PDEnotesPart2

PDEnotesPart2 - EGN 5422 PREVIOUS MOVIES SITE

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Unformatted text preview: EGN 5422 PREVIOUS MOVIES SITE http://netcast.usf.edu/browse.php?page=Classes/engineering/ snider/egn5422 NEW MOVIES SITE http://netcast.usf.edu/browse.php?page=Classes/engineering/ snider/PartialDiffEqs PREVIOUS MOVIE NAME egn5422pde_0106a egn5422pde_0106b egn5422pde_0108a egn5422pde_0108b egn5422pde_0111intro egn5422pde_0113a egn5422pde_0113b egn5422pde_0115a egn5422pde_0115b egn5422pde_0120a egn5422pde_0120b egn5422pde_0121a egn5422pde_0121b egn5422pde_0121c egn5422pde_0121d egn5422pde_0121e egn5422pde_0122a egn5422pde_0122b egn5422pde_0127a egn5422pde_0127b egn5422pde_0129a egn5422pde_0129b egn5422pde_0203a egn5422pde_0203b egn5422pde_0205a egn5422pde_0205b egn5422pde_0210a egn5422pde_0210b egn5422pde_0212a egn5422pde_0212b egn5422pde_0224a egn5422pde_0224b egn5422pde_0226a egn5422pde_0226b egn5422pde_0302a egn5422pde_0302b egn5422pde_0304a NEW MOVIE NAME egn5422pde_Lec1_ODEs egn5422pde_Lec2_cosh_sinh egn5422pde_Lec3_ConstCoef egn5422pde_Lec4_Equidimensional DELETE egn5422pde_Lec5_ChangeOf egn5422pde_Lec6_Variable egn5422pde_Lec7_Mass_Spring egn5422pde_Lec8_PowerSeries egn5422pde_Lec9_Frobenius egn5422pde_Lec10_Bessel egn5422pde_Lec11_Fourier1 egn5422pde_Lec12_Fourier2 egn5422pde_Lec13_Fourier3 egn5422pde_Lec14_Fourier4 egn5422pde_Lec15_Fourier5 egn5422pde_Lec16_Bessel egn5422pde_Lec17_Fourier egn5422pde_Lec18_OrthogonalFcns egn5422pde_Lec19_1D_WaveEq egn5422pde_Lec20_InitBndryConds egn5422pde_Lec21_StandingWaves egn5422pde_Lec22_DAlembert egn5422pde_Lec23_3D_Wave_Heat egn5422pde_Lec24_LaplaceEq egn5422pde_Lec25_AnalyticFcns egn5422pde_Lec26_FirstPDE egn5422pde_Lec27_Superposition egn5422pde_Lec28_BndryConds egn5422pde_Lec29_BndryConds egn5422pde_Lec30_SolutionCatalog egn5422pde_Lec31_Separation egn5422pde_Lec32_SphericalCoords egn5422pde_Lec33_Subproblems egn5422pde_Lec34_Eigenfunctions egn5422pde_Lec35_Expansions egn5422pde_Lec36_Expansions egn5422pde_0316a egn5422pde_0316b egn5422pde_0318a egn5422pde_0318b egn5422pde_0323a egn5422pde_0323b egn5422pde_0325a egn5422pde_0325b egn5422pde_0330a egn5422pde_0330b egn5422pde_0401a egn5422pde_0401b egn5422pde_0406a egn5422pde_0406b egn5422pde_0408a egn5422pde_0413a egn5422pde_0415a egn5422pde_0420a egn5422pde_physicalparameters egn5422pde_Lec37_RobinConds egn5422pde_Lec38_Expansions egn5422pde_Lec39_SturmLiouville egn5422pde_Lec40_PeriodicBCs egn5422pde_Lec41_Examples egn5422pde_Lec42_Examples egn5422pde_Lec43_PointAtInfinity