Corrections to Partial Differential Equations and Boundary-Value Problems with Applications by M. Pinsky, August 2001

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1 Corrections to Prtil Differentil Equtions nd Boundry-Vlue Problems with Applictions by M. Pinsky, August pge 4, exercise : u(x, y) = e kx e k y pge 5, line : Newton s lw of... pge 5, line : u(x i ; t) pge 7, line : sometimes pge 9, exercise : u(; t) = T, u x (L; t) = pge 9, exercise : u x (L; t) = Φ pge 9, exercise 3: (T u(l; t)) pge, line 6: x + y < R pge, line 3: t >, < x < L Y (y) Y (y) pge, line : = λ pge, line 7: A 4 sinky d pge 5, line 3: dx (eαx ) pge 7, line : y Y + yy = pge 7, line 3: y Y + yy + l Y = pge 8, line 5 : u(, y) = A (A 3 ) pge 8, line 4 : tht A = ; pge 8, line : Writing A = A A 3, pge 5, line 9 : = N i= ϕ i ( c i pge 9, line 8: interchnge cos x nd sin x pge 3, line 7 : hve ϕ, ψ = ā ϕ, ψ pge 33, exercise 3: Let f = pge 33, exercise 8: g(x) = b ϕ (x) + b ϕ (x) + pge 34, exercise 5: R = ϕ ψ ϕ, ψ pge 43, line 7: f( x) pge 45, exercise : π < x < π pge 46, line : x = πx/l pge 49, line 6: where x (, b) L < x < (delete minus sign before f pge 5, lines,4,6: x i+ x i (in five plces) pge 5, line 5: cos sin b pge 53, line 9: insert lim N before lst integrl pge 56, line : f(u) =... pge 56, exercise 4(e): (N )π X pge 58, line 7 : cos 5x + + cos(n )x pge 59, line : x k = pge 6, line : sin 3π/n 3 pge 65, line 7: A = L pge 65, line 8: A n = L pge 65, line 9: B n = L L L f (x)dx = f(l ) = f( L + ) L L f (x) cos(nπx/l) dx = nπ L L L f (x) sin(nπx/l) dx = nπ L B n L L f (x) sin(nπx/l) dx = nπ L L L f (x) cos(nπx/l) dx = nπ L A n pge 65, line : f(x), L < x < L pge 66, line : F ( π) = F (π) pge 67, line 3: = π nπ π pge 75, line : = π n= n ( n + pge 76, exercise : ϕ(n) n=n+ pge 76, exercise (): N s pge 78, line : We begin with Euler s formul... pge 79, line 8: my be proved using Euler s formul pge 79, line 6 : + i(e x sin bx + pge 8, line : we pply Euler s formul...

2 pge 8, line 8: Applying Euler s formul... pge 83, exercise 5: r sin x +r r cos x = n= rn sin nx pge 83, exercise 7: lim M,N N n =M α ne inπx/l Try exmple..4 t x = pge 85, line : = A cos(l λ) (insert prentheses) pge 87, line 8: = A( + hl) which requires...(delete L) pge 87, lines -: = h(ae µl + Be µl (delete µ twice) pge 87, line 5: B λ (delete prentheses) pge 9, line : t the end of the line, need Bµ sinh µ) = pge 9, line : t the end of the line, need Bµ sinh µb) = pge 9, line 6: in the intervl < L λ < pge 9, lines 8-9: The first one stisfies < L λ < while the second one stisfies L λ > pge 95, line 3: pge 95, line 5: b φ(x)[s(x)φ (x)] dx + b [λρ(x) q(x)] φ(x)φ(x) dx = b φ(x)[s(x) φ (x)] dx + b [ λρ(x) q(x)] φ(x)φ(x) dx = pge, line : delete minus sign before div(k grd u) pge, line : the integrnd tends to... pge, line 3: q x x(x, y, z ) pge 3, line 4: ht nd... (delete k) pge 3, line 5: krl/k kb (chnge + to ) pge 5, line : πkτ pge 8, exercise 4: conditions: [k( u/ z) h(u T )](x, y, ) =, [k( u/ z) + h(u T )](x, y, L) = pge 9, exercise 6: τ < t < τ pge 3, line 5 : u(z; t) = O(e t ) (cpitl O) pge, exercise 6: u z (; t) =, u z (L; t) = pge 3, line 6 : U (z) n=... pge 5, line 4 : u z (L; t) = pge 5, line : v(z; ) = T 3 T pge 6, line : B λ cos(l λ) (delete both minus signs) pge 7, line 7 : where h, T, T nd T 3 re... pge 7, line : U(z) = T + hz(t T )/( + hl) pge 3, line 3 : chnge v n to f n pge 3, line 3: u t Ku zz = pge 3, line 5: u n () = f n pge 3, line 7: u n (t) = f n e λnkt + pge 3, line : + L cos α cos β = (chnge sin to cos) pge 36, line : T ( r/ s)(; t) (, t) ( r/ s)(; t) = pge 39, line 3: substitute into (.4.9) pge 4, line : developed in Sec...4 pge 43, line : substitute this into (.4.) nd get pge 5, exercise 4:...solution of the wve eqution with... pge 5, exercise 6: = cf(x + ct) + cf(x ct) pge 54, line 6: = m,n= pge 54, line : u(x, y; t) = 4 π m,n=

