David Parra-Guevara et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.763-776 emission rate Q(t ) . The condition (67) means that there is no pollutant flow at x 0 (the closed boundary), and condition (68) means that at x 1 (the open boundary) the flow of pollutant is proportional to with 0 . As a result, the mass of pollutant in D increases due to the emission rate Q and decreases because of the outflow at the boundary point x 1 according to the following mass balance equation:

( x, t )dx Q(t ) (1, t ) t 0 1

2 A 2 x

2

( x, t ) dk Q( )e t d cos(k x) k 1

2 k

0

identity

A , g , A g

diffusion model (66)-(69) leads to the following adjoint model

g 2 g 2 p( x, t ), 0 x 1, 0 t T t x g (0, t ) 0, t 0 x g (1, t ) g (1, t ), t 0 x g ( x,T ) 0, 0 x 1 the

forcing

2 sin k ]1 , k 1, 2,K ,

is an orthogonal system of functions

(eigenfunctions), and the corresponding frequencies k (eigenvalues) are the roots of equation

cos( )-sin( ) = 0 . Such frequencies can be efficiently calculated by Newton’s method [35]. Note that for the numerical experiments of this section we used the Fejér series [36,37], calculated from the Fourier series (70), in order to have the pointwise convergence to ( x, t ) . That is, in practice we consider in (70) the Fejér coefficients: www.ijera.com

is and

(71) (72) (73) (74)

defined R is

as the

tj

(R, t j ) Q(t ) g j ( x0 , t )dt 0

that corresponds to equation (11). Here, the adjoint function g j can be expressed as

g j ( x0 , t ) ck e

k2 t j t

k 1

cos(k x)k 1

the

(70)

where

d k 2cos(k x0 )[1

to

monitoring point in domain D . As it was shown in Section II, the use of solution of adjoint problem (71)-(74) leads to the formula

By solving the Sturm-Liouville problem for the diffusion model (66)-(69), its non-negative solution can be expressed by Fourier series [34] as:

Lagrange

p( x, t ) ( x R ) (t t j ) ,

A , dx 2 (1, t ) x 0

instead of d k , where K is the truncation number of the series. On the other hand, the application of

where

is nonnegative:

t

) (k 1) d k 1 d k , k 1, 2,K K , K

Similarly to the general dispersion model (19)-(27), the diffusion model (66)-(69) is well posed, since its solution exists, is unique and continuously depends on the initial condition and forcing. This follows from the fact that the problem operator A defined as

1

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cos(k x0 ), t t j

(75)

where

ck 2cos(k R)[1

2 sin k ]1 , k 1, 2,K

are Fourier coefficients of series (75). In the numerical experiments, as it was mentioned before, we consider in the series (75) the Fejér coefficients

) (k 1) ck 1 ck , k 1, 2,K K , K instead of c k , where K is the truncation number of the series. In this example, due to the steady dispersion conditions, the basic kernel 773 | P a g e

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