egn5422pde_Lec44_SingularBCs egn5422pde_Lec45_Examples egn5422pde_Lec46_SolutionsTable egn5422pde_Lec47_3DExamples egn5422pde_Lec48_3DExamples egn5422pde_Lec49_Examples egn5422pde_Lec50_Examples egn5422pde_Lec52_Eigenmodes egn5422pde_Lec53_GreensFcns egn5422pde_Lec54_TimeDependence egn5422pde_Lec55_Examples egn5422pde_Lec51_Numerics Title: Jan 6 12:37 PM (1 of 10) Title: Jan 6 12:42 PM (2 of 10) Title: Jan 6 12:48 PM (3 of 10) Title: Jan 6 12:51 PM (4 of 10) Title: Jan 6 12:56 PM (5 of 10) Title: Jan 6 12:59 PM (6 of 10) Title: Jan 6 1:01 PM (7 of 10) Title: Jan 6 1:04 PM (8 of 10) Title: Jan 6 1:09 PM (9 of 10) Title: Jan 6 1:10 PM (10 of 10) Title: Jan 8 12:08 PM (1 of 9) Title: Jan 8 12:18 PM (2 of 9) Title: Jan 8 12:22 PM (3 of 9) Title: Jan 8 12:32 PM (4 of 9) Title: Jan 8 12:47 PM (5 of 9) Title: Jan 8 12:51 PM (6 of 9) Title: Jan 8 12:55 PM (7 of 9) Title: Jan 8 1:01 PM (8 of 9) Title: Jan 8 1:09 PM (9 of 9) Title: Jan 13 12:12 PM (1 of 11) Title: Jan 13 12:19 PM (2 of 11) Title: Jan 13 12:30 PM (3 of 11) Title: Jan 13 12:33 PM (4 of 11) Title: Jan 13 12:39 PM (5 of 11) Title: Jan 13 12:49 PM (6 of 11) Title: Jan 13 12:54 PM (7 of 11) Title: Jan 13 12:56 PM (8 of 11) Title: Jan 13 1:00 PM (9 of 11) Title: Jan 13 1:07 PM (10 of 11) Title: Jan 13 1:11 PM (11 of 11) Title: Jan 15 11:57 AM (1 of 11) Title: Jan 15 12:09 PM (2 of 11) Title: Jan 15 12:15 PM (3 of 11) Title: Jan 15 12:19 PM (4 of 11) Title: Jan 15 12:25 PM (5 of 11) Title: Jan 15 12:31 PM (6 of 11) Title: Jan 15 12:44 PM (7 of 11) Title: Jan 15 12:52 PM (8 of 11) Title: Jan 15 12:53 PM (9 of 11) Title: Jan 15 12:59 PM (10 of 11) Title: Jan 15 1:06 PM (11 of 11) Title: Jan 20 12:09 PM (1 of 8) Title: Jan 20 12:16 PM (2 of 8) Title: Jan 20 12:21 PM (3 of 8) Title: Jan 20 12:24 PM (4 of 8) Title: Jan 20 11:55 AM (5 of 8) Title: Jan 20 12:45 PM (6 of 8) Title: Jan 20 1:05 PM (7 of 8) Title: Jan 20 1:10 PM (8 of 8) Title: Jan 18 11:29 AM (1 of 58) Title: Jan 15 9:24 PM (2 of 58) Title: Jan 15 9:57 PM (3 of 58) Title: Jan 15 10:20 PM (4 of 58) Bank Account p(n) = principal on nth day r = daily interest rate P = equilibrium level (If p(n) = P and you withdraw rP then p(n+1) = P.) p(n+1) = p(n) + rp(n) rP + f(n) let x(n) = p(n)P x(n+1)=x(n)+rx(n) + f(n) Sinusoidal solutions: take f(n) = cos Wn try x(n) = A cos Wn + B sin Wn x(n+1) = A cos W(n+1) + B sin W(n+1) cos W(n+1) = cos Wn cos W sin Wn sin W sin W(n+1) = sin Wn cos W cos Wn sin W x(n+1) = A[cos Wn cos W sin Wn sin W] +B[sin Wn cos W cos Wn sin W] A[cos Wn cos W sin Wn sin