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Partial Differential Equations: Graduate Level Problems and ...

Partial Di erential equations : Graduate Level Problems and Solutions Igor Yanovsky 1. Partial Di erential equations Igor Yanovsky, 2005 2. Disclaimer: This handbook is intended to assist Graduate students with qualifying examination preparation. Please be aware, however, that the handbook might contain, and almost certainly contains, typos as well as incorrect or inaccurate solutions. I can not be made responsible for any inaccuracies contained in this handbook. Partial Di erential equations Igor Yanovsky, 2005 3. Contents 1 Trigonometric Identities 6. 2 Simple Eigenvalue Problem 8. 3 Separation of Variables: Quick Guide 9. 4 Eigenvalues of the Laplacian: Quick Guide 9. 5 First-Order equations 10. Quasilinear equations .. 10. Weak Solutions for Quasilinear equations .

8.3 Polar Laplacian inR2 forRadialFunctions ..... 32 8.4 Spherical Laplacian in R3 and Rn forRadialFunctions ... Partial Differential Equations Igor Yanovsky, 2005 10 5First-OrderEquations 5.1 Quasilinear Equations Consider the Cauchy problem for the quasilinear equation in two variables

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Transcription of Partial Differential Equations: Graduate Level Problems and ...

1 Partial Di erential equations : Graduate Level Problems and Solutions Igor Yanovsky 1. Partial Di erential equations Igor Yanovsky, 2005 2. Disclaimer: This handbook is intended to assist Graduate students with qualifying examination preparation. Please be aware, however, that the handbook might contain, and almost certainly contains, typos as well as incorrect or inaccurate solutions. I can not be made responsible for any inaccuracies contained in this handbook. Partial Di erential equations Igor Yanovsky, 2005 3. Contents 1 Trigonometric Identities 6. 2 Simple Eigenvalue Problem 8. 3 Separation of Variables: Quick Guide 9. 4 Eigenvalues of the Laplacian: Quick Guide 9. 5 First-Order equations 10. Quasilinear equations .. 10. Weak Solutions for Quasilinear equations .

2 12. Conservation Laws and Jump Conditions .. 12. Fans and Rarefaction Waves .. 12. General Nonlinear equations .. 13. Two Spatial Dimensions .. 13. Three Spatial Dimensions .. 13. 6 Second-Order equations 14. Classi cation by Characteristics .. 14. Canonical Forms and General Solutions .. 14. Well-Posedness .. 19. 7 Wave Equation 23. The Initial Value Problem .. 23. Weak Solutions .. 24. Initial/Boundary Value Problem .. 24. Duhamel's Principle .. 24. The Nonhomogeneous Equation .. 24. Higher Dimensions .. 26. Spherical Means .. 26. Application to the Cauchy Problem .. 26. Three-Dimensional Wave Equation .. 27. Two-Dimensional Wave Equation .. 28. Huygen's Principle .. 28. Energy Methods .. 29. Contraction Mapping Principle .. 30. 8 Laplace Equation 31.

3 Green's Formulas .. 31. polar Coordinates .. 32. polar Laplacian in R2 for Radial Functions .. 32. Spherical Laplacian in R3 and Rn for Radial Functions .. 32. Cylindrical Laplacian in R3 for Radial Functions .. 33. Mean Value Theorem .. 33. Maximum Principle .. 33. The Fundamental Solution .. 34. Representation Theorem .. 37. Green's Function and the Poisson Kernel .. 42. Partial Di erential equations Igor Yanovsky, 2005 4. Properties of Harmonic Functions .. 44. Eigenvalues of the Laplacian .. 44. 9 Heat Equation 45. The Pure Initial Value Problem .. 45. Fourier Transform .. 45. Multi-Index Notation .. 45. Solution of the Pure Initial Value Problem .. 49. Nonhomogeneous Equation .. 50. Nonhomogeneous Equation with Nonhomogeneous Initial Condi- tions.

