Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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168 Chapter 3
When solving boundary value problems, the first choice of differential equation to be solved is
generally the spatial differential equation. The reason for this is that the spatial equation, along
with the spatial boundary conditions, determines the character of the system eigenvalues.
These eigenvalues must be known before we can get a complete solution to the time-dependent
differential equation.
We now consider the solution to the time-dependent differential equation. This is a first-order
ordinary differential equation, which we solved in Section 1.2. The solution, for the allowed
values of λ given earlier, reads
T
n
(t) = C(n)e
−
n
2
π
2
t
10
Thus, the eigenfunction expansion for the solution to the problem reads
u(x, t) =
∞
n=1
C(n)e
−
n
2
π
2
t
10
√
2 sin(nπx)
The unknown coefficients C(n) for n = 1, 2, 3,... are to be determined from the initial
condition function that is imposed on the problem.
3.5 Initial Conditions for the Diffusion Equation in
Rectangular Coordinates
We now consider the initial conditions on the problem. Suppose that, at time t = 0, the rod has
an initial temperature distribution
u(x, 0) = f(x)
Substituting this into the following general solution
u(x, t) =
∞
n=0
X
n
(x)C(n)e
−kλ
n
t
at time t = 0 yields
f(x) =
∞
n=0
X
n
(x)C(n)
Since the eigenfunctions of the regular Sturm-Liouville problem form a complete set with
respect to piecewise smooth functions over the finite interval I, then the preceding is the
generalized Fourier series expansion of the function f(x) in terms of the eigenfunctions. The
terms C(n) are the corresponding Fourier coefficients. We now evaluate these coefficients.
The Diffusion or Heat Partial Differential Equation 169
As we did in Section 2.3 for the generalized Fourier series expansion, we take the inner product
of both sides of the preceding series with respect to the weight function w(x) = 1. Assuming
validity of the interchange between the summation and integration operations, we get
b
a
f(x)X
m
(x)dx =
∞
n=0
C(n)
⎛
⎝
b
a
X
m
(x)X
n
(x)dx
⎞
⎠
Taking advantage of the statement of orthonormality, this reduces to
b
a
f(x)X
m
(x)dx =
∞
n=0
C(n)δ(n, m)
Due to the mathematical character of the Kronecker delta function, only one term (m = n)in
the sum survives, and the preceding sum reduces to
C(m) =
b
a
f(x)X
m
(x)dx
Thus, we have evaluated the Fourier coefficients in the generalized expansion of f(x). We can
write the final generalized solution to the diffusion equation in one dimension, subject to the
given homogeneous boundary conditions and initial conditions, as
u(x, t) =
∞
n=0
X
n
(x)e
−kλ
n
t
⎛
⎝
b
a
f(s)X
n
(s)ds
⎞
⎠
All of the previous operations are based on the assumption that the infinite series is uniformly
convergent and that the formal interchange between the integration and the summation
operators is valid. It can be shown that if the function f(x) is piecewise smooth over the
interval I and it satisfies the same boundary conditions as the eigenfunctions, then the
preceding series is, indeed, uniformly convergent.
DEMONSTRATION: We now provide a demonstration of these concepts for the example
problem given in Section 3.2 for the case where the initial temperature distribution u(x, 0) is
f(x) = x(1 −x)
SOLUTION: From an earlier development, the unknown Fourier coefficients C(n) are
evaluated from the inner product between the orthonormalized eigenfunctions and f(x) over
the interval I ={x |0 <x<1} as shown:
C(n) =
1
0
x(1 −x)
√
2 sin(nπx)dx
170 Chapter 3
Evaluation of this integral yields
C(n) =−
2
√
2((−1)
n
−1)
n
3
π
3
for n = 1, 2, 3,.... Thus, the final series solution to the example problem reads
u(x, t) =
∞
n=1
⎛
⎝
−
4((−1)
n
−1)e
−n
2
π
2
t
10
sin(nπx)
n
3
π
3
⎞
⎠
The detailed development of the solution of this problem and its graphics are given in one of
the Maple worksheet examples given later.
