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Corrections to Third Edition of Pinsky ``Partial Differential  
Equations  and 
Boundary Value Problems with Applications'', 1998. (9 August, 1999)
\vskip 2em
Page xiv, line 13: Appendices

Page 1, line 7: in the appendices

Page 4, Exercise 2: $u(x,y)=e^{kx}e^{k^2y}$

Page 5, line 2: Newton's law of gravitational...

Page 5, line 22: $u(x_i;t)$

Page 7, line 21: is {\it sometimes} called...

Page 9, exercise 1: $u_x(L;t)=0$

Page 9, exercise 2: $u_x(L;t)=\Phi$

Page 9, exercise 3: $u_x(L;t)=h(T_1-u(L;t))$

Page 10, line 6: $x^2+y^2<R^2$

Page 10, line 13: $t>0$, $0<x<L$

Page 11, line 11: second item in display should read
${Y''(y)\over Y(y)}=-\lambda$

Page 12, line 17: ($A_1\sinh  kx+\cdots\sin ky)$

Page 15, line 3: ${d\over dx}(e^{\alpha x})=$

Page 17, line 20: $y^2Y'' +yY'=0$

Page 17, line 23: $y^2Y'' +yY' +l^2Y=0$

Page 18, line $5-$: $0=u(0;y)=A_1(\cdots$

Page 18, line $4-$: that $A_1=0$ and $0=u(L,y)=A_2\cdots$

Page 18, line $1-$: $A=A_2A_3$ (instead of $A=A_1A_3$)


Page 25, line 15: $\sum_{i=1}^N ||\varphi_i||^2\bigl(c_i-\cdots$

Page 29, line 8: $(1/\sqrt{2\pi}, \sin x/\sqrt{\pi}, \cos x/\sqrt{\pi})$

Page 32, line $7-$: ...we have $<\varphi,a\psi>={\bar a}<\varphi,  
\psi>$ for any complex constant


Page 33, Exercise 3: Let ${\bar f}$

Page 33, Exercise 8: $g(x)=b_1\varphi_1(x)+\cdots$

Page 43, line 17: $f_E(x)=f(-x)\quad -L<x<0$


Page 45, Exercise 10: $-\pi<x<\pi$

Page 46, line $1-$: $x'=n\pi x/L$

Page 49, line 6: where $x\in (-\pi,\pi)$

Page 50, lines 2,4,6: $\int_{x_i}^{x_{i+1}}$ instead of  
$\int_{x_i}^{x_i+1}$  (five different places)

Page 51, line 5: $\cos a \sin b={1\over 2}[\sin(a+b)-\sin(a-b)]$

Page 53, line $8-$: Insert ``$\lim_{N\to\infty}$ " before  
$\int_0^{\pi}D_N(u)\,du$

Page 56, Exercise 14(a): $f(u)=\ldots$

Page 56, Exercise 14(e): If
$   (N-{1\over 2})\pi
\le X
\le (N+{1\over 2})\pi$

Page 58, equation (1.3.1) : Insert $\cdots$  before $\cos(2n-1)x$

Page 59, last line:  $$\sum_{k=1}^n g(x_k')\Delta x_k = \cdots$$

Page 60, line 2:

$${2n\over \pi} {\pi\over n}\biggl[\sin (\pi/ 2n) + {\sin  
(3\pi/2n)\over 3} +\cdots$$

Page 65, lines 17,18,19:


$$A_0'={1\over 2L}\int_{-L}^L\cdots$$

$$A_n'={1\over L}\int_{-L}^L\cdots={n\over L}\int_{-L}^L\cdots$$

$$B_n'={1\over L}\int_{-L}^L\cdots =-{n\over L}\int_{-L}^L\cdots$$

Page 65, line 20: ....and used the continuity of $f(x), -L<x<L$

Page 66, line 13: $F(-\pi)=F(\pi)$

Page 67, line 3: $[{\cos nu-\cos n\pi\over n}]$

Page 75, line 2: $(a_n^2+\cdots$

Page 78, line 11: Change De Moivre's to Euler's

Page 79, line 8: Change De Moivre's to Euler's

Page 79, line $6-$: $e^{ax}$ instead of $e^{axx}$

Page 81, line $4-$:$\alpha_0= [F(L-0)-F(-L+0)]/2L$

Page 83, exercise 5: ${r \sin x\over 1+r^2-2r\cos  
x}=\sum_{n=1}^{\infty}r^n\sin nx$

