑢^𝑖+1_ℎ +𝛽^𝑖+1_ℎ 𝛿ℎ( 𝑢^𝑖+1_ℎ )

=𝑔_ℎ^𝑖+1, (3.2)

where L^ℎ is a finite difference approximation of the operator L, 𝛿_ℎ is the finite difference approximation of the operator _∂n^∂ . Then we have the following

Theorem 1. The Neumann–Ulam scheme is applicable to the finite difference mixed problem (3.1)–(3.2).

Proof. Let us consider the difference problem (1.5)–(1.7) and approximate it by 𝑥. We prove the theorem for the case Ω ∈ ℝ¹ ≡ [0,1]. Then we divide [0,1] in 𝑁 parts with step ℎ= 1/𝑁 by 𝑥. Then we derive the following finite-difference problem for𝑘= 1, . . . , 𝑁−1,

ℎ²𝑢^𝑖+1_𝑘 −2ℎ²𝑢^𝑖+1_𝑘 +ℎ²𝑢^𝑖−1_𝑘 −𝜏²𝑢^𝑖+1_𝑘+1+ 2𝜏²𝑢^𝑖+1_𝑘 −𝜏²𝑢^𝑖+1_𝑘−1+ℎ²𝜏²𝛾²𝑢^𝑖+1_𝑘 =ℎ²𝜏²𝑓_𝑘^𝑖. (3.3) The first-order of approximation of∂𝑡𝑢(𝑥,0) is𝑂(𝜏²). We use the following obvious equalities:∂𝜏𝑢(𝑥,0) =𝑑𝑢(𝑥,0)/𝑑𝑡=𝜓(𝑥) at𝑡= 0,𝑢(𝑥,0) =𝑢0(𝑥) =𝜑(𝑥) and

∂𝜏𝑢(𝑥,0) = 𝑑𝑢(𝑥,0)

𝑑𝑡 = 𝑢(𝑥, 𝜏)−𝑢(𝑥,0)

𝜏 =∂𝑡𝑢(𝑥,0) +𝜏

2∂_𝑡²𝑢(𝑥,0) +𝑂(𝜏²). (3.4) From (1.1) when𝑡= 0 and using the first initial condition we get:

∂_𝑡²𝑢(𝑥,0) =∂_𝑥²𝑢(𝑥,0)−𝛾²𝑢(𝑥,0) +𝑓(𝑥,0) = 𝑑²𝜑(𝑥)

𝑑𝑥² −𝛾²𝜑(𝑥) +𝑓(𝑥,0). (3.5) Now from (3.5) we derive

𝜏

2∂_𝑡²𝑢(𝑥,0) = 𝜏 2

(𝑑²𝜑(𝑥)

𝑑𝑥² −𝛾²𝜑(𝑥) +𝑓(𝑥,0))

. (3.6)

We put into (3.4) the expression𝜏∂_𝑡²𝑢(𝑥,0)/2 from (3.6), and we get

∂𝜏𝑢(𝑥,0)−𝜏 2

(𝑑²𝜑(𝑥)

𝑑𝑥² −𝛾²𝜑(𝑥) +𝑓(𝑥,0))

=∂𝜏𝑢(𝑥,0) +𝑂(𝜏²).

Hence

∂𝜏𝑢(𝑥,0)≈𝑢¹_𝑘−𝑢⁰_𝑘

𝜏 =𝜓𝑘+𝜏 2

(𝜑_𝑘+1−2𝜑_𝑘+𝜑_𝑘−1

ℎ² −𝛾²𝜑𝑘+𝑓𝑘

)+𝑂(𝜏²) or

𝑢¹_𝑘=𝜑_𝑘+𝜏𝜓_𝑘+𝜏 2

(𝜑𝑘+1−2𝜑𝑘+𝜑𝑘−1

ℎ² −𝛾²𝜑_𝑘+𝑓_𝑘)

+𝑂(𝜏²). (3.7) Thus, we show that∂_𝑡𝑢(𝑥,0) (second initial condition) is also approximated with accuracy of 𝑂(𝜏²). The finite-difference equation (3.7) with boundary condition

(1.8) on every time layer (𝑖+1) is solved by the sweep method. Indeed, from (3.3) for𝑖= 1,2, . . . , 𝑀−1, 𝑘= 1,2, . . . , 𝑁−1 we get

−𝜏²𝑢^𝑖+1_𝑘+1+(

2𝜏²+ℎ²+ℎ²𝜏²𝛾²)