The Maple worksheets given later are expressed in terms of generalized values for length,
diffusivity, and initial condition function u(x, 0) = f(x). The reason for the generalization is so
that the student or practitioner of the problem can change these values at will for different
applications.
3.6 Example Diffusion Problems in Rectangular
Coordinates
We now consider several examples of partial differential equations for heat or diffusion
phenomena under various homogeneous boundary conditions over finite, one-dimensional
intervals in the rectangular coordinate system. We note that all the spatial ordinary differential
equations are of the Euler type.
EXAMPLE 3.6.1: We seek the temperature distribution u(x, t) in a thin rod over the finite
interval I ={x |0 <x<1} whose lateral surface is insulated. Both the left and right ends of the
rod are held at a fixed temperature of zero. The initial temperature distribution f(x) is given
following, and the diffusivity is k = 1/10.
SOLUTION: The homogeneous diffusion equation is
∂
∂t
u(x, t) = k
∂
2
∂x
2
u(x, t)
The boundary conditions are type 1 at x = 0 and type 1 at x = 1
u(0,t)= 0 and u(1,t)= 0
The initial condition is
u(x, 0) = x(1 −x)
The Diffusion or Heat Partial Differential Equation 171
Ordinary differential equations obtained from the method of separation of variables:
d
dt
T(t) +kλT(t) = 0
and
d
2
dx
2
X(x)+λX(x) = 0
Boundary conditions on spatial equation
X(0) = 0 and X(1) = 0
Assignment of system parameters
> restart:with(plots):a:=0:b:=1:k:=1/10:
Allowed eigenvalues and orthonormal eigenfunctions are obtained from Example 2.5.1.
> lambda[n]:=(n*Pi/b)ˆ2;
λ
n
:= n
2
π
2
(3.1)
for n = 1, 2, 3,....
Orthonormal eigenfunctions
> X[n](x):=sqrt(2/b)*sin(n*Pi/b*x);X[m](x):=subs(n=m,X[n](x)):
X
n
(x) :=
√
2 sin(nπx) (3.2)
Statement of orthonormality with respect to the weight function w(x) = 1
> w(x):=1:Int(X[n](x)*X[m](x)*w(x),x=a...b)=delta(n,m);
1
0
2 sin(nπx) sin(mπx)dx = δ(n, m) (3.3)
Time-dependent solution
> T[n](t):=C(n)*exp(−k*lambda[n]*t);u[n](x,t):=T[n](t)*X[n](x):
T
n
(t) := C(n)e
−
1
10
n
2
π
2
t
(3.4)
172 Chapter 3
Eigenfunction expansion
> u(x,t):=Sum(u[n](x,t),n=1..infinity);
u(x, t) :=
∞
n=1
C(n)e
−
1
10
n
2
π
2
t
√
2 sin(nπx) (3.5)
Evaluation of Fourier coefficients for the specific initial condition
> f(x):=x*(1−x);
f(x) := x(1 −x) (3.6)
> C(n):=Int(f(x)*X[n](x)*w(x),x=a..b);C(n):=expand(value(%)):
C(n) :=
1
0
x(1 −x)
√
2 sin(nπx)dx (3.7)
> C(n):=radsimp(subs({sin(n*Pi)=0,cos(n*Pi)=(−1)ˆn},C(n)));
C(n) := −
2
√
2(−1 +(−1)
n
)
n
3
π
3
(3.8)
Generalized series terms
> u[n](x,t):=eval(T[n](t)*X[n](x));
u
n
(x, t) := −
4(−1 +(−1)
n
)e
−
1
10
n
2
π
2
t
sin(nπx)
n
3
π
3
(3.9)
Series solution
> u(x,t):=Sum(u[n](x,t),n=1..infinity);
u(x, t) :=
∞
n=1
−
4(−1 +(−1)
n
)e
−
1
10
n
2
π
2
t
sin(nπx)
n
3
π
3
(3.10)
First few terms of sum
> u(x,t):=sum(u[n](x,t),n=1..3):
ANIMATION
> animate(u(x,t),x=a..b,t=0..5,color=red,thickness=10);
The preceding animation command illustrates the spatial-time-dependent solution for u(x, t).
