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Partial Differential Equations with Fourier Series and Boundary Value Problems

Nakhle H. Asmar

Chapter 6

Sturm-Liouville Theory with Engineering Applications - all with Video Answers

Educators

Section 1

Orthogonal Functions

09:10

Problem 1

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$$
\text { 1, } \sin \pi x, \cos \pi x, \sin 2 \pi x, \cos 2 \pi x, \sin 3 \pi x, \cos 3 \pi x_{4} \ldots 4 w(x)=1 \text { on }[0,2] \text {, }
$$

Matthew Allcock
Matthew Allcock
Numerade Educator
01:07

Problem 2

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$f(x)$ is an even function, $g(x)$ is an odd function; $w(x)=1$ on any symmetric interval about 0 .

Raj Bala
Raj Bala
Numerade Educator
07:36

Problem 3

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
3. $1, x,-1+2 x^{2} ; w(x)=\frac{1}{\sqrt{1-x^{2}}}$ on $\left.\mid-1,1\right]$. (These are examples of Chebyshev polynomials of the first kind. See Exercises $6.2$ for further details.)

Matthew Allcock
Matthew Allcock
Numerade Educator
01:19

Problem 4

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$$
-3 x+4 x^{3}, 1-8 x^{2}+8 x^{4} ; w(x)=\frac{1}{\sqrt{1-z^{2}}} \text { on }[-1,1]
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
07:36

Problem 5

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$1,2 x,-1+4 x^{2} ; w(x)=\sqrt{1-x^{2}}$ on $[-1,1]$, (There are examples of Chebyshev polynomials of the second kind.)

Matthew Allcock
Matthew Allcock
Numerade Educator
06:10

Problem 6

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$1,1-x,\left(2-4 x+x^{2}\right) / 2 ; w(x)=e^{-x}$ on $(0, \infty)$, (These are examples of Laguerre) polynomials.)

Tanishq Gupta
Tanishq Gupta
Numerade Educator
06:10

Problem 7

In Brercises $1-8$, show that the given set of functions is orthogonal with respect to the given weight on the prescribed interval.
$1,2 x,-2+4 x^{2} ; w(x)=e^{-x^{\prime}}$ on $(-\infty, \infty)$, (These are examples of Hermite polynomials. |Hint: Exercise $33(a)$, Section $4.7$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
02:45

Problem 8


\left(2-4 x+x^{2}\right) / 2,-12 x+8 x^{3} ; w(x)=e^{-x} \text { on }[0, \infty)

Bryan Lynn
Bryan Lynn
Numerade Educator
00:48

Problem 9


Determine the constants $a$ and $b$ so that the functions $1, x$, and $a+b x+x^{2}$ become orthogonal on the interval $[-1,1]$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:01

Problem 10


If the functions $f(x)$ and $g(x)$ are orthogonal with respect to a weight $w(x)$ on $[0, L]$, what can be said about the functions $f(a x)$ and $g(a x)$ where $a>0 ?$,

Raj Bala
Raj Bala
Numerade Educator
01:38

Problem 11


Compute the norms of the functions in Exercise 1 .

Gregory Higby
Gregory Higby
Numerade Educator
00:16

Problem 12


Compute the norms of the functions in Exercise $5 .$

AG
Ankit Gupta
Numerade Educator
04:05

Problem 13


What is the orthonormal set corresponding to the Legendre polynomials on the interval $[-1,1 \mid ?$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:38

Problem 14


Show that if $f$ and $g$ are contintous functions on $[a, b]$ that are orthogonal with respect to the weight function 1 , then either $f$ or $g$ must vanish somewhere in $(a, b)$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
00:43

Problem 15

Show that if $f_{1}(x), f_{2}(x), \ldots$ are orthogonal on $[0,1]$ with renpect to the welght $x$, then $f_{1}(\sqrt{x}), f_{2}(\sqrt{x}), \ldots$ are orthogonal on $[0,1]$ with respect to the weight function $1 .$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:23

Problem 16

Parseval's identity for Fourier series. Specialize (9) to the trigoncmetric, system (of period $2 p$ ) to obtain (6) of Section 2.5.

Kajal Gautam
Kajal Gautam
Numerade Educator
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Problem 17

Parieval's identity for Legendre series. Use (9) to derive the identity
$$
\int_{-1}^{1} f(x)^{2} d x=\sum_{n=0}^{\infty} \frac{2 A_{n}^{2}}{2 n+1}
$$
where $A_{n}$ is the nth Legendre coefficient of $f .$ (See Section 5.6.)

Victor Salazar
Victor Salazar
Numerade Educator
08:06

Problem 18

Parseval's identity for Bessel series, Use (9) to derive the identity
$$
\int_{0}^{n} f(x)^{2} x d x=\sum_{j=1}^{\infty} \frac{R^{2} J_{p+1}^{2}\left(\alpha_{p j}\right)}{2} A_{j}^{2}
$$
where $A_{j}$ is the $j$ th coefliclent of the Bessel series expansion of $f$ of order $P$, and $\alpha_{p j}$ is the $j$ th poeitive zero of $J_{p}$. (See Section 4.8.)

WZ
Wen Zheng
Numerade Educator
01:24

Problem 19


Sums of reciprocals of squares of zeros of Bessel functions. Derive the following interesting formule: $\frac{1}{4(p+1)}=\sum_{j=1}^{\infty} \frac{1}{\alpha_{p j}^{2}}$. [Hint: Apply Exercise 18 with

Linh Vu
Linh Vu
Numerade Educator
00:47

Problem 20

By specializing Exercise 19 to the case $p=\frac{1}{2}$, derive the identity
$$
\frac{\pi^{2}}{6}=\sum_{j=1}^{\infty} \frac{1}{j^{2}}
$$

Graham Avers
Graham Avers
Numerade Educator
View

Problem 21

Show that the inner product satisfies the following properties:
(a) $(a f, g)=a(f, 9)$ for any number $a$,
(b) $(f+g, h)=(f, h)+(g, h)$
(c) $(f, f) \geq 0$ for any funetion $f$.

Victor Salazar
Victor Salazar
Numerade Educator