3 pge 57, line : 4 π m,n= pge 58, line 4: m,n= pge 58, line : m,n= pge 58, line : m,n= pge 58, line 5: m,n= pge 6, line 8: m,n= pge 63, line 6: 3 5 pge 65, line 3 : when x =, x = L pge 69, exercise : u(x, L; t) = pge 7, line 3 : U ϕ pge 73, line 4 : chnge u ρρ to u ϕϕ t end of line pge 75, line : k u ρ ρ=ρ = k ρ T T ln(ρ /ρ ) pge 76, line : boundry conditions u(, ϕ) = nd u(, ϕ) =... pge 77, line : where re the Fourier coefficients pge 78, line : R n B n pge 87, line 3: = ρ λ n= pge 89, line 4 : ie iθ e ix cos θ ) dθ (insert minus sign in exponent) pge 89, line 3 : = π (insert minus sign before integrl π pge 9, line 8: = x m pge 9, line : + d ρ y + pge 9, line 3:... first-order liner eqution for v, which... pge 96, lines -: since ρ θ(ρ) is unbounded when ρ. From... pge 97, line : ( xj m ) (x) = R(x) cos θ(x). The eqution cos βj m (x) + x sin βj m(x) = is... pge 98, line 5: (3..37) [(xy ) ] + (x nx m )(y ) = pge, line :...formuls (3..8)-(3..9) in the... pge, line 7 : For m =, β =,... pge, line : A n J (x x n ) x dx = pge 4, line 3: jj[3.5,4] pge 4, line 6 : jj[3.8,3.85] pge 8, exercise 4: = J (x) + J (x) + J 4 (x) + nd = J (x) + J 3 (x) + J 5 (x) + (chnge minus sign to plus sign in two plces pge 8, exercise 33: F 3 (ρ) = n= J (ρx n )/x 3 nj (x n ) pge, line : the initil conditions (3.3.)-(3.3.3). pge 3, line 9 : while A n = pge 4, line 3: we tke c =, =,... pge 4, line 5: cos(tx (m) n ) (delete c) pge 8, line 8: with β =, m =... pge 9, line : U(ρ mx ) = T. pge 9, line : U(ρ mx ) = T. pge, line 4: with cot β = hρ mx /k pge, line 5: pge, line 6: pge, line 8: = kρ mx h x = 8kρ mx h n= A n = n= J (x x n ) [x n + (hρ mx /k) ]J (x n ) J (x x n ) x n[x n + (hρ mx /k) ]J (x n ) ( ) [x n + (hρ mx /k) ]J (x n ) 3 < x < < x <

