Math-33 Chapter Partial Differetial Equatios November 6, 7 Chapter Partial Differetial Equatios ad Fourier Series
Math-33 Chapter Partial Differetial Equatios November 6, 7. Boudary Value Problems for d order ODE Oe-Dimesioal Boudary Value Problems + +, ( a,b) y p y q y g d order liear ODE Plae layer Domai: a < < b, boudary cosists of two poits: a ad b y y y y a y y b a b The possible types of boudary coditios at a ad b: k,h,h > Type of boudary coditio No-homogeeous boudary coditios: I st kid (Dirichlet) y( a) y y( b) y II d kid (Neuma) y ( a) y y ( b) y III 3 rd kid (Robi, mied) ky ( a) + h y ( a) y ky b + h y b y Homogeeous boudary coditios: I y( a) y b y b II y ( a) III ky ( a) + h y ( a) ky b + h y b
Math-33 Chapter Partial Differetial Equatios November 6, 7 3 Special Equatio (homogeeous d order liear ODE with costat coefficiets), (, ) y λ y Characteristic equatio m λ Geeral Solutio ca be writte i the followig forms: ) λ >, m y ce + ce ± ( ) + ( ) y c cosh c sih ( ) + ( ) y ccosh csih ) <, λ m ± i y c cos ( ) + c si( ) ( ) + ( ) y ccos c si 3) λ m, y c+ c + ( ) y c c Eigevalue Problem: fid the values of λ for which differetial has o-zero solutios, (, ) y λ y subject to oe of the ie possible combiatios of the homogeeous boudary coditios: y I y I y II y II III y + Hy III y + H y This problem has a o-zero solutio oly if λ < There eist ifiitely may real values < < < 3 <... (eigevalues) ad correspodig o-zero solutios y, y, y 3,... (eigefuctios) which satisfy the boudary coditios (i a case whe both b.c. s are of the d kid: y, y, there are also ad y ) Eigefuctios 3 m m y,y y y d for m y, y, y,... are mutually orthogoal:
Math-33 Chapter Partial Differetial Equatios November 6, 7 4 There ie possible eigevalue problems + subject to differet combiatios of boudary coditios: y y Eample : + (,) y y I y I y π,,... π y si y c cos + c si (p.58) y c si ( y,y ) y c cos + c si c y si π Eample : + (,). #8 y y II y II y π ( y,y) y y π cos ( y,y ) Eample 3: + (,) y y I y II y. #4 π +,,,... π y si + ( y,y ) Eample 4: + (,). #5,7 y y II y I y π +,,,... π y cos + ( y,y )
Math-33 Chapter Partial Differetial Equatios November 6, 7 5 The followig possible eamples are importat but they are ot cosidered i our class (here, H> is a costat): Eample 5: + (,) y y I y III y + Hy Eample 6: + (,) y y III I y y + Hy Eample 7: + (,) y y II y III y + Hy Eample 8: + (,) y y III y + Hy II y Eample 9: + (,) y y III y + Hy y + H y III
Math-33 Chapter Partial Differetial Equatios November 6, 7 6. Fourier Series Periodic fuctio f ( + T) f for all. T is a period. The smallest period is called the fudametal period. Eve fuctio f ( ) f Odd fuctio u( ) u Properties: et f,g be eve fuctios, ad let u,v be odd fuctios, the f + g eve f g eve v u + odd v u eve f u odd f ( ) is eve, the f d u( ) is odd, the u d f d Ier product of two fuctios o [ a,b ] : ( u,v) b uvd a b Fuctios u ad v are said to be orthogoal if: u,v u v d a Geeralized Fourier Series: If { yk } is a complete set of fuctios mutually orthogoal o [ a,b ], the f ca be represeted by f c y + c y +... + c y +... c y, where k k k k k c k ( f,yk ) ( y,y ) k k
Math-33 Chapter Partial Differetial Equatios November 6, 7 7 Set of fuctios π π, cos, si is orthogoal o [, ]. Verify orthogoality d mπ π cos cos d m m π cos d π si π mπ π si si d m m π si d mπ π cos si d The Euler-Fourier Formulas (p.587): Fourier Series a mπ mπ f + a cos + b si m m (9) m a f d () a m π f cos d (3) b m π f si d (4).3 Covergece of the Fourier Series trucated Fourier series Gibb's pheomea limit of Fourier series et f ( ) be a piece-wise cotiuous with a fiite umber of fiite jumps o [,), the a mπ mπ + + amcos b si coverges to m f ( ), if f + + f f is cotiuous at, if f ( ) is discotiuous at
