Abstract

In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.

1. Introduction

Random differential equations (RDE) are defined as differential equations involving random inputs. In recent years, increasing interest in the numerical solution of (RDE) has led to the progressive development of several numerical methods. This paper is interested in studying the following random differential initial value problem (RIVP) of the form:

(1.1)

Randomness may exist in the initial value or in the differential operator or both. In [1,2], the authors discussed the general order conditions and a global convergence proof is given for stochastic Runge-Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. In [3,4], the authors discussed the random Euler method and the conditions for the mean square convergence of this problem. In [5], the authors considered a very simple adaptive algorithm based on controlling only the drift component of a time step. Platen, E. [6] discussed discrete time strong and weak approximation methods that are suitable for different applications. Other numerical methods are discussed in [7-12].

In this paper the random Euler and random RungeKutta of the second order methods are used to obtain an approximate solution for equation (1.1). This paper is organized as follows. In Section 2, some important preliminaries are discussed. In Section 3, the existence and uniqueness of the solution of random differential initial value problem is discussed and the convergence of random Euler and random Runge-Kutta of the second order methods is discussed. In Section 4, the statistical properties for the exact and numerical solutions are studied. Section 5 presents the solution of some numerical examples of first order random differential equations using random Euler and random Runge-Kutta of the second order methods showing the convergence of the numerical solutions to the exact ones (if possible). The general conclusions are presented in the end section.

2. Preliminaries

2.1. Mean Square Calculus [13]

Definition1. Let us consider the properties of a class of real r.v.’s whose second moments, are finite. In this case, they are called“second order random variables”, (2.r.v’s).

Definition 2. The linear vector space of second order random variables with inner product, norm and distance, is called an -space.

A s.p. is called a “second order stochastic process” (2.s.p) if for, the r.v’s are elements of -space.

A second order s.p. is characterized by

,.

2.1.1. The Convergence in Mean Square

A sequence of r.v’s converges in mean square (m.s) to a random variable

if i.e. or

where lim is the limit in mean square sense.

2.1.2. Mean-Square Differentiability

The random process is mean-square differentiable at t if exists, and is denoted by

3. Random Initial Value Problem (RIVP)

3.1. Existence and Uniqueness

Let us have the random initial value problem

(3.1)

where is second order random process. This equation is equivalent to integral equation

(3.2)

Theorem (3.1.1)

If we have the random initial value problem (3.1) and suppose the right-hand side function is continuous and satisfies a mean square (m.s) Lipschitz condition in its second argument:

(3.3)

where C is a constant or

(3.4)

where c(t) is a continuous function {because in every finite interval c(t) ≤ constant}.

then the solution of equation (3.1) exists and is unique.

The proof

The existence can be proved by using successive approximations. Let

(3.5)

and for n 1

(3.6)

For n 1 we obtain:

where

(3.7)

For n > 1 we obtain:

(3.8)

Successively, we can obtain the following:

Hence

(3.9)

Since:

is convergent for finite t,

(3.10)

hence we can have the following

(3.11)

Accordingly,

Hence:

This yield

Then exists. i.e.

(3.12)

Since is the general solution of equation (3.6) and is the general solution of equation (3.2).

To prove the uniqueness of the solution, let is a solution of the initial-value problem (3.1), or, which is the same, of the integral equation (3.2), and is the solution of

(3.13)

to prove the uniqueness of the solution we want to prove that

(3.14)

By subtraction (3.2) and the corresponding integral equation for

Since then:                     

(3.15)

(3.16)

i.e;                        

(3.17)

where.                      

 (3.18)

From equation (3.17) we have:

(3.19)

Note that: at we obtain then:

From (3.19) must satisfy the following condition:

(3.20)

which is in contradiction with being an independent free constant, hence the only solution of the integral equation (3.17) is 

(3.21)

Hence i.e., the solution of equation (3.1) exists and is unique.