4 pge, line 3: π ϕ π pge, line : theory of Sec pge 3, line 6: m,n R(m) n (ρ) pge 3, line : u(ρ; t) = U(ρ) + A n R n (ρ)e λnkt n= pge 3, line : where λ n = λ () n, R n = R n (),... pge 3, line 4: pge 3, line 5: pge 3, line 6: πa n = ρ ρ [f(ρ) U(ρ)]R n (ρ) ρ dρ ρ M ρ R n (ρ) dρ ρ (ρ ρ ) / M pge 3, line 8: the series n= A nr n (ρ)e λnkt pge 5, line 8: λ = iω/k (delete minus sign) pge 6, exercise 4: u t = K u + σ (insert u) pge 8, line : (3.5.8) γ = ν (delete = m ) pge 3, line 4: = 8 π (chnge z to ) pge 38, line : Thus w t = ru t, w r = ru r + u,... (chnge u to w) pge 4, line : f(r) = T,... pge 44, lines 7-8: if nd only if tnh(µ) = kµ/(k h) pge 44, line 5 : (4..9) [rf(r) W (r)] (chnge ϕ to f pge 47, line : ( ) C n cos nπct + D n sin nπct pge 47, line 6 : tht C n = (C/nπ)( ) n+,... pge 48, line 3: ( ) C n cos nπct + D n sin nπct pge 48, line 7 : E cos(nπct/) (chnge C to E) pge 55, line 5 : ( d ds )k (s ) k = pge 57, line 7 : (n )! = s n [(n )!] sn + lower order terms pge 57, line : j+ cn j = pge 58, line 9: The integrls (4..7) cn be used... pge 59, line : in the form [( s )P k ] + k(k + )P k = pge 59, line 6: A k = k + k(k + ) [( )P k() ( b )P k(b)] pge 6, line 3 fter tble: Θ (m) (chnge rgument to superscript) pge 64, line 4: (4..7) cos βf() + sin βf () = (chnge R to in three plces pge 64, line 8: pge 64, line : = (π/4) J k+/ ( λ) (chnge R to in three plces) pge 64, line : (π/ )(λ + cot β m / )J k+/ ( λ) (chnge R to in three plces) pge 69, line 5: u(, θ, ϕ) = cos ϕ P + 3 cos ϕ P pge 69, line : = πa km k π pge 69, line : = πb km k π pge 7, line 3: u(r, θ) = (r cos θ)/( + ) 4

5 pge 73, line 3: r pge 73, line 4: u/ r = A /r + pge 73, line 7: A 4π = pge 74, line : = δ 6 (3 + 3 π b + 4b + 3πb3 8 ) pge 74, line 4: δ cos θ r 3 sin θ pge 74, line 5: = δ π 8 ( + b cos θ)4 cos θ sin θ dθ pge 74, line 8 : A = r 4 sin θ pge 74, line 7 : = δ π ( + b sin θ)5 pge 74, line 3 : A = δ [5 4 bπ + 63 b + 5 b 3 π + 64b b5 π ] 8 pge 8, line : e iµ F (µ) is the Fourier trnsform... pge 86, lines 3, 4 : delete which is stisfied...< nd dd period fter f() =. pge 89, line 8: x +y +z =R pge 9, line 3: = C A (insert minus sign fter equlity) pge 37, line 8: x eh(x ξ) (h + )e ξ dξ (delete minus sign) pge 37, line : e ξ (insert minus sign) pge 3, line : delete = t end of eqution pge 36, line : F (µ) = π pge 39, line : f k (x) dx (chnge i to k subscript) pge 39, line 4: F k (µ) dµ pge 3, line 3 : gives z(ξ, η) = ψ(η) + pge 38, line : ( f)(p + ξ)ds (chnge dξ to ds) pge 33, line : u(x, y) = e iµξ (insert minus sign in exponent) pge 338, line 3 : ( = vt [c v xx (α β )v] + c ) v x v xt + vv t dx (insert prentheses where needed pge 346, line : pge 346, line : nd n n+ log x dx < < log x dx n n+ log x dx log x dx + log(n + ) pge 346, line 4: n log n n + < log + + log n < n log n n + + log(n + ) pge 346, line 5: log n n < log + + log n n pge 35, lines, : we cn llow b =, > in cse... pge 35, line 7: bove expression is O(e th /t) when... pge 356, line : (6.3.6) h(x) H (chnge to ) pge 356, line 7: O( eth t ). pge 356, line 8 : O( eth t ). pge 356, line : t e th pge 357, line 3: O( eth t ). pge 359, line : F (µ) = π 5 < log n + O( log n n )

