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THE STUDY OF ANALYTICAL SOLUTIONS OF
PARABOLIC EQUATIONS
By
name(id)
DEPARTMENT OF MATHEMATICS
VIRTUAL UNIVERSITY OF PAKISTAN
2021
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Contents
Irreversible Process ................................................................................................................................... 3
Time irreversibility of Heat Equation ........................................................................................................... 4
Reference ...................................................................................................................................................... 7
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Introduction
Irreversible Process
In science, process that is not reversible is called irreversible. This concept arises
frequently in the thermodynamics.
In thermodynamics, a change in the thermodynamics state of a system and all of its
surroundings cannot be precisely restored to its initial state by infinitesimal changes in some
property of the system without expenditure of energy. A system that undergoes an irreversible
process may still be capable of returning to its initial state. However, the impossibility occurs in
restoring the Thermodynamic to its own initial conditions. An irreversible process increases the
entropy of the universe. Because entropy is state function, the change in entropy of the system is
the same, whether the process is reversible or irreversible. The thermodynamics’ second law can
be used to determine whether a process is reversible or not.
The entropy equation can be written as
Q
dS
T

=+
where
is the entropy production which can be stated as
0 internally irreversible process,
0 internally reversible process,
0 impossible process.



=
Quantity availability (also called energy, work potential, has been used by engineers to
assess the maximum work that can be obtained for a combined system, closed or open, in a given
environmental condition. It is known that all macro-processes are irreversible, and the
availability is destroyed. The destruction of availability is called irreversibility.
Wall stated that energy is motion or ability to produce motion and energy is work or
ability to produce work. He also stated that time is experienced when energy is destroyed, i.e., an
irreversible process, which creates a motion in a specific direction, i.e., the direction of time. The
idea that availability can be destroyed is useful. This concept essentially brings together the first
and second laws.
Example:
Both the Heat equation and the diffusion equation describe processes which are
irreversible, because both equations have an odd time derivative.
[Keenan JH. Thermodynamics,1957]
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4
Time irreversibility of Heat Equation
Explanation:
The irreversibility associated with the heat equation, and alluded to it in terms of entropy,
disorder and randomness, maybe it is time to look back at its physical origins, particularly as it relates to
stochastic processes.
Consider a walker on the real line that, at regular intervals
t
, moves to the left or right,
with equal probabilities, a fixed distance
x
. If the walker starts walking at time
0t =
from
0x =
, what is the probability
of finding it in position
j
x x j x= =
at time
n
t t n t= =
? We
can answer this question recursively: in order to be in
j
xx=
at time
n
t
, the waker must have
been one step to the left or right the step before. Then
( )
11
1
..... 1
2
nn
jj
n
j
PP
P
−+
+
+
=
where the
1
2
is the probability of stepping in each direction. We can reorganize this expression
in the form
( )
1
2
11
2
2
1
..... 2
2
n n n n n
j j j j j
P P P P P
x
t t x
+
+−
+
=
To actually take the limit as
x
and
t
get small, we need some extra considerations.
First, we need to re-interpret the discrete
n
j
P
in terms of a function defined on the real line. The
natural choice is a probability density
( )
,xt
. In terms of this,
n
j
P
can be interpreted as the
integral of
( )
,
n
xt
between
1
2
j
x
and
1
2
j
x
+
. Then we need to take a succession of walks, with
decreasing time intervals and step-sizes. Yet this needs to be done so that the quotient
2
1
2
x
r
t
=
remains finite, or else the evolution equation will trivialize. Therefore the stepsize
x
needs to
scale as the square-root of the time interval
t
. With all this done, we can now the limit of (2),
and obtain the one-dimensional heat equation
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t xx

=
where the diffusivity
is the limit of r as
t
and
x
go to zero.
Let us now generalize this to arbitrary dimensions and more general walks that do not
have a fixed step
x
. Instead, consider a probability distribution
( )
,,y x t t
for all possible
steps
y
in
n
R
. The distribution may depend on the position x and time t from which we are
walking away, and is temporarily associated with a fixed time-interval
t
. Then the probability
density
( )
,x t t
+
is given by the following generalization of (1):
( ) ( )
( )
( )
, , , , , .... 3x t t x y t y x y t t dy
+ =
So,
( ) ( ) ( )
( )
( )
( )
( )
, , , , , , , ,
4
x t t x t x y t y x y t t x t y x t dy

+ =

where we have used the fact that
integrates to one.
We need to impose some requirement on
as
t
gets smaller. First, we should expect
the distribution of steps to be concentrated near
0y =
. Then we can approximate the right hand-
side of (4) using the first two terms in the Taylor expansion, and the left-hand size with a time
derivative:
( ) ( ) ( )
2
1 , 1
,
1
2
nn
j j k
j j k
j j k
xt
t y y y dy
t x x x
 