W] +B[sin Wn cos W cos Wn sin W] = {A cos Wn + B sin Wn}{1+r} + cos Wn {cos W (1+r)}A + {sin W}B = 1 {sin W}A + {cos W (1+r)}B = 0 x(n) = {cos W (1+r)}/{[cos W (1+r)] sin W} cos Wn + {sin W}/{[cos W (1+r)] sin W} sin Wn Title: Jan 16 3:30 PM (5 of 58) Can we express A cos t + B sin t more conveniently? Answer #1. A cos t + B sin t = C cos (t + ) if C = (A2+B2) , = tan1 (B/A) Answer #2. Title: Jan 16 3:57 PM (6 of 58) Title: Jan 16 4:20 PM (7 of 58) Can an arbitrary function f(t) be expressed as a superposition f(t) = A cos t + B sin t or A cos t + B sin t Title: Jan 16 4:37 PM (8 of 58) Title: Jan 18 11:29 AM (9 of 58) Title: Jan 16 4:55 PM (10 of 58) Title: Jan 16 5:03 PM (11 of 58) Title: Jan 16 11:55 PM (12 of 58) Title: Jan 16 5:08 PM (13 of 58) Title: Jan 16 5:19 PM (14 of 58) Title: Jan 16 5:27 PM (15 of 58) Title: Jan 17 12:20 AM (16 of 58) Title: Jan 17 12:21 AM (17 of 58) Title: Jan 17 12:22 AM (18 of 58) Vector Interpretation of f(t) g(t) dt v = ai + bj + ck w = di + ej + fk vw = ad + be + cf t f(t) g(t) 0.0 # # 0.1 # # 0.2 # # 0.3 # # 1.0 # # f(t) g(t) dt [## + ## + ## + ## + + ##] 0.1 = " fg " = <f,g> "inner product" So we say {cos 2nt/T, sin 2nt/T : n = 0, 1, 2, 3,...} are "orthogonal" <cos 2n t/T, cos 2n T> = 0 if n n <sin 2n t/T, sin 2n T> = 0 if n n <cos 2n t/T, sin 2n T> = 0 (always) Title: Jan 16 11:57 PM (19 of 58) Title: Jan 17 12:27 AM (20 of 58) Title: Jan 18 11:30 AM (21 of 58) Title: Jan 17 11:33 AM (22 of 58) Title: Jan 17 8:48 PM (23 of 58) Title: Jan 17 11:49 AM (24 of 58) Title: Jan 17 11:51 AM (25 of 58) Title: Jan 17 11:57 AM (26 of 58) Title: Jan 17 12:00 PM (27 of 58) Title: Jan 17 12:02 PM (28 of 58) Title: Jan 17 8:51 PM (29 of 58) Title: Jan 17 12:37 PM (30 of 58) Title: Jan 17 12:39 PM (31 of 58) Title: Jan 18 11:31 AM (32 of 58) Title: Jan 17 8:31 PM (33 of 58) Title: Jan 17 9:05 PM (34 of 58) Title: Jan 17 9:09 PM (35 of 58) Title: Jan 17 9:12 PM (36 of 58) Title: Jan 17 9:14 PM (37 of 58) Title: Jan 18 10:50 AM (38 of 58) Title: Jan 18 10:52 AM (39 of 58) Title: Jan 18 10:56 AM (40 of 58) Title: Jan 18 11:01 AM (41 of 58) Title: Jan 18 11:07 AM (42 of 58) Title: Jan 18 11:09 AM (43 of 58) Title: Jan 18 11:14 AM (44 of 58) Title: Jan 18 11:32 AM (45 of 58) Title: Jan 17 9:17 PM (46 of 58) Title: Jan 17 9:25 PM (47 of 58) Title: Jan 17 9:31 PM (48 of 58) Title: Jan 17 9:40 PM (49 of 58) Title: Jan 17 9:43 PM (50 of 58) Title: Jan 17 9:45 PM (51 of 58) Title: Jan 17 10:05 PM (52 of 58) Title: Jan 17 10:11 PM (53 of 58) Title: Jan 17 10:18 PM (54 of 58) Title: Jan 17 10:22 