6 pge 36, line : π th γ pge 36, line 8 : = (τ + ) (insert ) pge 364, line 7: J m (t) = pge 367, line 5: pge 373, line : pge 384, line : πt cos( ) I = φ + (µ) =, φ (µ) = π 4 eiπ/4 (chnge + to ) (7..) Y n+ = Y n + h(f(t n, Y n ) + (insert h in numertor of frction ) pge 387, line 4 :...we recll the big O... pge 39, line :...error bound (7..7) for... pge 39, line 5: ɛ n+ = ɛ n + h pge 39, line 7: ɛ n = nh pge 39, line 9: ɛ n = t n h pge 393, lines 8-9:...mintined t equl tempertures. pge 393, line 6 :...for i =,,..., n + pge 4, line 3: u i (t + t) + u i (t t) = u i+ (t) + u i (t) (chnge = to+) pge 45, line 5: We hve K t/h = pge 4, line 5 : ϕ(p ) dp (delete w(p ) before dv pge 46, line 5 : 4 x 4 y ) (chnge = to ) pge 46, line 4 : = (6/9)ρc 6 (chnge 8 to 6 in exponent) pge 47, line 3:...of the form u = c cos(πx/) cos(πy/) nd... (insert c fter =) pge 47, line 7: Likewise ρu dx dy = ρc(4/π) pge 47, line 9: is ttined t c = 3ρ /π 4 (delete slsh) pge 47, line : Φ(c) = (56/π 6 )ρ 4 pge 48, line : N u y = c j (x) ( j + )π sin pge 48, line 3: = N j= j= [ c j (x) + ( j + ) (π ) cj (x) ρc j (x) ( )j j + ( )] dx π pge 48, line 5: pge 48, line 7: pge 4, line 5: c j (x) = ( j + ) (π = c j (x) = [ ρ( ) j b b (j + ) cj (x) ρ( )j 3 j + ( ) 3 ]( ) π ( ) π [( ( x ) (b y )] 6

7 pge 4, line 7: 5ρ/8( + b ) pge 4, line 5: u(x, y) = 64 b π 4 pge 44, line 8: C = ρ/(k ) pge 46, line : if L / < y < L, L (L y)/l < x < L y/l (insert < sign) pge 46, line 7: if L / < y < L, L (L y)/l < x < L y/l (insert < sign) pge 43, line 4: Theorem 8.. Suppose tht zero is not n eigenvlue of (8..). Then... pge 43, line 7: C = ( 9ξ/5 + ξ3 /)f(ξ) dξ pge 437, line 7: lim ɛ (P Q) n pge 437, line 8: = 4π P Q (chnge Q to Q 3 ) pge 438, line 3: = (chnge to =) D ɛ D ɛ pge 439, line 5 :...point Q is suitbly chosen with Q / D nd... pge 44, line 8: C = /4π Q pge 44, line 8: P = (ρ cos ϕ sin θ, ρ sin ϕ sin θ, ρ cos θ) G pge 44, line : τ (chnge r to τ) pge 445, line 4 : u g/ n u(p ) (delete minus sign) pge 445, line : u(p ) = D pge 453, line 7: = L (chnge odd to ) pge 453, line 8: = L (chnge odd to ) pge 454, line 3: pge 455, line 4: = pge 455, line 5 : pge 455, line : 4πKt <m< (e (x y ml) /(4Kt) e (x+y (m+)l) /(4Kt) ) h(p ; t) = H(µ; t) dµ u(p ; t) = ds e i µ,ξ dξ xi =cs = e i µ,p +ξ H pge 456, line 3: delete the second dq t the end of the line pge 456, line : pge 457, line 4: pge 457, line 4 : pge 457, line : = t M c(t s) ((t s)h (s) )(P ) ds u(p ; t) = + t (chnge = to +) 4π + ( f)(p + ξ)dξ 4πt ξ =ct + ( f)(p + ξ)dξ 4πt ξ =ct 7