𝑢^𝑖+1_𝑘 −𝜏²𝑢^𝑖+1_𝑘−1= 2ℎ²𝑢^𝑖_𝑘−ℎ²𝑢^𝑖−1_𝑘 +ℎ²𝜏²𝑓_𝑘^𝑖. (3.8) Sufficiency condition of convergence for the sweep method of system (3.8) is satis- fied because the inequality is true:∣2𝜏²+ℎ²+ℎ²𝜏²𝛾²∣>∣−𝜏²∣+∣−𝜏²∣, where𝜏is the step by time𝑡,𝜏 >0,ℎis the step by space variable𝑥,ℎ >0, and parameter 𝛾²>0. Thus, the solution of system (3.8) exists and it can be solved numerically by the sweep method. Now, (3.8) is written for each (𝑖+1) in the matrix form

Au=F, (3.9)

where matrixAhas 3 diagonals: diagonal elements are 2𝜏²+ℎ²+ℎ²𝜏²𝛾², upper and lower diagonal elements are−𝜏², the rest of the elements are zeros. The column vectorFis known because of the right side of the equation, initial and boundary conditionsF=F(𝑢^𝑖_𝑘, 𝑢^𝑖−1_𝑘 , 𝑓_𝑘^𝑖), and the column vectoruis an unknown vector to be found. The solution of system (3.9) can be written in the form

u=A⁻¹F. (3.10)

Referring to [7] we can write out all eigenvalues of the matrixA:

𝜆_𝑘( A)

= 2𝜏²+ℎ²+ℎ²𝜏²𝛾²−2√

𝜏²𝜏²cos(𝑘𝜗), 𝑘= 1,2, . . . , 𝑁, 𝜗= 𝜋 𝑁+ 1. If𝑁≫1, then𝜆_𝑘(

A)

≥ℎ²+ℎ²𝜏²𝛾²>0. From the condition∣𝜆_𝑘( A)

∣<1 we get a relation between the steps𝜏 and ℎ:

−√ 1−ℎ² ℎ𝛾 < 𝜏 <

√1−ℎ² ℎ𝛾 .

When it is considered that 𝜏 >0 and for 𝛾 > 0 we get the condition (Courant type condition)

𝜏 <

√1−ℎ²

ℎ𝛾 . (3.11)

An analogous approach was carried out in the [8]. If condition (3.11) is true, then the iterational process for the solution of system (3.9) (ex., Jacobi method) converges. The existence of a solution of system (3.9) (the same for system (3.8)), the convergence of the numerical method of sweep (inversion of matrixA) for that system, as well as the discrete solution are constructed on absorbing Markov’s chain, that terminates on 𝜀-border, since for (3.9) the iterational process con- verges and (1.8) is true. The mixed boundary condition (1.8) consists of two parts:

Dirichlet’s and Neumann’s; due to Dirichlet’s condition Markov’s chain is absorbed on the𝜀-border, that is, it terminates. This proves the usability of the Neumann–

Ulam scheme for the solution of (3.8). This system can be solved by “random walk on lattices”, that is, by the probability difference method. The theorem is proved.

Now we prove the applicability of the Neumann–Ulam scheme to (3.8) dif- ferently. We write (3.8) on time layers (𝑖+1) in the form

u_r=Ku_l+Φ, (3.12)

whereKis a two-diagonal matrix (operator): its upper and lower diagonal elements are𝜏²/(2𝜏²+ℎ²+ℎ²𝜏²𝛾²); the rest of the elements are zero,u_r=u_r(u^𝑖+1_𝑘 ),u_l= u_l(u^𝑖+1_𝑘+1,u^𝑖+1_𝑘−1),Φ=Φ(

u^𝑖_𝑘,u^𝑖−1_𝑘 ,f_𝑘^𝑖)

. Now we show that the operator (matrix)K is a compressive operator. We calculate the eigenvalues 𝜆𝑘(K) of the matrix K, cf. [7]:

𝜆_𝑘( K)

= −2𝜏²

2𝜏²+ℎ²+ℎ²𝜏²𝛾²cos(𝑘𝜗), 𝑘= 1,2, . . . , 𝑁, 𝜗= 𝜋 𝑁+ 1. It is easy to see, that for𝜏 >0,ℎ >0,𝛾²>0, the condition for the spectral radius of the matrixK,

𝜌( K)

<1 (3.13)

is true for any𝜏 >0 andℎ >0. In that case the iterational process (by 𝑘, where 𝑘= 0 and for𝑘=𝑁 the boundary conditions are true) converges for (3.12) also;

condition (3.13) is now the necessary and sufficient condition of convergence of the iterational process for (3.12). It means that we can use the Neumann–Ulam scheme for the system (3.12), that is, that the system (and (3.8) also) can be solved by the probability difference methods: by constructing the Markov chains converging to the𝜀-border, and all its trajectories terminate on the𝜀-border because of Dirich- let’s conditions present on the borders. Thus, the theorem is proved again. □

Analogous algorithms can be found in [8], [9], [10].