3.2. The Convergence of Euler Scheme for Random Differential Equations in (m.s.) Sense

Let us have the random differential equation

(3.22)

where X₀ is a random variable and the unknown as well as the right-hand side f (X,t) are stochastic processes defined on the same probability space.

Definitions [6,7]

• Let g: is an m.s. bounded function and let h > 0 then The “m.s. modulus of continuity of g” is the function

• The function g is said to be m.s uniformly continuous in T if:

Note that:

(The limit depends on h because g is defined at every t so we can write)

In the problem (3.22), we find that the convergence of this problem depends on the right hand side (i.e. then we want to apply the previous definition on hence:

Let be defined on where S is bounded set in

Then we say that f is “randomly bounded uniformly continuous” in S, if

(note that)

3.2.1. Random Mean Value Theorem for Stochastic Processes

The aim of this section is to establish a relationship between the increment of a 2-s.p. and its m.s. derivative for some lying in the interval for. This result will be used in the next section to prove the convergence of the random Euler method.

Lemma (3.3.2) [6,7]

Let is a 2-s.p., m.s. continuous on interval . Then, there exists such that

(3.25)

The proof

Since is m.s. continuous, the integral process

is well defined and the correlation function

is well defined, is a deterministic continuous function on T × T.

For each fixed r, the function is continuous and by the classic mean value theorem for integrals, it follows that:

(3.26)

Note that by definition of expression (3.26) can be written in the form

Since

We must prove that for the value satisfying (3.26) one get:

(3.27)

The proof of (3.27)

As

(3.28)

and since:

then by substituting in (3.28)

(3.29)

And since:

then by substituting in (3.28) we have:

i.e. we obtain

Theorem (3.3.1) [6,7]

Let be a m.s. differentiable 2-s.p. in and m.s. continuous in. Then, there exists such that,

The proof

The result is a direct consequence of Lemma (3.3.2) applied to the 2-s.p.

And the integral formula

(3.30)

The proof of (3.30)

Let be a m.s. differentiable on T and let the ordinary function be continuous on whose partial derivative exist If                

 (3.31)

Then

(3.32)

Let in Equations (3.31) and (3.32) we have the useful result that:

If is m.s. Riemann integrable on T then:

Then we have:

3.2.2. The Convergence of Random Euler Scheme

In this section we are interested in the mean square convergence, in the fixed station sense, of the random Euler method defined by

(3.33)

where and are 2-r.v.’s , , and f:, satisfies the following conditions:

C1: is randomly bounded uniformly continuousC2: satisfies the m.s. Lipschitz condition

where

(3.34)

Note that under hypothesis C1 and C2, we are interested in the m.s. convergence to zero of the error

(3.35)

where is the theoretical solution 2-s.p. of the problem (3.22),.

Taking into account (3.22), and Theorem (3.3.1), one getsSince from (3.22) we have at then

Note  and we can use instead of and from Theorem (3.3.1) at then we have:

then

Note that we deal with the interval and hence was the starting in the problem (3.22) and here is the starting and since Euler method deal with solution depend on previous solution and if we have instead of then we can use instead of.

Then the final form of the problem (3.22) is

, for some

(3.36)

Now we have the solution of problem (3.22) is

At then and the solution of Euler method (3.33) is

Then we can define the error

By (3.33) and (3.36) it follows that

This implies

Hence

(3.37)

Since:

(3.38)

Since the theoretical solution is m.s. bounded in

, and Under hypothesis C1, C2 We obtain

•

• (*)

Since is Lipschitz constant (from C2) and from Theorem (3.3.1) we have and note that the two points are and in (*) then we have:

Since and

•

Then by substituting in (3.38) we have

(3.39)

Then by substituting in (3.37) we have

Since:

is geometrical sequence.

Then:

Then we get

Taking into account that where

.

(3.40)

Note that:

The term: as

(as)      

(3.41)

And the second term:

we have:

(3.42)

The first limit in (3.42) equal zero and:

The computation of as follows:

Let then by tacking the logarithm of the two sides we have:

By using the (L’Hospital’s Rule):

(3.43)

Then which implies that

hence

By substituting in (3.42):

(3.44)

By substituting from (3.44) and (3.42) in (3.40) hence i.e., converge in m.s to zero as

hence.