8 pge 459, line 4 :...right side of the wve eqution is written... pge 46, line 7:...w xx = v xx e (ky/c) pge 46, line 7: [f (x + ct) + + k ct f (x ξ)i move c from first term to second term nd chngef tof c ct pge 47, line 5:... = e t dy ds + et d Y ds = sdy ds + s d Y ds = sy + s Y pge 473, line 4 : y = n= (n + )ny nt n pge 477, line 6 : = ( M p )/(p ) pge 48, line 7 : we hve f n (x) =... pge 48, line 4 : we hve f n (/n) = pge 48, line 3 : f n (x n ) f(x n ) pge 484, line 3: < ( x N+ )M + ɛ/ pge 484, line 4: If x N+ < ɛ/4m pge 485, line 8 : insert comm fter first ppernce of t pge 485, line 7 : insert comm fter first ppernce of t pge 53, line : nswer to (d) is...y + (λ )Y = pge 54, line 8: nswer to 5 with λ < is u(x, y) = [A cos( λ ln x ) + A sin( λ ln x )](A 3 e y λ + A 4 e y λ ) pge 54, line 7: Answer to no. 3 in..4 is u n (x, y) = A n sin(nπx/l)e (nπy/l, n =,,... pge 54, line 8 : nswer to no. (d) in.3 is 7 (not 3) 6 pge 55, line 6: (nπ) 3 pge 55, line 9: nswer to no. 5 in. is cos 4x pge 55, line : nswer to no. 6 in. is cos 3x cos x pge 55, line 9: nswer to 6(c) needs + (nπ/l) in denomintor pge 56, line : nswer to 6(d) is π /4 pge 56, line 5 : σn (sine series) = = O(N 5 ) pge 57, line 7: e x L = inπ L +n π (delete extr fctor of n) pge 57, line 8: re = ix n= pge 57, line : nswer to no. in.6 should red...is root of the trnscendentl eqution h λ cos(l λ) + (h λ) sin(l λ) =, n =,,... pge 57, line : nswer to no. 3 in. should red U(z) = Φ(z L) + T pge 57, line : nswer to no. 4 in. should red U(z) = /(K + hl) pge 58, line : nswer to no. 5 should end with sinh(l z) β/k sinh L β/k pge 58, lines 8-9: solution to no, chnge t s to tu s in six plces, thus u(z; t) = A + A exp [ π ] (πt π z cos z ) Kτ τ Kτ +A exp [ π ] (πt π z cos z ) Kτ τ Kτ pge 58, line 5: delete c before sin, thus cos(βt cz) + sin(βt cz) pge 59, line 8: nswer to no 3 in.3 need plus sign before L, thus A n = + L (T T ) pge 5, line 4: in nswer to no. 9, insert ds fter g (s) 8

9 pge 5, lines -3: u(x, y : t) = exp [ ( nπ L ) Kt ] (chnge L to L, A n = (T 3 T )( ( ) n ) nπ + (T T )( ) n nπ (deletel pge 5, line 5: nswer to no. 5 in 3. is u(ρ, ϕ) = ln ρ ln π [ρ n ρ n ] n n n= pge 5, line 4 : nswer to no in 3.3 is U(ρ) = (g/4c )(ρ ) pge 5, line, 3 replce R by everywhere, thus pge 5, line : u(ρ, ϕ; t) = A n = n= A n J (ρx n /) cos( ctx n, J (x n ) = x n J (x n ) F (ρ)j ( ρx n ) ρ dρ u(ρ, ϕ; t) = n= A n J (ρx n /) sin( ctx n, A n = cx n J (x n ) pge 53, line 9: nswer to no. 4 in Sec. 4. should red (cot θ)/r. pge 53, line 3 : nswer to no. 8, chnge 6 to exponent of, thus exp[ ( nπ ) Kt] + T pge 53, line : chnge 9 to left prenthesis in exponent, thus exp[ ( n π ) Kt] pge 54, line 4: nswer to no. in Sec 4. should red,,, 3 8 pge 54, lines 9-: chnge. to.,. to 3., 3. to 4., 4. to 5. pge 54, line : u(r, θ) = (r/)3 pge 55, line 4: nswer to no. 5 in 5. should hve pge 57, line F (µ) = [ π + ( + µ) + ] ( + hz) [I + h(l x)] h( + hl) pge 58, line : chnge 6. to 5. nd chnge k to k in denomintor of second formul pge 58, line 3: chnge 7. to 6. pge 58, line 4: chnge 8. to 7. pge 58, line : in solution to no. in Sec 8.4, insert ds t end of line pge 58, line : in solution to no. in Sec 8.4, insert ds t end of line pge 58, line 3: in solution to no. 3 in Sec 8.4, insert dη ds t end of line pge 5, line 5 of left column: insted of de Moivre s formul.., it should red Euler s formul, 78, 8 nd be moved to line 7 of the sme pge. 9