II INTERNATIONAL SCIENTIFIC CONFERENCE OF YOUNG RESEARCHERS
Baku Engineering University
12
27-28 April 2018, Baku, Azerbaijan
q
z
p
z
2
0
1
2
p
q
z
p
z
With solutions
1
1
z
and
p
q
z
2
.This gives, for
q
p
, the following general solution to (2)
k
k
k
p
q
C
C
A
2
1
1
where the constants
1
C
and
2
C
are determined
by the boundary conditions
0
0
A
,
0
2
1
C
C
,
1
1
1
b
a
p
q
C
1
b
a
A
,
1
2
1
b
a
p
q
C
C
,
1
1
2
b
a
p
q
C
and
b
a
k
k
b
a
b
a
k
p
q
p
q
p
q
p
q
p
q
A
1
1
1
1
1
1
b
a
a
a
p
q
p
q
A
1
1
For
q
p
, we get
1
1
2
1
2
1
k
k
k
A
A
A
The characteristic polynomial
0
1
2
2
z
z
has the double roots
1
2
1
z
z
, so that we
need one more solution.
k
A
k
satisfies the given equation so that
k
C
C
A
k
k
2
1
1
0
0
A
0
1
C
1
b
a
A
b
a
C
1
2
Finally,
we obtain
b
a
k
A
k
b
a
a
A
a
In general, we can write that the probability that A ruins B (the particles is absorbed at
b
x
) is
2
1
if
2
1
if
1
1
p
b
a
a
p
p
q
p
q
A
b
a
a
a
Another interesting fact related to random walk is Brownian motion. In 1827,Robert Brown ,who
is
English botanist, found that pollen seeds suspended in water move around randomly as shown.
