1. Introduction

Development in recent years in area of researches globally, in Mathematics and science related subjects such as, Engineering, Physics, Biology, Ecology, Hydrology, Economics, Investment, Finance, and population dynamics, just to mention but a few, have realised the importance of application of stochastic differential equations and the importance of random effects in most real problems. These are difficult to handle by the deterministic models [1, 2]. In view of this, there have been an increase in the need to construct stochastic models to simulate systems that deal with real life situations that contain uncertainties.

Many physical systems are modelled by stochastic differential equations where random effects (noise) are being modelled by a Brownian motion or what is called Wiener process [3]. Such differential equations are rarely solved analytically due to diffusion term involved. So numerical methods required and should be constructed in line with the principles of stochastic processes, to handle stochastic processes effects [4]. These models can offer more realistic representation of the physical system. Interesting enough, Runge-Kutta methods proves effective in handling stochastic differential equation theories that fits or handle stochastic processes, over some of the analytic methods [5, 6]. Therefore, there is a high need to develop and implement some stochastic Runge-Kutta methods for solving stochastic differential equations [7]. In this paper, an explicit Stochastic Rational Runge-Kutta method is derived based on the modified approach of stochastic Runge-Kutta methods for solving stochastic ordinary differential equations.

Consider ordinary differential equations of the form,

(1)

The general Rational Runge-Kutta schemes for the solution of (1) according to [8, 9, 10, 11, 12] is expressed as:

(2)

where

and

Consider the non-autonomous, one Wiener SDE of Stratonovich type:

(3)

(4)

We shall assume a solution of the form

(5)

This now motivate us to formulate an s-stage explicit stochastic Rational Runge-Kutta (SRRK) methods for the solution of (3) as

(6)

Where , N is a positive integer,

and

where are constants to be determined. We can classify SRRK methods, as follows:

If then the method is called semi-implicit.

If then the method is called explicit.

Otherwise it is called implicit.

Definition 1: (Strong convergence)

We say that a discrete time approximation converges strongly with order at time if there exist a positive constant , which does not depend on the maximum step size h, and , such that

holds for each is the number of subintervals of the interval, , is the exact solution at is the approximate solution at We shall employ the strong convergence principles in analysing our results.

2. Derivation of the Methods

In order to derive the one-stage explicit SRRK methods, let in (6) to obtain

(7)

where

, N is a positive integer,

With

Expand the R.H.S of (7) binomially, simplify and neglect terms in h of order two and higher to obtain

(8)

Expanding using Taylor series about we have:

(9)

(10)

and

(11)

Substituting(9), (10) and (11) in the R.H.S of (8) we get

(12)

But

Then (12) becomes

(13)

From Taylor series expansion of about we get

(14)

If we use the notations

(15)

since and let be the solution of the stochastic part denoted by ys and when:

Therefore, (14) becomes

(16)

For consistency,

(17)

Which is the simultaneous equations to be solved in three unknowns , where c₁ is a free parameter.

To determine a particular scheme of one-stage explicit Stochastic Rational Runge-Kutta Methods (SRRK):

Case 1, let

This gives

(18)

where

Where is Gaussian random normal distribution with mean zero and standard deviation one

Case 2, if

, we obtain

(19)

where are as defined in (18)

Case 3, if

, then,

(20)

where are as defined in (18)

Case 4, if

(21)

where are as defined in (18)

Thus, the schemes (18), (19), (20) and (21) are the one-stage SRRK methods.

3. Stability Analysis

Theorem 1: (Convergence, [13])

(i) Let the function be continuously jointly as a function of its three arguments, in the region defined by

(ii) Let satisfy a Lipchitz condition of the form for all points in then the method is convergent if and only if it is consistent.

Theorem 2 [18]: For the test equation the Euler-Maruyama method is numerical stable in mean if .

Note that the stability condition is the same as Euler method for ODEs according to Lambert (1991). In general the numerical stability conditions for the additive case are coincident with them for ODEs (Hermandez and Spigler, 1992).

Theorem 3 [18]: If the Euler-Maruyama method satisfies the numerical stable condition in mean, i.e. , the Euler-Maruyama method is asymptotically consistent in mean square.

For the stability analysis of the derived schemes, we shall use the Shur method and utilize the principles of [14, 15, 16, 17, 18, 19], that states, the stability analysis of stochastic methods correspond with its deterministic counterpart. Using the test equation

(22)

From (18)

Expanding and truncating terms in h of second and higher orders, we obtain

(23)

But and since

substituting these in (23), we get

is the characteristic polynomial

(24)

Therefore, (-2.0, 0) is the interval of the absolute stability of the one-stage scheme (18). Similarly, we can see that (18), (19), (20) and (21) all have the same interval of absolute stability. Which is called numerical stable in mean in stochastic sense.

4. Numerical Problems and Results

Problem 1

Consider the SDE [6]

With the exact solution given by:

Problem 2

Consider the SDE [20]

with exact solution

Therefore, the numerical solution using the explicit SRRK methods for the one-stage schemes as obtained in this work with absolute errors are given in the Tables 1 and 2 as seen below. The following notations will be used to represents results in the tables. PL, RAe1-3: Results from [6], Soheil, Results from [20] and JAk, Results obtained by our new methods.

Table 1. Numerical results of one-stage JAk explicit SRRK in comparison with [6] for Problem 1

Table 2. Numerical results of one-stage JAk explicit SRRK in comparison with [20] for Problem 2

5. Discussion of Results

With the one- stage explicit stochastic rational Runge-Kutta schemes (SRRK) denoted JAk in the numerical results tables. We derived family of four schemes of one-stage SRRK methods The family schemes for the one- stage were tested to solve numerical Problems 1and 2 from [6] and [20] respectively for some of the derived schemes as presented in tables 1 and 2. Matlab software (version 2010) was employed to run the simulations, based on normal distributed random numbers with mean zero and variance (standard deviation) one, i.e N(0,1). From Tables 1and 2 we can see the performance of our one stage schemes with the existing schemes in [6] and [20]. Also stability analysis of the one-stages developed were carried out using Schur method, in line with what we call mean and mean square stability principles in stochastic stability analysis discussed by some authors in section 3.0 and the stability analysis of the derived schemes showed they are bounded by the interval (-2.0, 0), just as that of deterministic one-stage explicit Runge-Kutta methods.

6. Conclusions

Clearly our one- stage schemes performs better in terms of convergence and accuracy, also the derived one -stage schemes was able to recover the results of their two stage schemes better. This means that our explicit SRRK methods is an alternative schemes to be used solve such problems. Also one is left with an options of choosing a scheme to work with in our case, unlike Euler Runge-Kutta method that has just a single scheme.