Mathematical Models in Environmental Problems by G. I. Marchuk

By G. I. Marchuk

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J 1 S i m i l a r l y t o t h e above p r o c e d u r e , we s h a l l d e t e r m i n e t h e r e g i o n u c G i n which i n e q u a l i t y ( 1 . 4 4 ) remains v a l i d . 45). Figure 2 . 4 i s a g r a p h i c a l s o l u t i o n f o r t h i s problem. F i g u r e 2 . 5 g i v e s FIGURE 2 . 4 t h e i s o l i n e s of t h e f u n c t i o n ( 1 . 4 0 ) , with x @ ( x , t ) , which a r e s o l u t i o n s f o r d i r e c t problem found from t h e s o l u t i o n of t h e a d j o i n t problem.

On t h e o t h e r hand, s o l u t i o n of t h e a d j o i n t problem g i v e s i n f o r m a t i o n on how much of t h e p o l l u t a n t e m i t t e d by a p o l l u t i o n s o u r c e a t an a r b i t r a r y moment w i l l f a l l down a t a f i x e d p o i n t and a t a given moment. 25) of t h e f u n c t i o n a l . 7765 Value 20 of l e v e l 1 . 1 5 5 3 30 28 26 24 22 20 78 76 74 72 70 a , , , , , , ~ , , , , ~ , , , , ~, ,, , ,, ,,, ~, , , 1 2 4 6 8 70 , / , , I/ , 72 74 76 , , 1 , , , , I/ 78 20 / , I 1 1 I I I I I C ~ ~ I f i ~ ~ , I , , / ~ 22 24 26 2 8 30 FIGURE 2 .

5) 0 The second condition follows from the requirement that the solution must be continuous everywhere, including the point x = at $+ = $- x 0’ x = x . 3) and ( 4 . 5) and ( 4 . 1. 1. process results in the distribution which falls exponentially and symmetrically in both directions from the source that at x = x . It is a simple matter to verify 28 Chapter 1 I m $ (x) dx = Q, 4 We now turn to a more interesting case where the air stream velocity is non-zero. Suppose that it is constant and positive.

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