Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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148 Chapter 2
(c) The Laguerre equation from quantum mechanics
x
d
2
dx
2
y(x)
+(1 −x)
d
dx
y(x)
+λy(x)= 0
(d) The Hermite equation from quantum mechanics
d
2
dx
2
y(x) −2 x
d
dx
y(x)
+2 λy(x)= 0
(e) The Tchebycheff equation
(1 −x
2
)
d
2
dx
2
y(x)
−x
d
dx
y(x)
+λy(x)= 0
(f) The Bessel differential equation of order m
x
2
d
2
dx
2
y(x)
+x
d
dx
y(x)
−m
2
y(x) +λ
2
x
2
y(x) = 0
In the following problems, you will be asked to generate a set of eigenfunctions for a
Sturm-Liouville problem over an interval I with a particular set of boundary conditions.
You are also asked to develop a generalized Fourier series expansion for three functions
over the interval. The first, f 1(x), should not satisfy any of the boundary conditions.
The second, f 2(x), should satisfy one of the boundary conditions (left or right). The
third, f 3(x), should satisfy both boundary conditions (left and right). Pay particular
attention to the quality of the convergence of each of the series depending on the
boundary condition behavior of the particular function f(x) over that interval.
For problems 2.2 through 2.21, you are asked to do the following:
(a) Evaluate the eigenvalues and corresponding eigenfunctions.
(b) Normalize the eigenfunctions with the respective weight function.
(c) Write out the statement of orthonormality.
(d) Write out the generalized Fourier series expansion for function f(x).
(e) Write out the integral for the Fourier coefficients F(n).
2.2. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ(0) = 0,ϕ(1) = 0
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 149
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x
f 3(x) = x(1 −x)
With the first five terms in each series above, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.3. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ(0) = 0,ϕ
x
(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) =
x −
1
2
2
f 2(x) = x
f 3(x) = x
1 −
x
2
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.4. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ
x
(0) = 0,ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
f 2(x) = 1
f 3(x) = 1 −x
2
150 Chapter 2
With the first five terms in each series above, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.5. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ
x
(0) = 0,ϕ
x
(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
f 2(x) = 1 −x
2
f 3(x) = x
2
1 −
2x
3
With the first five terms in each series above, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.6. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ(0) = 0,ϕ
x
(1) +ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x
f 3(x) = x
1 −
2x
3
With the first five terms in each series above, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.7. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 151
with boundary conditions
ϕ
x
(0) −ϕ(0) = 0,ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = 1 −x
f 3(x) =−
2x
2
3
+
x
3
+
1
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.8. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ
x
(0) = 0,ϕ
x
(1) +ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
f 2(x) = 1
f 3(x) = 1 −
x
2
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.9. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |0 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x)= 0
with boundary conditions
ϕ
x
(0) −ϕ(0) = 0,ϕ
x
(1) = 0
152 Chapter 2
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
f 2(x) = 1
f 3(x) =−
(x −1)
2
3
+1
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.10. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ(1) = 0,ϕ(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x −1
f 3(x) = (x −1)(2 −x)
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.11. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ
x
(1) = 0,ϕ(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x −1
f 2(x) = 1
f 3(x) = x(2 −x)
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 153
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.12. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ(1) = 0,ϕ
x
(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) =
x −
3
2
2
f 2(x) = x −1
f 3(x) =
(x −1)(3 −x)
2
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.13. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ
x
(1) = 0,ϕ
x
(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x −1
f 2(x) = x(2 −x)
f 3(x) = 3 x
2
−
2x
2
3
−4 x +
5
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
154 Chapter 2
2.14. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ(1) = 0,ϕ
x
(2) +ϕ(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x −1
f 3(x) =
(x −1)(5 −2 x)
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.15. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x)= 0
with boundary conditions
ϕ
x
(1) −ϕ(1) = 0,ϕ(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = 2 −x
f 3(x) =−
2x
2
3
+
5x
3
−
2
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.16. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x) = 0
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 155
with boundary conditions
ϕ
x
(1) = 0,ϕ
x
(2) +ϕ(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x −1
f 2(x) = 1
f 3(x) =
2
3
−
x
2
3
+
2x
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.17. Consider the Sturm-Liouville problem with the Cauchy-Euler operator over the interval
I ={x |1 <x<2}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λϕ(x) = 0
with boundary conditions
ϕ
x
(1) −ϕ(1) = 0,ϕ
x
(2) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x −1
f 2(x) = 1
f 3(x) =−
x
2
3
+
4x
3
−
1
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.18. Consider the Sturm-Liouville problem with the Euler operator over the interval
I ={x |−1 <x<1}.
d
2
dx
2
ϕ(x) +λϕ(x) = 0
with periodic boundary conditions
ϕ(−1) = ϕ(1), ϕ
x
(−1) = ϕ
x
(1)
156 Chapter 2
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
f 2(x) = 1 −x
2
f(3) =|x|
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.19. Consider the Sturm-Liouville problem with the Bessel operator of order zero over the
interval I ={x |0 <x<1}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λ
2
x
2
ϕ(x) = 0
with boundary conditions
ϕ(0)<∞,ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = x
2
f 2(x) = 1
f 3(x) = 1 −x
2
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.20. Consider the Sturm-Liouville problem with the Bessel operator of order zero over the
interval I ={x |0 <x<1}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λ
2
x
2
ϕ(x) = 0
with boundary conditions
|ϕ(0)| < ∞,ϕ
x
(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x
2
f 3(x) = x
2
−
x
4
2
Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 157
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.21. Consider the Sturm-Liouville problem with the Bessel operator of order zero over the
interval I ={x |0 <x<1}.
x
2
d
2
dx
2
ϕ(x)
+x
d
dx
ϕ(x)
+λ
2
x
2
ϕ(x) = 0
with boundary conditions
|ϕ(0)| < ∞,ϕ
x
(1) +ϕ(1) = 0
Evaluate the generalized Fourier series expansion for the following:
f 1(x) = 1
f 2(x) = x
2
f 3(x) = 1 −
x
2
3
With the first five terms in each preceding series, generate the curve of f(x) and its
corresponding series on the same set of axes.
2.22. Consider the non-self-adjoint boundary value problem
d
2
dx
2
y(x) +4
d
dx
y(x)
+(4 +9λ)y(x) = 0
with boundary conditions
y(0) = 0,y(1) = 0
(a) Transform into the self-adjoint form and identify the weight function w(x).
(b) Solve for the eigenvalues and the corresponding eigenfuctions.
(c) Normalize the eigenfunctions and write out the statement of orthonormality.
(d) Evaluate the generalized Fourier series expansion of the function f(x) = x −x
2
over the interval I ={x |0 <x<1}.
2.23. Consider the non-self-adjoint boundary value problem
d
2
dx
2
y(x) +2
d
dx
y(x)
+(1 −λ)y(x) = 0
with boundary conditions
y(0) = 0,y(1) = 0