==

+ =




( ) ( )
2
,
1 , 1
1
,
2
nn
j j k
j j k
j j k
c
x x x

==
−+

Where,
( ) ( )
, , ; , ,x t t y y x t t dy

=
is the expected value of the displacement, and
( ) ( )
,
, , ; , ,
j k j k
c x t t y y y x t t dy
=
its covariance matrix. In order to make nontrivial contributions to the limit as
0t→
, both of
these need to scale proportionally to
t
:
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( ) ( )
( ) ( )
,,
, , , , ,
, , , , ,
j k j k
x t t u x t t
c x t t d x t t


in which case we obtain in the limit the Fokker-Planck equation
( )
( )
( )
2
,
,1
1
. ..... 5
2
n
jk
jk
jk
d
u
t x x
=
+=
If the drift
u
vanishes and the covariance matrix d equals twice the identity, we recover
the normalized heat equation.
Notice the similarity between the left-hand side of (5) and the equation for conservation
of mass in fluids. Indeed they are the same, though we are talking here of probability rather than
fluid densities. The drift term
u
represents the deterministic part of the walk, and plays the same
role as a fluid velocity: if, in the mean, the walker tends to go from one place to another, then the
probability of finding it in the latter will increase over time. The terms on the right-hand side
correspond instead to the fluctuating part of the velocity field, yielding diffusion.
[Hoover, W.G.; Hoover, C.G. 2018]
Conclusion:
The heat equation
0
t xx
uu−=
is not time-reversible because it involves an odd-order
derivative of t. Under time reversal
tt
, we get
tt
uu
. So if
( )
,u x t
is a solution to the
heat equation, then
( )
,u x t
is a solution to a different equation, namely
0
t xx
uu =
. The only
way for u to solve both equations is if
0
xx
u =
everywhere, which is not the case for most initial
conditions of interest, such as your
( )
x
.
By contrast, the wave equation
0
tt xx
uu =
is time-reversible because it involves only even-
order derivatives of t. Under time reversal
tt
, we get
tt xx
uu
. So if
( )
,u x t
is a solution to
the wave equation, then
( )
,u x t
is also a solution to the wave equation.
[Hoover, W.G.; Hoover, C.G. 2018]
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7
Reference
Keenan JH. Thermodynamics. New York: John Wiley & Sons, Inc.; 1957
Hatsopoulos GN, Keenan JH. Principles of general thermodynamics. New York: John Wiley & Sons,
Inc.; 1965.
Moran MJ. Availability analysis. New York: ASME Press; 1989.
Hoover, W.G.; Hoover, C.G. Time-irreversibility is hidden within Newtonian mechanics. Mol. Phys.
2018, 116, 30853096.

