PARTIAL DIFFERENTIAL EQUATIONS:

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PARTIAL DIFFERENTIAL EQUATIONS
THE VIBRATING STRING
Suppose that a flexible string is pulled taut on
the x-axis and fastened at two points, which we
denote by x 0 and x L. The string is then
drawn aside into a certain curve y f(x) in the
xy plane and then released from rest.
Problem
To find the vertical displacement y(x,t) of the
string at the point x from the origin at time t.
It
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is proved in Physics that y(x,t) satisfies the
one-dimensional wave equation
where a is a constant (depending on the mass of
the string and the tension of the string). We
thus want to solve the boundary value-problem
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Sepration of Variables.
Method of sepration of variables for solving
boundary value problems involving partial
differential equations.
Consider the one-dimensional wave equation
Subject to the boundary conditions
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and the initial conditions
where f(x) and g(x) are assumed known. Physically
this problem consists of finding the equation of
motion of an elastic string which is stretched
along the x-axis from 0 to ?, clamped at its
endpoints, given initial position f(x), intial
velocity g(x).
Formal solution of (1) by the method of
sepration of variable
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which are factorable into a product of functions
each of which depends on only one of the
independent variable.
Substituting (6) into (1), we get
The left-hand side of this expresion is a
function of x alone, while the right-hand side
involves t only. Thus each is a constant ?, and
is equivalent to the pair of ordinary
differential equations
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The first of which must satisfy the endpoint
condition u(0) u(?) 0. The general solution
of (7) is
Use the condition u(0) u(?) 0, we find the
only nontrivial solution of this problem are
corresponding to the eigenvalues ?nn2.
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Moreover, when ?n2 the general solution of
is
where An and Bn are arbitrary constants. Hence
forming the product of the functions, we see that
is a solution of the one-dimentional wave
equation which vanishes when x 0 and x ?
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Let us now apply the initial conditions (4) (5)
to the series
So yn(x, t) satisfied equation (1) and condition
(2) and (3) and it is easily verified that the
same is true for any finite sum of constant
multiple of the yn. Therefor it appears that is
infinite series of the form
For suitable values of An and Bn.
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Now when t 0, this series reduces to
where
The only condition which remains to be satisfied
is the second initial condition. From (11) we get
Now t 0, we get
where
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The physical significance of this solution can be
given in the special case where g(x) 0. Then
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Example
Solve the wave equation when
Thus
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The heat equation
We consider the flow of heat in a thin
cylindrical rod of cross-sectional area A whose
lateral surface is perfectly insulated so that no
heat flows through it. The word thin means that
the temperature is uniform on any cross- section
and thus is a function of the time t and its
horizontal distance x from one end,
u(x,t). Noting the laws of heat conduction (in
Physics), namely that heat flows in the direction
of decreasing temperature, the rate of
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heat flow across an area is proportional to the
area and the rate of change in temperature w.r.t.
the distance is in a direction perpendicular to
the area, etc., we can get the heat equation
describing the temperature u(x,t) as
Here the constant a2 depends on three things
density of the rod, specific heat of the
substance and thermal conductivity of the
substance.
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We assume the boundary conditions
This means that the ends are maintained at zero
temperature at all times. We also assume that the
rod has an initial temperature distribution f(x)
at time t 0 i.e. u(x,0) f(x) 0 lt x lt
L Again separating the variables, we find
that u w v where w satisfies the eigenvalue
problem
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(1)
and that v satisfies first order d.e
(2)
Thus for nontrivial solutions,
Corresponding eigenfunctions are
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For each n1,2,3,
is a solution of (2) with
Hence for each n 1,2,3,
is a solution of the heat equation.
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Again we assume that the infinite linear
combination
is also a solution. Putting t 0, we get
Hence bn nth Fourier sine coefficient of f(x)
in 0, L
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Steady state temperature We assume that when t is
large, w? a function independent of t, w(x) say.
This is called the steady state temperature. And
noting that in steady state
we get
Therefore
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Suppose that the ends are maintained at
temperature u1,u2 at all times. Hence
Hence the steady state temperature is
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Problem
Find the solution of the one-dimensional heat
equation satisfying the boundary and initial
conditions
for all t.
0 ? x ? L
Solution
We write the temperature distribution as
steady state temperature Transient
temperature.
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We impose the boundary conditions us(0)100,
us(L)0 on us(x) Thus
We impose the boundary conditions uT(0,t)uT(L,t)
0 for all t and the initial condition
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That is uT(x, t) satisfies the boundary
conditions uT(0,t)uT(L,t) 0 for all t and
the initial condition
Thus
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where
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Now
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And
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Thus
BEST OF LUCK IN THE COMPRE
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The two-dimensional heat equation A thin
rectangular plate has its boundary kept at
temperatures 0,0,0 and f(x) at all times. Find
the steady temperature distribution in the plate.
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Thus we have to solve the Laplaces equation
subject to the boundary conditions
We again solve by the method of separation of
variables. We assume w(x,y) X(x) Y(y)
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Substituting and separating the variables we get
(a constant)
Thus X satisfies the boundary value problem
X(0) X(L) 0
We know nontrivial solutions exist when
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and the corresponding eigen function is
For each n, Y satisfies the BV problem
Now
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Taking
we find
Thus for each n,
is a solution satisfying the first three boundary
conditions. Hence
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is also a solution. Putting y 0, we get
Hence
nth Fourier sine coefficient of f(x) in 0, L
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Thus we have found out the steady state
temperature distribution.
THE END