Math-33 Chapter Partial Differetial Equatios November 6, 7 8 > :; Maple Fourier series epasio of > f:heaviside(+)-heaviside(-); < < f < < < < : > a[]:/*it(f,-..); f : Heaviside ( + ) Heaviside ( ) a : > a[k]:/*it(f*cos(k*pi/*),-..); > b[k]:/*it(f*si(k*pi/*),-..); The Fourier Series: si k π a k : k π b k : > u:a[]/+sum(a[k]*cos(k*pi/*)+b[k]*si(k*pi/*),k..): > plot({f,u},-..); > plot({abs(f-u)},-..); > plot({u},-5*..5*); periodic etesio: Useful facts : si π cos π ( ) si π cos π
Math-33 Chapter Partial Differetial Equatios November 6, 7 9.4 The Fourier Series of Odd ad Eve Fuctios (Sie Fourier Series ad Cosie Fourier Series)
Math-33 Chapter Partial Differetial Equatios November 6, 7.5 The Heat Equatio Homogeeous Boudary Coditios ( I I) Iitial-Boudary Value Problem: u u iitial temperature distributio, t Heat Equatio u u a t Iitial coditio: u (,) u < <, t > u(,t) curret temperature distributio, t > Boudary coditios: u(,t) t > (I) u(,t) t > (I) ) Assume u(,t) T ( t) u,t T t u,t T t t T ) Separate variables T T λ a a T u,t ( T ) ( t) u(,t) T ( t) 3) Solve eigevalue problem λ λ (I) (I) 4) Solve T λ a T π,,... π si ( ) si T + at at T t e u,t c T t where 5) Solutio: u d π c u si d d
Math-33 Chapter Partial Differetial Equatios November 6, 7.5 The Heat Equatio ( I I) Sigle term solutio u kπ u si u(,t) iitial temperature distributio at t is i the shape of oe of eigefuctios k k si π with some value k curret temperature distributio, t > Heat Equatio u u a t Iitial coditio: < <, t > kπ u, si Boudary coditios: u(,t) t > (I) u(,t) t > (I) ) Assume u(,t) T ( t) u,t T t u,t T t t T ) Separate variables T T λ a a T u,t ( T ) ( t) u(,t) T ( t) 3) Solve eigevalue problem λ λ (I) (I) 4) Solve T λ a T π,,... π si ( ) si T + at at T t e u,t c T t where 5) Solutio: kπ kπ ck si si d c for all k Therefore, u (,t ) T ( t) k k
Math-33 Chapter Partial Differetial Equatios November 6, 7.5 The Heat Equatio Homogeeous Boudary Coditios (I II) Iitial-Boudary value Problem: u u iitial temperature distributio, t Heat Equatio u u a t Iitial coditio: u (,) u < <, t > u(,t) curret temperature distributio, t > Boudary coditios: u(,t) t > (I) u (,t) t > (II) ) Assume u(,t) T ( t) u,t T t u,t T t t substitute ito equatio ad boudary coditios T ) Separate variables T T λ a a T u,t ( T ) ( t) u,t T t 3) Solve eigevalue problem λ λ (I) (II) π +,,,... π si( ) si + T 4) Solve λ T + at a T at T t e u,t c T t, 5) Solutio: u d π c u si + d d
Math-33 Chapter Partial Differetial Equatios November 6, 7 3.6 The Heat Equatio No-Homogeeous Boudary Coditios Iitial-Boudary Value Problem: u u iitial temperature distributio, t Heat Equatio u u a t < <, t > u(,t) us curret temperature distributio, t > steady state temperature distributio Iitial coditio u (,) u Boudary coditios u (,t) u t > I u u,t t > I ) Steady stateproblem u s Boudary coditios: s u u (I) u u (I) s [ defiitio of steady state: u limu,t ] s t u u u + u Steady State Solutio: s ) Defie trasiet solutio: U (,t) u (,t) u s Iitial-Boudary problem U U a t < <, t > Iitial coditio U (,) u u s Boudary coditios U,t t > U,t t > (Homogeous B.C. s) 3) Solutio of IBVP: u (,t) U (,t) u + (cosists of trasiet ad steady state solutios) s u u + +, Solutio: u (,t) u c T ( t) c u us d d, at T t e
