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here we consider some aspects as say prof dr mircea orasanu and prof horia orasanu concerning the followed as

LAGRANGIAN AND STOCHASTIC QUESTION

ABSTRACT

1 Motivation and problem formulation

The time varying behavior of many physical phenomena can be described by deterministic ordinary differential equations. If we define the state of the physical system as x(t) we have

(1)

However when there are uncertainties, physical system behavior often can only be described in terms of probability and has to be described by means of a stochastic model. Therefore in this Chapter we discuss a stochastic differential equation as a model for a stochastic process Xt. We consider models of the following type:

(2)

where we have introduced a stochastic process Nt to model uncertainties in the underlying deterministic differential equation. The initial condition X0 is also assumed to be a random variable.

Example 4.1

Consider again the model for Biochemical-Oxygen Demand (BOD) in stream bodies as described in Chapter 2:

(3)

1 INTRODUCTION

Consider the stochastic BOD model (5) from Example 4.1:

or in terms of the Wiener process:

Since the white noise process in this stochastic model is a mathematical approximation of a noise process with a relatively short correlation scale, this SDE has to be interpreted in the Stratonovitch sense. Since the Euler scheme (14) can only be used for Ito equations, the model above is rewritten as an Ito SDE:

Different realizations of Bt can be generated using the next program.

ex7a

The next program can determine the mean and standard deviation of the BOD.

ex7b

4.7 Fokker Planck equation

Consider the Ito stochastic differential equation (10). To gain insight into the probability of exceedence of the process Xt we need to know the probability density function of Xt. This function can be obtained by solving the Fokker-Planck equation also known as the Kolmogorov forward equation:

(18)

where:

is the probability density function of Xt at time t given Xt=x0 at t0. The initial condition for equation (18) is:

Using Bayes’ rule it is easy to include the uncertainty due to the initial condition and to compute the probability distribution p(x,t) of Xt:

where p(x0,t0) is the probability distribution of the initial condition Xt at t0.

The Fokker-Planck equation is a deterministic partial differential equation that in general has to be solved numerically. For vector systems with dimension larger than, say, 3 this is very time consuming. In this case the probability density can be determined more efficiently by generating a large number of tracks of the underlying SDE.

Example 4.9

Consider again the SDE from Example 4.6:

The Fokker-Planck equation for this SDE is:

The initial condition for this equation is:

It is easy to verify that in this case the solution of the Fokker-Planck equation is:

4.8 Numerical approximation of stochastic differential equations

4.8.1 Order of convergence of a numerical scheme

Consider first the deterministic equation:

We can approximate this equation numerically with the Euler scheme:

where is the time step. Recall that the order of convergence of a numerical scheme for a deterministic differential equation is defined as follows (see Chapter 2):

Definition: The order of convergence is j if there exists a positive constant K and a positive constant such that for fixed :

for all .

We known that the local error of the Euler scheme is (see Chapter 2). The global error EN for fixed can be found easily by computing:

So for deterministic models we have that the global error is since we make times a local error of .

Now consider the stochastic case:

with the Euler scheme introduced in Section 4.5:

or with t=nt:

2 . FORMULATION

For deterministic differential equations the Taylor series expansion is an important method to analyze the order of convergence of a numerical scheme. Let us now study the stochastic case and derive a stochastic version of the Taylor expansion. Consider first again the Ito differential rule introduced in Section 4.6 for scalar systems:

(19)

where the operators L0 and L1 are defined as:

Ito’s differential rule holds for arbitrary functions . So we can apply the rule also for the functions f and g:

which results in:

and:

which results in:

Substituting these results in the stochastic equation (11) yields:

(19)

where we have assumed that the functions f and g are sufficiently smooth. Equation (20) is the stochastic Taylor expansion. By applying Ito’s rule again to the various integrants higher order terms of the expansion can be obtained.

The first terms of the stochastic Taylor expansion represents the stochastic Euler scheme discussed in Section 4.8.1:

or with t=nt:

By analyzing the error terms of equation (20) the order of convergence of the Euler scheme can be determined heuristically. Consider first the error term:

This deterministic error term introduces a local error of O(t2) and, as a consequence, a global error of O(t). For the two stochastic terms we have:

Both these stochastic terms introduce a strong local error of O(t1.5) and, as a consequence, a strong global error of O(t) (see the discussion in Section 4.8.1). Finally consider the last error term:

This stochastic term introduces a strong local error of O(t) and a strong global error O(t1/2). This last error term dominates and determines the strong order of convergence of the Euler scheme.

For weak order convergence many realizations are generated and averaged to determine an approximation of (see definition of weak order of convergence in Section 4.8.1):

Because of the averaging procedure all random error terms cancel out and vanish for increasing number of realizations. As a result for weak order of convergence only the first deterministic error term has to be taken into account resulting in a weak order of convergence of the Euler scheme of O(t). This implies that if we use the Euler scheme and generate many tracks then the individual tracks are only half order accurate (strong convergence) while for example the results on the mean and variance of the tracks are first order accurate (weak convergence). This is caused by the fact that the stochastic errors in the track wise computations cancel out in computing ensemble mean quantities like mean and variance.