PM (55 of 58) Title: Jan 17 10:28 PM (56 of 58) Title: Jan 17 10:32 PM (57 of 58) Title: Jan 18 9:57 AM (58 of 58) Universal Fourier Formulae: for any choice of the symbol , 1 - j t j t f (t ) = f ( )e d , f ( ) = f (t )e dt = F .T . of f (t ) 2 - - or v(t ) = V ( f )e - j 2f t df , V( f ) = - v(t )e - j 2f t dt = F .T . of v(t ) Exchange Rule: 1 f (- ) = F .T . of f (t ) 2 2 or v(- f ) = F .T . of V (t ) Convolution Rule: f g ( ) = F .T . of f (t ) g (t ) , 2f ( ) g ( ) = F .T . of f (t ) g (t ) or V ( f ) W ( f ) = F .T . of v(t ) w(t ) , V ( f )W ( f ) = F .T . of v(t ) w(t ) Parseval's Theorem: - f (t ) g (t )dt = 2 2 - f ( ) g ( )d or - v(t ) w(t )dt = V ( f )W ( f )df - _ _ Poisson Sum Formula: 1 or n = - 1 f ( + n2 n )= 1 f (n )e - jn = F .T . of f sampled (t ) 2 n = - - jn 2f 1 n2 f (t + ) =n f (n)e jn n = - = - For periodic functions: If f (t ) = n = - V ( f + ) = v(n )e n = - = F .T . of v sampled (t ) ; or 1 f n = - v(t + n ) = V (nf )e jn 2f f n = - k = - T f (k )e jk 2 t / T and g (t ) = k = - g (k )e jk 2 t / T then 1 T 1 T 0 T f (t ) g (t )dt = k = - f (k ) g (k ) (Parseval) (Convolution) (Convolution) 0 f (t - t ' ) g (t ' )dt ' = k = - f (k ) g (k )e j 2kt / T k f (k - k ' ) g (k ' )e j 2kt / T = f (t ) g (t ) k = - '= - Fourier Formulae using frequency v(t ) = V ( f )e - j 2f t df , V( f ) = - v(t )e - j 2f t dt = F .T . of v(t ) Exchange Rule: v(- f ) = F .T . of V (t ) Convolution Rule: V ( f ) W ( f ) = F .T . of v(t ) w(t ) Parseval's Theorem: , V ( f )W ( f ) = F .T . of v(t ) w(t ) - v(t ) w(t )dt = V ( f )W ( f )df - _ _ Start from here Poisson Sum Formula: 1 n = - f ( + n2 )= 1 f (n )e - jn = F .T . of f sampled (t ) 2 n = - 1 n2 f (t + ) = f (n)e jn n = - n = - For periodic functions: If f (t ) = k = - f 1 T 1 T (k )e jk 2 t / T and g (t ) = k = - g (k )e jk 2 t / T then T 0 T f (t ) g (t )dt = k = - f (k ) g (k ) (Parseval) 0 f (t - t ' ) g (t ' )dt ' = k = - f (k ) g (k )e j 2kt / T (Convolution) k = - f k '= - (k - k ' ) g (k ' )e j 2kt / T = f (t ) g (t ) (Convolution) Proof of Convolution Rule Title: Jan 22 12:17 PM (1 of 13) Title: Jan 22 12:28 PM (2 of 13) Title: Jan 22 12:29 PM (3 of 13) Title: Jan 22 12:36 PM (4 of 13) Title: Jan 22 12:39 PM (5 of 13) Title: Jan 22 12:46 PM (6 of 13) Title: Jan 22 12:51 PM (7 of 13) Title: Jan 22 12:56 PM (8 of 13) Title: Jan 22 1:01 PM (9 of 13) Title: Jan 22 1:05 PM (10 of 13) Title: Jan 22 1:09 PM (11 of 13) Title: Jan 22 1:12 PM (12 of 13) Title: Jan 22 1:14 PM (13 of 13) ...
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