3.3. The Convergence of Runge-Kutta of Second Order Scheme for Random Differential Equations in Mean Square Sense

In this section we are interested in the mean square convergence, in the fixed station sense, of the random Runge-Kutta of second order method defined by

(3.45)

where and are 2-r.v.’s, , and f:, satisfies the following conditions:

C1: is randomly bounded uniformly continuousC2: satisfies the m.s. Lipschitz condition

where

(3.46)

Note that under hypothesis C1 and C2, we are interested in the m.s. convergence to zero of the error

(3.47)

where is the theoretical solution 2-s.p. of the problem (3.22),.

Taking into account (3.22), and Theorem (3.3.1), one getsSince from (3.22) we have at then

Note and we can use instead of

And from Theorem (3.3.1) at then we obtain

Note that we deal with the interval and hence was the starting in the problem (3.22) and here is the starting  and since Euler method deal with solution depend on previous solution and if we have instead of we can use instead of then the final form of the problem (3.22) is

, for some

(3.48)

Now we have the solution of problem (3.22) is

At then and the solution of Runge-Kutta of 2 order method (3.45) is

Then we can define the error

By (3.45) and (3.48) it follows that

Then we obtain:

By taking the norm for the two sides:

(3.49)

Since:

(3.50)

Since the theoretical solution is m.s. bounded in

, and Under hypothesis C1, C2 We have

•

• (*)

where Is Lipschitz constant (from C2) and:

From Theorem (3.3.1) we have

and note that the two points Are and in (*) then

where and

•

Then by substituting in (3.50) we have

(3.51)

And another term:

Since:

•

•

where. Is Lipschitz constant (from C2) and:

From Theorem (3.3.1) we have

and note that the two points are and in (*) then we have:

where and

And the last term:

Then by substituting in (3.49) we have

Then we have:

Since:

is geometrical sequence then we have:

Then we get:

Taking into account that where

(3.52)

Note that:

The term:

and the second term:

we have:

(3.53)

The first limit in (3.53) equals zero and:

The computation of

is as follows:

Let then by tacking the logarithm of the two sides we have:

By using the (L’Hospital’s Rule):

(3.54)

Then hence:

By substituting in (3.53):

(3.55)

By substituting from (3.55) and (3.53) in (3.51) then we obtain i.e. converges in m.s to zero as hence

4. Some Results

Theorem 4.1

Let {X_n, n = 0, 1, ···}, {Y_n, n = 0, 1, ···} be sequences of 2-r.v’s over the same probability space and let a and b be deterministic real numbers.

Suppose: and

Then:

1)

2)

3)

4)

5)

Definition 4.1 [13]. “The convergence in probability”

A sequence of r.v’s converges in probability to a random variable as if

Definition 4.2 [13]. “The convergence in distribution”

A sequence of r.v’s converge in distribution to a random variable as if

Lemma (4.1) [13]

The convergence in m.s implies convergence in probability

Lemma (4.2) [13]

The convergence in probability implies convergence in distribution

Theorem 4.2

If then PDF of PDF of i.e.;

Proof Since we have shown that If then

i.e., if then

Then we have: then

5. Numerical Examples

Example (5.1)

The differential equation with random term in it and random initial condition

K, D are independent Poisson random variables with joint PDF

1) The exact solution,

2) The numerical solution

Using the Random Euler Method:

at n = 1

at n = 2

at n = 3

at n = 4

and so on…

Then the general numerical solution is

i.e.,.

This can be written in another form:

.

We can prove that:

1)

Proof Since (if and only if)

Then:

where

i.e;

We can verify theorem (4.1) as follows

2)

Proof

Then:

i.e.;.

3)

Proof Since

Then we have:

Then by taking the limit:

i.e..

4)

Proof

i.e.,.

5)

Proof Since

Let us define Z = D. Then the inverse transformation is:

, D = Z then we have D = Z and

Then:

Since then hence

.