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THE STUDY OF ANALYTICAL SOLUTIONS OF PARABOLIC EQUATIONS By name(id) DEPARTMENT OF MATHEMATICS VIRTUAL UNIVERSITY OF PAKISTAN 2021 1 Contents Irreversible Process ................................................................................................................................... 3 Time irreversibility of Heat Equation ........................................................................................................... 4 Reference ...................................................................................................................................................... 7 2 Introduction Irreversible Process In science, process that is not reversible is called irreversible. This concept arises frequently in the thermodynamics. In thermodynamics, a change in the thermodynamics state of a system and all of its surroundings cannot be precisely restored to its initial state by infinitesimal changes in some property of the system without expenditure of energy. A system that undergoes an irreversible process may still be capable of returning to its initial state. However, the impossibility occurs in restoring the Thermodynamic to its own initial conditions. An irreversible process increases the entropy of the universe. Because entropy is state function, the change in entropy of the system is the same, whether the process is reversible or irreversible. The thermodynamics’ second law can be used to determine whether a process is reversible or not. The entropy equation can be written as dS = Q T +  where  is the entropy production which can be stated as   0 internally irreversible process,  = 0 internally reversible process,   0 impossible process. Quantity availability (also called energy, work potential, has been used by engineers to assess the maximum work that can be obtained for a combined system, closed or open, in a given environmental condition. It is known that all macro-processes are irreversible, and the availability is destroyed. The destruction of availability is called irreversibility. Wall stated that energy is motion or ability to produce motion and energy is work or ability to produce work. He also stated that time is experienced when energy is destroyed, i.e., an irreversible process, which creates a motion in a specific direction, i.e., the direction of time. The idea that availability can be destroyed is useful. This concept essentially brings together the first and second laws. Example: Both the Heat equation and the diffusion equation describe processes which are irreversible, because both equations have an odd time derivative. [Keenan JH. Thermodynamics,1957] 3 Time irreversibility of Heat Equation Explanation: The irreversibility associated with the heat equation, and alluded to it in terms of entropy, disorder and randomness, maybe it is time to look back at its physical origins, particularly as it relates to stochastic processes. Consider a walker on the real line that, at regular intervals t , moves to the left or right, with equal probabilities, a fixed distance x . If the walker starts walking at time t = 0 from x = 0 , what is the probability Pjn of finding it in position x = x j = j x at time t = tn = nt ? We can answer this question recursively: in order to be in x = x j at time t n , the waker must have been one step to the left or right the step before. Then Pjn +1 = where the Pjn−1 + Pjn+1 2 ..... (1) 1 is the probability of stepping in each direction. We can reorganize this expression 2 in the form Pjn +1 − Pjn t n n n 1 x 2 Pj +1 − 2 Pj + Pj −1 = 2 t x 2 ..... ( 2 ) To actually take the limit as x and t get small, we need some extra considerations. First, we need to re-interpret the discrete Pjn in terms of a function defined on the real line. The natural choice is a probability density  ( x, t ) . In terms of this, Pjn can be interpreted as the integral of  ( x, tn ) between x j− 1 2 and x j+ 1 2 . Then we need to take a succession of walks, with decreasing time intervals and step-sizes. Yet this needs to be done so that the quotient 1 x 2 r= 2 t remains finite, or else the evolution equation will trivialize. Therefore the stepsize x needs to scale as the square-root of the time interval t . With all this done, we can now the limit of (2), and obtain the one-dimensional heat equation 4 t =  xx where the diffusivity  is the limit of r as t and x go to zero. Let us now generalize this to arbitrary dimensions and more general walks that do not have a fixed step x . Instead, consider a probability distribution  ( y x, t , t ) for all possible steps y in R n . The distribution may depend on the position x and time t from which we are walking away, and is temporarily associated with a fixed time-interval t . Then the probability density  ( x, t + t ) is given by the following generalization of (1):  ( x, t + t ) =   ( x − y, t )  ( y x − y , t , t )dy, .... ( 3) So,  ( x, t + t ) −  ( x, t ) =    ( x − y, t )  ( y x − y, t , t ) −  ( x, t )  ( y x, t )dy, (4 ) where we have used the fact that  integrates to one. We need to impose some requirement on  as t gets smaller. First, we should expect the distribution of steps to be concentrated near y = 0 . Then we can approximate the right handside of (4) using the first two terms in the Taylor expansion, and the left-hand size with a time derivative: t  n  ( x, t )  (  ) 1    − y j + t x j 2  j =1 n  (  j ) j =1 x j − 1 + 2 n  2 (  c j ,k ) j , k =1 x j xk  n  j , k =1 y j yk  2 (  )   dy = x j xk  , Where,  ( x, t , t ) =  y  ( y; x, t , t ) dy is the expected value of the displacement, and c j ,k ( x, t , t ) =  y j yk  ( y; x, t , t ) dy its covariance matrix. In order to make nontrivial contributions to the limit as t → 0 , both of these need to scale proportionally to t : 5  ( x, t , t ) u ( x, t , ) t , c j ,k ( x, t , t ) d j ,k ( x, t , ) t , in which case we obtain in the limit the Fokker-Planck equation  1 + .( u ) = t 2 n  2 ( d j ,k  ) j , k =1 x j xk  ..... ( 5 ) If the drift u vanishes and the covariance matrix d equals twice the identity, we recover the normalized heat equation. Notice the similarity between the left-hand side of (5) and the equation for conservation of mass in fluids. Indeed they are the same, though we are talking here of probability rather than fluid densities. The drift term u represents the deterministic part of the walk, and plays the same role as a fluid velocity: if, in the mean, the walker tends to go from one place to another, then the probability of finding it in the latter will increase over time. The terms on the right-hand side correspond instead to the fluctuating part of the velocity field, yielding diffusion. [Hoover, W.G.; Hoover, C.G. 2018] Conclusion: The heat equation ut − uxx = 0 is not time-reversible because it involves an odd-order derivative of t. Under time reversal t −t , we get ut −ut . So if u ( x, t ) is a solution to the heat equation, then u ( x, −t ) is a solution to a different equation, namely −ut − uxx = 0 . The only way for u to solve both equations is if u xx = 0 everywhere, which is not the case for most initial conditions of interest, such as your  ( x ) . By contrast, the wave equation −utt − uxx = 0 is time-reversible because it involves only evenorder derivatives of t. Under time reversal t −t , we get utt uxx . So if u ( x, t ) is a solution to the wave equation, then u ( x, −t ) is also a solution to the wave equation. [Hoover, W.G.; Hoover, C.G. 2018] 6 Reference Keenan JH. Thermodynamics. New York: John Wiley & Sons, Inc.; 1957 Hatsopoulos GN, Keenan JH. Principles of general thermodynamics. New York: John Wiley & Sons, Inc.; 1965. Moran MJ. Availability analysis. New York: ASME Press; 1989. Hoover, W.G.; Hoover, C.G. Time-irreversibility is hidden within Newtonian mechanics. Mol. Phys. 2018, 116, 3085–3096. 7 Name: Description: ...
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