1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

2.13 Comments and References 205

i

m− 3 − 2 − 10123

00 0 0 1 0 0

10 0 0. 500 .50 0

20 0. 25 0 0. 50 0. 25 0

30. 125 0 0. 375 0 0. 375 0 0. 125

Table 3 Random-walk probabilities

probabilities are related by the equation

ui(m+ 1 )=

1

2 ui−^1 (m)+

1

2 ui+^1 (m).

Theu’s are completely determined once an initial probability distribution is
given. For instance, if the particle is initially at point zero(u 0 ( 0 )=1,ui( 0 )=0,
fori=0), theui(m)are formed by successive applications of the difference
equation, as shown in Table 3.
The equation may be transformed into a close relative of the heat equation.
First, subtractui(m)from both sides:

ui(m+ 1 )−ui(m)=

1

2

(

ui+ 1 (m)− 2 ui(m)+ui− 1 (m)

)

.

Next divide by x^2 /2ontherightandby t·( x^2 / 2 t)on the left to obtain

ui(m+ 1 )−ui(m)

t

2    t

(   x)^2 =

ui+ 1 (m)− 2 ui(m)+ui− 1 (m)
( x)^2.
If both the time interval tand the step length xare small, we may think
ofui(m)as being the value of a continuous functionu(x,t)atx=i x,t=
m t. In the limit, the difference quotient on the left approaches∂u/∂t.The
right-hand side, being a difference of differences, approaches∂^2 u/∂x^2 .The
heat equation thus results if, in the simultaneous limit as xand ttend
to zero, the quantity 2 t/( x)^2 approaches a finite, nonzero limit. In this
context, the heat equation is called theFokker–Planck equation.Moredetails
and references may be found in Feller,Introduction to Probability Theory and
Its Applications.
We h a v e u s e d t h e t e r mlinear partial differential equationseveral times. The
most general such equation, of second order in two independent variables, can
be put in the form

A∂

(^2) u
∂x^2 +B
∂^2 u
∂x∂t+C
∂^2 u
∂t^2 +D
∂u
∂x+E
∂u
∂t+Fu+G=^0 ,
whereA,B,...,Gare known — perhaps functions ofxandtbut not ofuor
its derivatives. IfGis identically zero, the equation is homogeneous. Of course,