The animation sequence shown in Figure 3.1 shows snapshots at times t = 0, 1, 2, 3, 4, and 5.
The Diffusion or Heat Partial Differential Equation 173
ANIMATION SEQUENCE
> u(x,0):=subs(t=0,u(x,t)):u(x,1):=subs(t=1,u(x,t)):
> u(x,2):=subs(t=2,u(x,t)):u(x,3):=subs(t=3,u(x,t)):
> u(x,4):=subs(t=4,u(x,t)):u(x,5):=subs(t=5,u(x,t)):
> plot({u(x,0),u(x,1),u(x,2),u(x,3),u(x,4),u(x,5),},x=a..b,thickness=10);
x
0 0.2 0.4 0.6 0.8 1
0
0.05
0.10
0.15
0.20
Figure 3.1
EXAMPLE 3.6.2: We seek the temperature distribution u(x, t) in a thin rod over the finite
interval I ={x |0 <x<1} whose lateral surface is insulated. The left end is at a fixed
temperature of zero and the right end is insulated. The initial temperature distribution f(x) is
given following, and the diffusivity is k = 1/10.
SOLUTION: The homogeneous diffusion equation is
∂
∂t
u(x, t) = k
∂
2
∂x
2
u(x, t)
The boundary conditions are type 1 at x = 0 and type 2 at x = 1
u(0,t)= 0 and u
x
(1,t)= 0
The initial condition is
u(x, 0) = x −
x
2
2
Ordinary differential equations obtained from the method of separation of variables:
d
dt
T(t) +kλT(t) = 0
and
d
2
dx
2
X(x)+λX(x) = 0
174 Chapter 3
Boundary conditions on spatial equation
X(0) = 0 and X
x
(1) = 0
Assignment of system parameters
> restart:with(plots):a:=0:b:=1:k:=1/10:
Allowed eigenvalues and orthonormal eigenfunctions are obtained from Example 2.5.2.
> lambda[n]:=((2*n−1)*Pi/(2*b))ˆ2;
λ
n
:=
1
4
(2n −1)
2
π
2
(3.11)
for n = 1, 2, 3,....
Orthonormal eigenfunctions
> X[n](x):=sqrt(2/b)*sin((2*n−1)*Pi/(2*b)*x);X[m](x):=subs(n=m,X[n](x)):
X
n
(x) :=
√
2 sin
1
2
(2n −1)πx
(3.12)
Statement of orthonormality with respect to the weight function w(x) = 1
> w(x):=1:Int(X[n](x)*X[m](x)*w(x),x=a..b)=delta(n,m);
1
0
2 sin
1
2
(2n −1)πx
sin
1
2
(2m −1)πx
dx = δ(n, m) (3.13)
Time-dependent solution
> T[n](t):=C(n)*exp(−k*lambda[n]*t);
T
n
(t) := C(n)e
−
1
40
(2n−1)
2
π
2
t
(3.14)
Generalized series terms
> u[n](x,t):=T[n](t)*X[n](x);
u
n
(x, t) := C(n)e
−
1
40
(2n−1)
2
π
2
t
√
2 sin
1
2
(2n −1)πx
(3.15)
Eigenfunction expansion
> u(x,t):=Sum(u[n](x,t),n=1..infinity);
u(x, t) :=
∞
n=1
C(n)e
−
1
40
(2n−1)
2
π
2
t
√
2 sin
1
2
(2n −1)πx
(3.16)
The Diffusion or Heat Partial Differential Equation 175
Evaluation of Fourier coefficients for the specific initial condition
> f(x):=x−xˆ2/2;
f(x) := x −
1
2
x
2
(3.17)
>C(n):=Int(f(x)*X[n](x)*w(x),x=a..b);C(n):=expand(value(%)):
C(n) :=
1
0
x −
1
2
x
2
√
2 sin
1
2
(2n −1)πx
dx (3.18)
> C(n):=subs({sin(n*Pi)=0,cos(n*Pi)=(−1)ˆn},C(n));
C(n) :=
8
√
2
π
3
(8n
3
−12n
2
+6n −1)
(3.19)
Generalized series terms
> u[n](x,t):=eval(T[n](t)*X[n](x));
u
n
(x, t) :=
16e
−
1
40
(2n−1)
2
π
2
t
sin
1
2
(2n −1)πx
π
3
(8n
3
−12n
2
+6n −1)
(3.20)
Series solution
> u(x,t):=Sum(u[n](x,t),n=1..infinity);
u(x, t) :=
∞
n=1
16e
−
1
40
(2n−1)
2
π
2
t
sin
1
2
(2n −1)πx
π
3
(8n
3
−12n
2
+6n −1)
(3.21)
First few terms of sum
> u(x,t):=sum(u[n](x,t),n=1..4):
ANIMATION
> animate(u(x,t),x=a..b,t=0..5,thickness=3);
The preceding animation command illustrates the spatial-time-dependent solution for u(x, t).