Math-33 Chapter Partial Differetial Equatios November 6, 7 4.7 The Wave Equatio Homogeeous Boudary Coditios Iitial-Boudary Value Problem: u u iitial shape, t Wave Equatio u u a t < <, t > u u(,t) iitial velocity, t curret strig deflectio, t > Iitial coditio: u (,) u u, u t Boudary coditios: u,t t > I u,t t > I ) Assume u(,t) T ( t) u,t T t u,t T t t substitute ito equatio ad boudary coditios T ) Separate variables T T λ a a T u,t ( T ) ( t) u(,t) T ( t) 3) Solve eigevalue problem λ λ (I) (I) 4) Solve T λ a T π,,,... π si ( ) si T + at ( ) + ( ) T t c cos a t c si a t,, π u,t a cos a t b si a t + si, 5) Solutio: ( ) ( ) π a u si d, b u si d a π π
Math-33 Chapter Partial Differetial Equatios November 6, 7 5 The Wave Equatio Homogeeous Boudary Coditios Stadig Waves
Math-33 Chapter Partial Differetial Equatios November 6, 7 6.8 The aplace Equatio Basic Boudary Value Problem 3 homogeeous, o-homogeous b.c. s : y f The aplace Equatio u u + y < <, < y < M M Boudary coditios: [ I,II,III ] [ I,II,III ] [ I,II,III ] y [ I,II,III ] f y M ) Assume u(,y) Y ( y) substitute ito equatio ad b.c. s ) Separate variables Y + Y Y λ Y u (,y) Y ( y) u (,y) Y ( y) u (,) Y Y 3) Solve eigevalue problem No-zero solutios eist oly if λ λ λ (I, II or III) (I, II or III) 4) Solve Y λ Y Y Y ( ) + ( ) Y y c cosh y c sih y,, Y Y ( y) c, sih( y) 5) Solutio: u (, y) a sih( y), a sih ( M ) f d d
Math-33 Chapter Partial Differetial Equatios November 6, 7 7 Semi-ifiite layer domai: >, boudary is defied by a sigle poit: y y y y
Math-33 Chapter Partial Differetial Equatios November 6, 7 8 The aplace Equatio No-homogeeous boudary coditios (superpositio priciple) Dirichlet Problem (I) M f ( y) f 3 u ( y) f 4 Split ito supplemetal basic problems: f f u ( y) u f 3 ( y) u 3 u 4 f 4 f Solutio of supplemetal basic problems: π π u(, y) a si sih ( y M ) a π f si d π sih M π π u(, y) b si sih y b π f si d π sih M π π u3(, y) c sih ( ) si y M M c M π 3 M M f y si y dy π sih M π π u4(, y) d sih si y M M d M π f4 ( y) si y dy M M π sih M Solutio of Dirichlet problem superpositio of supplemetal solutios: (, y) u (, y) + u (, y) + u (, y) u (, y) u 3 + 4
Math-33 Chapter Partial Differetial Equatios November 6, 7 9 POISSON'S EQUATION M y [ u] f y M u u + + F (,y) y (,y) (,) (,M ) [ u] f ( y) 3 u+ F [ u] f y [ u] f ( y) 4 [ u] S f brackets mea that coditio at the boudary ca be either st d rd of I, II or III kid ( i a case if 3-D problem is cosidered ) Supplemetal Sturm-iouville Problems [ ] [ ] SP λ λ Y ηy [ Y] y [ Y] y M SP Y ν Y m m m η ν m Ym ( y) m Z γ Z [ Z] z [ Z] z K SP Z ω Z γ ω k Zk k k k ( z) k APACE'S EQUATION (homogeeous eq, o-homogeeous boudary coditios) f 3 f u 5 f f 4 u Y a ( ) u,y a Y f d Y f Solutio of Basic Cases of aplace's Equatio with oe o-homegeeous b.c. by Separatio of Variables ( SV ) f 3 f u u 3 u 4 f 4 Y m Y m m b ( ) u,y b Y c ( ) m u,y c Y 3 m m m M f Y dy 3 m m m Ym f d Y M d ( ) m u,y d Y 4 m m m M f Y dy 4 m m m Ym Y m u5 u + u + u3 + u4 Superpositio Priciple ( SP) Solutio of Basic Case of Poisso's Eq (homogeeous b.c.'s) u 6 F Y m u,y A Y 6 m m m A M ( λ + ν ) F Y ddy m m m Ym Eigevalue Epasio ( EE) Solutio of Poisso's Equatio with o-homogeeous b.c.'s u(,y) u5 + u6 Superpositio Priciple ( SP)
Math-33 Chapter Partial Differetial Equatios November 6, 7 Eigevalue Problem +, (,) subject to homogeeous boudary coditios. y y There eist ifiitely may real values < < 3 <... (eigevalues) ad correspodig o-zero solutios y, y, y 3,... which satisfy the boudary coditios. (eigefuctios) Boudary cs. Eigevalues, eigefuctios Fourier series I y π π f b si I y π y si π b f si d II y, π a π f + a cos II y y, y π cos π a f cos d I y π + π f c si + II y π y si + π c f si d + II y π + π f d cos + I y π y cos + π d f cos d + f cy The Geeralized Fourier series ( ) ( y,y ) f,y c f ( ) y d Orthogoality of eigefuctios ( m ) m y, y y y d if m