For a numerical solution:

since

Let then

Then:

where

Then by taking the limit we have

i.e.;

B. Using the Random Runge-Kutta method:

at n = 0

At n = 1

At n = 2

Then the general solution is:

,

,

.

This can be written in another form:

.

We can prove that:

1)

Proof

Since (if and only if)

where

i.e.;.

Verification of Theorem (4.1):

2)

Proof

i.e.;.

3)

Proof Since then:

i.e.;.

4)

Proof

i.e.

5)

Since

Let us define Z = D. Then the inverse transformation is:

D = z then we have D = z and

Since then this implies

.

Numericallysince

Let then

where

i.e.;.

Example (5.2)

Solve the problem

The exact solution

The numerical solution by the Euler method:

AT n = 1

At n = 2

At n = 3

And so on….

Then the general numerical solution:.

It is clear that:

1)

Since

Verification of Theorem (4.1)

It is clear that:

2)

3)

Since

4)

5)

Then which implies

Then which implies

Then:

6. Conclusions

The initially valued first order random differential equations can be solved numerically using the random Euler and random Runge-Kutta methods in mean square sense. The existence and uniqueness of the solution have been proved. The convergence of the presented numerical techniques has been proven in mean square sense. The results of the paper have been illustrated through some examples.

7. References

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[2] K. Burrage and P. M. Burrage, “General Order Conditions for Stochastic Runge-Kutta Methods for Both Commuting and Non-commuting Stochastic Ordinary Equations,” Applied Numerical Mathematics, Vol. 28, No. 2-4, 1998, pp. 161-177. doi:10.1016/S0168-9274(98)00042-7

[3] J. C. Cortes, L. Jodar and L. Villafuerte, “Numerical Soluion of Random Differential Equations, a Mean Square Approach,” Mathematical and Computer Modelling, Vol. 45, No. 7, 2007, pp. 757-765. doi:10.1016/j.mcm.2006.07.017

[4] J. C. Cortes, L. Jodar and L.Villafuerte, “A Random Euler Method for Solving Differential Equations with Uncertainties,” Progress in Industrial Mathematics at ECMI, Madrid, 2006.

[5]H. Lamba, J. C. Mattingly and A. Stuart, “An adaptive Euler-Maruyama Scheme for SDEs, Convergence and Stability,” IMA Journal of Numerical Analysis, Vol. 27, No. 3, 2007, pp. 479-506. doi:10.1093/imanum/drl032

[6] E. Platen, “An Introduction to Numerical Methods for Stochastic Differential Equations,” Acta Numerica, Vol. 8, 1999, pp. 197-246. doi:10.1017/S0962492900002920

[7] D. J. Higham, “An Algorithmic Introduction to Numerical Simulation of SDE,” SIAM Review, Vol. 43, No. 3, 2001, pp. 525-546. doi:10.1137/S0036144500378302

[8] D. Talay and L. Tubaro, “Expansion of the Global Error for Numerical Schemes Solving Stochastic Differential Equation,” Stochastic Analysis and Applications, Vol. 38, No. 4, 1990, pp. 483-509. doi:10.1080/07362999008809220

[9] P. M. Burrage, “Numerical Methods for SDE,” Ph.D. Thesis, The University of Queensland, 1999.

[10] P. E. Kloeden, E. Platen and H. Schurz, “Numerical Solution of SDE Through Computer Experiments,” Second Edition, Springer, Berlin, 1997.

[11] M. A. El-Tawil, “The Approximate Solutions of Some Stochastic Differential Equations Using Transformation,” Journal of Applied Mathematics and Computing, Vol. 164, No. 1, 2005, pp. 167-178. doi:10.1016/j.amc.2004.04.062

[12] P. E. Kloeden and E. Platen, “Numeical Solution of Stochastic Differential Equations,” Springer, Berlin, 1999.

[13] T. T. Soong, “Random Differential Equations in Science and Engineering,” Academic Press, New York, 1973.