The algorithm of constructing the Markov chains from [11] allows us to con- struct Markov chains, and the estimate discrete solution of the system (3.8),

𝑢^ℎ=E𝑥

[_𝑁

ℎ−1

∑

𝑖=0

𝐶_𝑖^ℎ𝑓( 𝜉^ℎ_𝑖)

△𝑡^ℎ_𝑖𝐼_Ω^ℎ( 𝜉_𝑖^ℎ)

+𝐶_𝑁^ℎ_ℎ−1𝑟( 𝜉_𝑁^ℎ_ℎ)

+

𝑁∑ℎ−1 𝑖=0

𝐶_𝑖^ℎ(

−𝑔( 𝜉^ℎ_𝑖))

𝑑𝜇^ℎ_𝑖 ]

is the unique discrete solution of the problem (the same discrete solution of problem (3.1), (3.2)), whereE𝑥is an expectation.

Remark 2. The common error of the probability difference method depends on the following parameters:

1) 𝜏 step by time 𝑡,𝜏 >0, as𝑂(𝜏²);

2) ℎstep by space variable𝑥,ℎ >0, as𝑂(ℎ²);

3) parameter 𝛾²>0, and

4) it depends on𝜀-border linearly.

References

[1] Michael Ruzhansky and James Smith,Dispersive and Strichartz estimates for hy- perbolic equations with constant cofficents.MSJ Memoirs,Vol. 22,(2010).

[2] Kanat K. Shakenov.Solution of Mixed Problem for Elliptic Equation by Monte Carlo and Probability-Difference Methods.CP1076. 7th International Summer School and Conference. Melville,New York. American Institute of Physics (AIP). Conference Proceedings,Volume 1076,(2008) 213–218.

[3] A. Haji-Sheikh and E.M. Sparrow,The floating random walk and its application to Monte Carlo solutions of heat equations. SIAM. Journal on Applied Mathematics, Vol. 14,2,Printed in U.S.A. (1966),370–389.

[4] K. Shakenov,Solution of one problem of linear relaxational filtration by Monte Carlo methods. International Conference on Computational Mathematics (ICCM 2002), Part I. ICM & MG Publisher. Novosibirsk,Russia,(2002),276–280.

[5] K. Shakenov,Solution of one mixed problem for equation of a relaxational filtration by Monte Carlo methods. Notes on Numerical Fluid Mechanics and Multidisciplinary Design,Springer Berlin – Heidelberg. Volume 93/2006. Book. Advances in High Performance Computing and Computational Sciences. Chapter 71. (2006),205–210.

[6] K.K. Shakenov, Solution of problem for one model of relaxational filtration by probabilitly-difference and Monte Carlo methods.Polish Academy of Sciences. Com- mittee of Mining. Archives of Mining Sciences. Volume 52,Issue 2,Krakow,(2007), 247–255.

[7] R. Bellman,Introduction to matrix analysis.McGraw–Hill Book Company,Inc. New York Toronto London (1960).

[8] S.M. Ermakov,K.K. Shakenov,On the applications of the Monte Carlo method to Navier–Stokes equations.Bulletin of Leningrad State University. Series Mathematics, Mechanics,Astronomy. No. 6267-B86. Leningrad,(1986). 1–14.

[9] S.M. Ermakov,Monte Carlo method and adjacent questions.Nauka,Moscow (1975).

[10] S.M. Ermakov,V.V. Nekrutkin,A.S. Sipin,Random process for solution of classical equations of mathematical physics.Nauka,Moscow (1984).

[11] Harold J. Kushner,Probability Methods of Approximations in Stochastic Control and for Elliptic Equations.Academic Press. New York – San Francisco – London. (1977).

Kanat Shakenov al-Farabi avenue,71

al-Farabi Kazakh National University 050040,Almaty,Kazakhstan

e-mail:shakenov2000@mail.ru