Page 83, exercise 7: $\lim_{M,N\to\infty}\sum_{n=-M}^N \alpha_n  
e^{in\pi x/L}$

Page 83, exercise 7: Try example 1.1.4 at $x=0$

Page 85, line 20: $0=A\cos(L\sqrt{\lambda}) +\cdots$

Page 87, line 3 of Solution of Example 1.6.3: $=A(1+hL)$

Page 87, lines 6-7 of Solution of Example 1.6.3: $0=h(Ae^{\mu  
L}+Be^{-\mu L})+\cdots$


Page 87, line 10 of Solution of Example 1.6.3:  $B\sqrt{\lambda}$

Page 90, Last two lines:

$$\cos \alpha(A \sinh \mu a +\cdots+B\mu \sinh \mu a)=0$$
$$\cos \beta(A \sinh \mu b + \cdots + \mu B \sinh \mu b)=0$$

Page 92, line 16: This is in the interval  
$0<L\sqrt{-\lambda}<\sqrt{|\cot\alpha\cot\beta|}$

Page 92, lines 18-19: The first one  satisfies  
$0<L\sqrt{-\lambda_1}<\sqrt{|\cot\alpha\cot\beta|}$
while the second one satisfies
$L\sqrt{-\lambda_2}>\sqrt{|\cot\alpha\cot\beta|}$


Page 93, bulleted summary:

Second bullet should read:  .....This   satisfies  
$0<L\sqrt{-\lambda_1}<\sqrt{|\cot\alpha\cot\beta|}$

Third bullet should read:..... The first one satisfies  
$0<L\sqrt{-\lambda_1}<\sqrt{|\cot\alpha\cot\beta|}$
while the second one satsifies $ L\sqrt{-\lambda_2}
>\sqrt{|\cot\alpha\cot\beta|}$

Page 99, line $4-$: ${\rm grad}\! u$ (decrease space)

Page 100, line $11-$: delete minus sign before ``div"

Page 101, line 12: When $x_2\to x_1$, the {\it integrand} tends to....


Page 101, line 13: ...resulting integral tends to $q_x^x(x_1,y_1,z_1)$


Page 103, line 4: $-hT_1$ (delete $/k$)

Page 103, line 5: $-kB$ (insert minus sign)

Page 105, line 22: $\sqrt{\pi K \tau}$

Page 108, Exercise 3, lines 2-3: $\dots+h(u-T_0)](x,y,L)=0$

Page 113, line $5-$: $O(e^{-at})$ (change 0 to O)

Page 120, exercise 6, line 2: $u_z(L:t)=0$


Page 123, line $6-$: $U''(z)-\sum_{n=1}^{\infty}$ (change $=$ to $-$)

Page 125, line $14-$: $u_z(L;t)=0$

Page 125, line $1-$: $(4'): v(z;0)=T_3-T_1$

Page 126, line $10-$: $B\sqrt{\lambda}\cos L\sqrt{\lambda}$

Page 127, example 2.3.2: ...where $h,T_1,T_2$ and $T_3$ are...

Page 127, line $1-$: $U(z)=T_1+hz(T_2-T_1)/(1+hL)$ (change $t_2$ to $T_2$

Page 130, line $3-$: $f_n$,  and  $u_n(t)$ are obtained....

Page 131, line 4: $u_n(0)=f_n$

Page 131, line 6: $u_n(t)=f_n e^{-\lambda_n Kt}+\cdots$

Page 131, line $10-$: $\cos\alpha \sin\beta + \sin\alpha \cos\beta  
+ \cos\alpha\cos\beta\ne 0$

Page 136, line 1: Should read

$$-T^-(a:t){
(\partial {\bf r}/\partial s)(a;t)\over|\partial {\bf r}/\partial  
s)(a;t)|}=\cdots$$ (add  minus sign and change $b$ to $a$ in three  
places

Page 154, line 	$14-$
: $\sum_{m,n=1}^{\infty}$ instead of single sum

Page 154, line 	$10-$
: $\sum_{m,n=1}^{\infty}$ instead of single sum

Page 157, line $11-$:  $\sum_{m,n=1}^{\infty}$ instead of single sum

Page 158. lines 13, 17, 19,22:
 $\sum_{m,n=1}^{\infty}$ instead of single sum

Page 160, line 8:  $\sum_{m,n=1}^{\infty}$ instead of single sum

Page 163, line 16 of table: $2^3\cdot 5$

Page 165, line $3-$: $x=L_1$

Page 172, line $4-$: $u_{\varphi\varphi}$ at end

Page 175, solution of example 3.1.3 :
$$-k{\partial u\over \partial \rho}|_{\rho=\rho_2}= {-k\over  
\rho_2}{T_2-T_1\over \ln(\rho_2/\rho_1)}$$