4 50. The Fundamental Solution .. 50. 10 Schro dinger Equation 52. 11 Problems : Quasilinear equations 54. 12 Problems : Shocks 75. 13 Problems : General Nonlinear equations 86. Two Spatial Dimensions .. 86. Three Spatial Dimensions .. 93. 14 Problems : First-Order Systems 102. 15 Problems : Gas Dynamics Systems 127. Perturbation .. 127. Stationary Solutions .. 128. Periodic Solutions .. 130. Energy Estimates .. 136. 16 Problems : Wave Equation 139. The Initial Value Problem .. 139. Initial/Boundary Value Problem .. 141. Similarity Solutions .. 155. Traveling Wave Solutions .. 156. Dispersion .. 171. Energy Methods .. 174. Wave Equation in 2D and 3D .. 187. 17 Problems : Laplace Equation 196. Green's Function and the Poisson Kernel .. 196. The Fundamental Solution.

5 205. Radial Variables .. 216. Weak Solutions .. 221. Uniqueness .. 223. Self-Adjoint Operators .. 232. Spherical Means .. 242. Harmonic Extensions, Subharmonic Functions .. 249. Partial Di erential equations Igor Yanovsky, 2005 5. 18 Problems : Heat Equation 255. Heat Equation with Lower Order Terms .. 263. Heat Equation Energy Estimates .. 264. 19 Contraction Mapping and Uniqueness - Wave 271. 20 Contraction Mapping and Uniqueness - Heat 273. 21 Problems : Maximum Principle - Laplace and Heat 279. Heat Equation - Maximum Principle and Uniqueness .. 279. Laplace Equation - Maximum Principle .. 281. 22 Problems : Separation of Variables - Laplace Equation 282. 23 Problems : Separation of Variables - Poisson Equation 302. 24 Problems : Separation of Variables - Wave Equation 305.

6 25 Problems : Separation of Variables - Heat Equation 309. 26 Problems : Eigenvalues of the Laplacian - Laplace 323. 27 Problems : Eigenvalues of the Laplacian - Poisson 333. 28 Problems : Eigenvalues of the Laplacian - Wave 338. 29 Problems : Eigenvalues of the Laplacian - Heat 346. Heat Equation with Periodic Boundary Conditions in 2D. (with extra terms) .. 360. 30 Problems : Fourier Transform 365. 31 Laplace Transform 385. 32 Linear Functional Analysis 393. Norms .. 393. Banach and Hilbert Spaces .. 393. Cauchy-Schwarz Inequality .. 393. Ho lder Inequality .. 393. Minkowski Inequality .. 394. Sobolev Spaces .. 394. Partial Di erential equations Igor Yanovsky, 2005 6. 1 Trigonometric Identities . L. n x m x 0 n = m cos cos dx =. cos(a + b) = cos a cos b sin a sin b L L L L n=m L.

7 Cos(a b) = cos a cos b + sin a sin b n x m x 0 n = m sin(a + b) = sin a cos b + cos a sin b sin sin dx =. L L L L n=m sin(a b) = sin a cos b cos a sin b L. n x m x sin cos dx = 0. L L L. cos(a + b) + cos(a b). cos a cos b =. 2 . sin(a + b) + sin(a b) . sin a cos b =. L. n x m x 0 n = m 2 cos cos dx = L. 0 L L 2 n=m cos(a b) cos(a + b) . sin a sin b = L. 0 n = m 2 n x m x sin sin dx = L. 0 L L 2 n=m cos 2t = cos2 t sin2 t sin 2t = 2 sin t cos t . L. 0 n = m 1 1 + cos t einx eimx dx =. cos2 t = 0 L n=m 2 2 . 1 1 cos t . sin2 t =. L. 0 n = 0. 2 2 einx dx =. 0 L n=0. 1 + tan2 t = sec2 t . x sin x cos x 2. cot t + 1 = csc t 2 sin2 x dx =.. 2 2.. x sin x cos x eix + e ix cos2 x dx = +. cos x = 2 2. 2 . eix e ix tan2 x dx = tan x x sin x =.