Exercise 4.9

Consider the same Ito SDE as in Exercise 4.8. Now we use the Euler scheme to compute the mean of Xt using 1000 samples and compare the result with the exact mean.

ex9

From the figure we see (more or less) that the Euler scheme is O(t) in the weak sense. Repeat the experiment to demonstrate that the effect of statistical fluctuations is very large. This shows that a huge amount of sample is required before the O(t1/2) errors cancel out by the averaging and becomes relatively small compared to the remaining O(t) errors.

Consider again the error term that dominated the strong order of convergence of the Euler scheme:

and apply Ito’s differential rule to the integrant:

or:

and substitute the result in the Taylor expansion (19):

From this result we see that a more accurate scheme for scalar stochastic differential equations has been obtained:

(21) .

This scheme is called the Milstein scheme and is in the strong sense for scalar equations. For vector systems it generally only (except for very special differential equations when its accIt states that the circulation of a vector field around a closed path is equal to the integral of over the surface bounded by this path. It may be noted that this equality holds provided and are continuous on the surface.

Let us consider an area S that is subdivided into large number of cells as shown in the figure below.

Let cell has surface area and is bounded path while the total area is bounded by path . As seen from the figure that if we evaluate the sum of the line integrals around the elementary areas, there is cancellation along every interior path and we are left the line integral along path . Therefore we can write,

As we can write,

which is the Stoke’s theorem.

ES202

A SUMMARY ON INTEGRAL THEOREMS

DIVERGENCE THEOREM: Let V be a bounded region in R with a boundary S consisting of a piecewise smooth surface. Let be the outward unit normal to S and (x,y,z) be a vector field with components having partial derivatives that are continuous at all points of V and S, then

Region R

Surface S enclosing region R

STOKES’S THEOREM: Let S be a piecewise smooth orientablesurface with unit normal n and having as boundary the simple closed curve C. Let F(x,y,z) be a vector field whose components have continuous first order partial derivatives on S, then

S (surface)

C (boundary)

provided that the orientations of C and S are compatible.

GREEN’S THEOREM: If V is a volume enclosed by a regular surface S and if u(x,y,z) and (x,y,z) are scalar functions and if then divergence theorem on will give

Since , the integral of the right-hand side of the above equation will give the first term of Green’s theorem:

(a)

If we interchange u and in equation (a) we’ll have

(b)

Now subtract (b) from (a), we’ll obtain the second form of Green’s theorem

Special Cases:

i) If u = = , then equation (a) will be

where = and note that , so:

ii) If u= and =1, then and

GREEN’S THEOREM IN PLANE: Let R be a closed region in the x-y plane having

its boundary as simple, closed, piecewise smooth curve C. Let M(x,y) and

N(x,y) be continuous functions of x and y with continuous first order partial

derivatives in R. Then

where C is traversed in positive direction.

Corollary: The are bounded by a simple closed curve C is given by

Proof of Green’s theorem in plane: Let be a vector function defined in a closed region S in x-y plane with boundary C. For this special case we get :

; if , so Stokes’s theorem, yields

y

C

where . x

Introduction to Partial Differential Equations – Math 21a

If you took Math 1b here at Harvard, then you have already been introduced to the idea of a differential equation. Up until now, however, if you have already worked with differential equations then they’ve probably all been ordinary differential equations (ODEs), involving “ordinary” derivatives of a function of a single variable. Now that you have worked with functions of several variables in Math 21a, you are ready to explore a new area of differential equations, one that involves partial derivatives. These equations are aptly named partial differential equations (PDEs). During this short section of Math 21a, you will get a chance to see some of the most important PDEs, all of which are examples of linear second-order PDEs (the terminology will be explained shortly).

First, however, in case you haven’t worked with too many differential equations at this point, let’s back up a bit and review some of the issues behind ordinary differential equations.

Ordinary Differential Equations

A differential equation, simply put, is an equation involving one or more derivatives of a function y = f(x). These equations can be as straightforward as

(1) y = 3,

or more complicated, such as

(2) y + 12y = 0

or

(3) (x2 y ) + ex y – 3xy = (x3 + x).

There are a number of ways of classifying such differential equations. At the least, you should know that the order of a differential equation refers to the highest order of derivative that appears in the equation. Thus these first three differential equations are of order 1, 2 and 3 respectively.

Differential equations show up surprisingly often in a number of fields, including physics, biology, chemistry and economics. Anytime something is known about the rate of change of a function, or about how several variables impact the rate of change of a function, then it is likely that there is a differential equation hidden behind the scenes. Many laws of physics take the form of differential equations, such as the classic force equals mass times acceleration (since acceleration is the second derivative of position with respect to time). Modeling means studying a specific situation to understand the nature of the forces or relationships involved, with the goal of translating the situation into a mathematical relationship. It is quite often the case that such modeling ends up with a differential equation. One of the main goals of such modeling is to find solutions to such equations, and then to study these solutions to provide an understanding of the situation along with giving predictions of behavior.

In biology, for instance, if one studies populations (such as of small one-celled organisms), and their rates of growth, then it is easy to run across one of the most basic differential equation models, that of exponential growth. To model the population growth of a group of e-coli cells in a Petri dish, for example, if we make the assumption that the cells have unlimited resources, space and food, then the cells will reproduce at a fairly specific measurable rate. The trick to figuring out how the cell population is growing is to understand that the number of new cells created over any small time interval is proportional to the number of cells present at that time.

This means that if we look in

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