Page 178. line  $15-$: $R^{-n}B_n=\cdots$

Page 187, line 3: ${a_0\over \rho\sqrt{\lambda}}\sum\ldots$

Page 189, line $4-$: $e^{-i\theta}$ instead of $e^{i\theta}$

Page 189, line $3-$: $=-\int_0^{\pi}\cdots$

Page 190, line 8: $=-{x\over m}$

Page 191, line 13; This is a first-order linear equation for $v'$,  
which may be solved.....

Page 196, lines 1-2: $\theta(\rho)$ is unbounded when $\rho\to\infty$.

Page 197, line@: $=+R(x)\ldots$ and $\cos \beta J_m(x) +x\sin x  
J_m'(x)=0$

Page 198, line 5: Add $\lambda=x_n^2$ at end of line

Page 200, line 1: ...use the formulas (3.2.18)-(3.2.19)....

Page 201, line $7-$: $\beta=0$

Page 202, line 11: $\int_0^1xJ_0(xx_n)^2\,dx$

Page 204, line 3: ${\bf jj[3.5,4]}$

Page 204, line 5: ${\bf jj[3.8, 3.85]}$


Page 211, line  $12-$: initial conditions (3.3.2)-(3.3.3)

Page 213, line $9-$: and $A_{0n}=0$ for $n>0$, while...


Page 218, line 8: $\beta=0$

Page 219, line 12: $U(\rho_{max})=T_1$

Page 221, line 14 [will check dimensions here and elsewhere]

Page 222, line 3: $-\pi\le\varphi\le \pi$

Page 223: lines 6,11,12, 14,15: Change $\Phi$ to $R$ in each case

Page 223, line 16: chnage $R_2^2-R_1^2$ to $\rho_2^2-\rho_1^2$

Page 225, line 8: $\lambda=i\omega/K$

Page 228, line 20: delete $-m^2$

Page 232, line 14: $1={8\over \pi}\ldots$

Page 238, line 1: $w_r=ru_r+u$

Page 241, line 10: $f(r)=T_2$

Page 244,  line 8: $ka\mu/(k-ah)$

Page 244, line $5-$: $rf(r)-W(r)$

Page 247, line $10-$: change $L$ to $a$ in two places

Page 247, line $6-$: $(2aC/n\pi)$

Page 247, line $4-$: change $L$ to $a$

Page 259, line 6: $[(1-a^2)P_k'(a)-(1-b^2)P_k'(b)]$

Page 280, line $11-$: $e^{-ia\mu}F(\mu)$ is the Fourier....

Page 286, line $3-$: ....satisfies  
$\int_0^{\infty}|F_s(\mu)|\,d\mu<\infty$

\centerline{ANSWERS}

Page 503, sec 0.2.3, exercise 1(d): $Y'+(\lambda-1)Y=0$

Page 504, Exercise 5: first line should read:
$$\lambda<0: u(x,y)=[A_1\cos(\sqrt{-\lambda} \ln |x|) +
 A_2\sin(\sqrt{-\lambda} \ln |x|)]
(A_3e^{y\sqrt\lambda} +
 A_4e^{-y\sqrt{-\lambda}})$$
\vskip 3em


Page 504, solution of Exercise no. 2 in section 0.2.4: should be  
$A_n\sin(\ldots)$
instead of $A_n\cos(\ldots)$


Page 504, Exercise 1(d) in section 0.3: answer to part (d) should be (7)



Page 505, Exercise 5 in section 1.1: should be ${1\over 2}-{1\over  
2} \cos(4x)$

Page 505, Exercise 6 in section 1.1: should be $ {1\over 2}\cos  
3x+{3\over 4}\cos(4x)$


Page 505, line 19 (exercise 16(c)): change $L/n\pi$ into $n\pi/L$  
in the denominator

Page 507, line $-2$: Change $\Phi_0$ to $\Phi$.

Page 511, Section 3.3, Exercise 1: $U(\rho)=(g/4c^2)(\rho^2-a^2)$

Page 511, line $1-$: exercise 5, last part is incorrect formula for
$A_n$. Need $A_n={2\over acx_nJ_1(x_n)^2}\int_0^R\cdots$

\vskip 2em
\centerline{End of corrections, August 9, 1999}
 \end{document}