The animation sequence shown in Figure 3.2 shows snapshots at times t = 0, 1, 2, 3, 4, and 5.
ANIMATION SEQUENCE
> u(x,0):=subs(t=0,u(x,t)):u(x,1):=subs(t=1,u(x,t)):
> u(x,2):=subs(t=2,u(x,t)):u(x,3):=subs(t=3,u(x,t)):
176 Chapter 3
> u(x,4):=subs(t=4,u(x,t)):u(x,5):=subs(t=5,u(x,t)):
> plot({u(x,0),u(x,1),u(x,2),u(x,3),u(x,4),u(x,5)},x=0..b,thickness=10);
xx
0 0.2 0.4 0.6 0.8 1
0.1
0.3
0.2
0.4
0
Figure 3.2
EXAMPLE 3.6.3: We again consider the temperature distribution u(x, t) in a thin rod over the
interval I ={x |0 <x<1} whose lateral surface is insulated. Both the left and right ends of the
rod are insulated. The initial temperature distribution f(x) is given following, and the
diffusivity is k = 1/10.
SOLUTION: The homogeneous diffusion equation is
∂
∂t
u(x, t) = k
∂
2
∂x
2
u(x, t)
The boundary conditions are type 2 at x = 0 and type 2 at x = 1
u
x
(0,t)= 0 and u
x
(1,t)= 0
The initial condition is
u(x, 0) = x
Ordinary differential equations obtained from the method of separation of variables
d
dt
T(t) +kλT(t) = 0
and
d
2
dx
2
X(x)+λX(x) = 0
Boundary conditions on spatial equation
X
x
(0) = 0 and X
x
(1) = 0
The Diffusion or Heat Partial Differential Equation 177
Assignment of system parameters
> restart:with(plots):a:=0:b:=1:k:=1/10:
Allowed eigenvalues and orthonormal eigenfunctions are obtained from Example 2.5.3.
> lambda[0]:=0;
λ
0
:= 0 (3.22)
for n = 0.
Orthonormal eigenfunction
> X[0](x):=1/sqrt(b);
X
0
(x) := 1 (3.23)
For n = 1, 2, 3,...,
> lambda[n]:=(n*Pi/b)ˆ2;
λ
n
:= n
2
π
2
(3.24)
Orthonormal eigenfunctions
> X[n](x):=sqrt(2/b)*cos(n*Pi/b*x);X[m](x):=subs(n=m,X[n](x)):
X
n
(x) :=
√
2 cos(nπx) (3.25)
Statement of orthonormality with respect to the weight function w(x) = 1
> w(x):=1:Int(X[n](x)*X[m](x)*w(x),x=a...b)=delta(n,m);
1
0
2 cos(nπx) cos(mπx)dx = δ(n, m) (3.26)
Time-dependent solution
For n = 1, 2, 3,...,
> T[n](t):=C(n)*exp(−k*lambda[n]*t);
T
n
(t) := C(n)e
−
1
10
n
2
π
2
t
(3.27)
Generalized series terms
> u[n](x,t):=T[n](t)*X[n](x);
u
n
(x, t) := C(n)e
−
1
10
n
2
π
2
t
√
2 cos(nπx) (3.28)