8 2i . cos2 x sin x cos x dx = . 2. ex + e x cosh x =. 2. ex e x ln(xy) = ln(x) + ln(y). sinh x = x 2 ln = ln(x) ln(y). y d ln xr = r lnx cosh x = sinh(x). dx d . sinh x = cosh(x). dx ln x dx = x ln x x . cosh2 x sinh2 x = 1 x2 x2. x ln x dx = ln x . 2 4.. du 1 u = tan 1 + C . a2. + u2 a a 2 . e z dz = . du u = sin 1 + C R. a2 u2 a z2 . e 2 dz = 2 . R. Partial Di erential equations Igor Yanovsky, 2005 7.. a b 1 1 d b A= , A =. c d det(A) c a Partial Di erential equations Igor Yanovsky, 2005 8. 2 Simple Eigenvalue Problem X + X = 0. Boundary conditions Eigenvalues n Eigenfunctions Xn n 2. X(0) = X(L) = 0 sin n . L x n = 1, 2, .. L1 2. (n 2 ) (n 1 ) . X(0) = X (L) = 0 L sin L2 x n = 1, 2, .. (n 12 ) 2 (n 1 ) . X (0) = X(L) = 0 L cos L2 x n = 1, 2.

9 N 2. X (0) = X (L) = 0 cos n L x n = 0, 1, 2, .. 2n . L . 2 2n . X(0) = X(L), X (0) = X (L) L sin L x n = 1, 2, .. cos 2n L x n = 0, 1, 2, .. n 2. X( L) = X(L), X ( L) = X (L) L. n . sin L x n = 1, 2, .. cos n L x n = 0, 1, 2, .. X X = 0. Boundary conditions Eigenvalues n Eigenfunctions Xn n 4. X(0) = X(L) = 0, X (0) = X (L) = 0 sin n . L x n = 1, 2, .. n . L . 4. X (0) = X (L) = 0, X (0) = X (L) = 0 L cos n . L x n = 0, 1, 2, .. Partial Di erential equations Igor Yanovsky, 2005 9. 3 Separation of Variables: 4 Eigenvalues of the Lapla- Quick Guide cian: Quick Guide Laplace Equation: u = 0. Laplace Equation: uxx + uyy + u = 0. X (x) Y (y). X Y . = = . + + = 0. ( = 2 + 2 ). X(x) Y (y) X Y. X + X = 0. X + 2 X = 0, Y + 2 Y = 0. X (t) Y ( ).

10 = = . X(t) Y ( ). Y ( ) + Y ( ) = 0. uxx + uyy + k2 u = 0. Wave Equation: utt uxx = 0. X Y . = + k 2 = c2 . X (x) T (t) X Y. = = . X(x) T (t) X + c2 X = 0, X + X = 0. Y + (k2 c2 )Y = 0. utt + 3ut + u = uxx . T T X . +3 +1 = = . T T X uxx + uyy + k2 u = 0. X + X = 0. Y X . utt uxx + u = 0. = + k 2 = c2 . Y X. T X Y + c2 Y = 0, +1 = = . T X X + (k2 c2 )X = 0. X + X = 0. utt + ut = c2 uxx + uxxt, ( > 0). X . = , X. 1 T T T X . + = 1 + . c2 T c2 T c2 T X. 4th Order: utt = k uxxxx. X 1 T . = = . X k T. X X = 0. Heat Equation: ut = kuxx . T X . = k = . T X.. X + X = 0. k 4th Order: ut = uxxxx . T X . = = . T X. X X = 0. Partial Di erential equations Igor Yanovsky, 2005 10. 5 First-Order equations Quasilinear equations Consider the Cauchy problem for the quasilinear equation in two variables a(x, y, u)ux + b(x, y, u)uy = c(x, y, u), with parameterized by (f (s), g(s), h(s)).


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