Partial Differential Equations: Graduate Level Problems and Solutions Igor Yanovsky 1 Partial Differential 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 Differential 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 5First-OrderEquations 10 5.1 Quasilinear Equations . 10 5.2 Weak Solutions for Quasilinear Equations . 12 5.2.1 Conservation Laws and Jump Conditions . 12 5.2.2 FansandRarefactionWaves..................... 12 5.3GeneralNonlinearEquations........................ 13 5.3.1 TwoSpatialDimensions....................... 13 5.3.2 ThreeSpatialDimensions...................... 13 6 Second-Order Equations 14 6.1ClassificationbyCharacteristics....................... 14 6.2CanonicalFormsandGeneralSolutions.................. 14 6.3Well-Posedness................................ 19 7WaveEquation 23 7.1TheInitialValueProblem.......................... 23 7.2WeakSolutions................................ 24 7.3 Initial/Boundary Value Problem . 24 7.4Duhamel’sPrinciple............................. 24 7.5TheNonhomogeneousEquation....................... 24 7.6HigherDimensions.............................. 26 7.6.1 Spherical Means . 26 7.6.2 ApplicationtotheCauchyProblem................ 26 7.6.3 Three-DimensionalWaveEquation................. 27 7.6.4 Two-DimensionalWaveEquation.................. 28 7.6.5 Huygen’sPrinciple.......................... 28 7.7EnergyMethods............................... 29 7.8ContractionMappingPrinciple....................... 30 8 Laplace Equation 31 8.1Green’sFormulas............................... 31 8.2PolarCoordinates.............................. 32 8.3 Polar Laplacian in R2 forRadialFunctions................ 32 8.4 Spherical Laplacian in R3 and Rn forRadialFunctions.......... 32 8.5 Cylindrical Laplacian in R3 forRadialFunctions............. 33 8.6MeanValueTheorem............................. 33 8.7MaximumPrinciple............................. 33 8.8 The Fundamental Solution . 34 8.9RepresentationTheorem........................... 37 8.10Green’sFunctionandthePoissonKernel.................. 42 Partial Differential Equations Igor Yanovsky, 2005 4 8.11PropertiesofHarmonicFunctions...................... 44 8.12EigenvaluesoftheLaplacian......................... 44 9HeatEquation 45 9.1ThePureInitialValueProblem....................... 45 9.1.1 FourierTransform.......................... 45 9.1.2 Multi-IndexNotation........................ 45 9.1.3 Solution of the Pure Initial Value Problem . 49 9.1.4 NonhomogeneousEquation..................... 50 9.1.5 Nonhomogeneous Equation with Nonhomogeneous Initial Condi- tions.................................. 50 9.1.6 The Fundamental Solution . 50 10 Schr¨odinger Equation 52 11 Problems: Quasilinear Equations 54 12 Problems: Shocks 75 13 Problems: General Nonlinear Equations 86 13.1TwoSpatialDimensions........................... 86 13.2ThreeSpatialDimensions.......................... 93 14 Problems: First-Order Systems 102 15 Problems: Gas Dynamics Systems 127 15.1Perturbation................................. 127 15.2StationarySolutions............................. 128 15.3PeriodicSolutions.............................. 130 15.4EnergyEstimates............................... 136 16 Problems: Wave Equation 139 16.1TheInitialValueProblem.......................... 139 16.2 Initial/Boundary Value Problem . 141 16.3SimilaritySolutions.............................. 155 16.4TravelingWaveSolutions.......................... 156 16.5Dispersion................................... 171 16.6EnergyMethods............................... 174 16.7WaveEquationin2Dand3D........................ 187 17 Problems: Laplace Equation 196 17.1Green’sFunctionandthePoissonKernel.................. 196 17.2 The Fundamental Solution . 205 17.3RadialVariables............................... 216 17.4WeakSolutions................................ 221 17.5Uniqueness.................................. 223 17.6Self-AdjointOperators............................ 232 17.7 Spherical Means . 242 17.8 Harmonic Extensions, Subharmonic Functions . 249 Partial Differential Equations Igor Yanovsky, 2005 5 18 Problems: Heat Equation 255 18.1HeatEquationwithLowerOrderTerms.................. 263 18.1.1HeatEquationEnergyEstimates.................. 264 19 Contraction Mapping and Uniqueness - Wave 271 20 Contraction Mapping and Uniqueness - Heat 273 21 Problems: Maximum Principle - Laplace and Heat 279 21.1HeatEquation-MaximumPrincipleandUniqueness........... 279 21.2LaplaceEquation-MaximumPrinciple.................. 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 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 29.1 Heat Equation with Periodic Boundary Conditions in 2D (withextraterms).............................. 360 30 Problems: Fourier Transform 365 31 Laplace Transform 385 32 Linear Functional Analysis 393 32.1Norms..................................... 393 32.2BanachandHilbertSpaces......................... 393 32.3Cauchy-SchwarzInequality......................... 393 32.4 H¨olderInequality............................... 393 32.5MinkowskiInequality............................. 394 32.6SobolevSpaces................................ 394 Partial Differential Equations Igor Yanovsky, 2005 6 1 Trigonometric Identities L nπx mπx 0 n = m cos cos dx = cos(a + b)=cosa cos b − sin a sin b − L L Ln= m L − cos(a b)=cosa cos b +sina sin b L nπx mπx 0 n = m sin(a + b)=sina cos b +cosa sin b sin sin dx = −L L L Ln= m sin(a − b)=sina cos b − cos a sin b L nπx mπx sin cos dx =0 −L L L cos(a + b)+cos(a − b) cos a cos b = 2 − sin(a + b)+sin(a b) L nπx mπx 0 n = m sin a cos b = cos cos dx = 2 L 0 L L n = m cos(a − b) − cos(a + b) 2 sin a sin b = L nπx mπx 0 n = m 2 sin sin dx = L L L 0 2 n = m cos 2t =cos2 t − sin2 t sin 2t =2sint cos t L inx imx 0 n = m 2 1 1+cost e e dx = cos t = Ln= m 2 2 0 − 2 1 1 cos t L sin t = inx 0 n =0 2 2 e dx = 0 Ln=0 1+tan2 t =sec2 t x sin x cos x cot2 t +1 = csc2 t sin2 xdx= − 2 2 2 x sin x cos x eix + e−ix cos xdx= + cos x = 2 2 2 2 eix − e−ix tan xdx=tanx − x sin x = 2i cos2 x sin x cos xdx= − 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 xdx = x ln x − x 2 − 2 x2 x2 cosh x sinh x =1 x ln xdx = ln x − 2 4 du 1 u = tan−1 + C 2 2 √ a + u a a −z2 du u e dz = π √ =sin−1 + C R 2 2 2 √ a − u a − z e 2 dz = 2π R Partial Differential Equations Igor Yanovsky, 2005 7 ab − 1 d −b A = ,A1 = cd det(A) −ca Partial Differential Equations Igor Yanovsky, 2005 8 2 Simple Eigenvalue Problem X + λX =0 Boundary conditions Eigenvalues λ Eigenfunctions X n n nπ 2 nπ X(0) = X(L)=0 L sin L xn=1, 2,... (n− 1 )π 2 (n− 1 )π X(0) = X (L)=0 2 sin 2 xn=1, 2,... L L (n− 1 )π 2 (n− 1 )π X (0) = X(L)=0 2 cos 2 xn=1, 2,... L L nπ 2 nπ X (0) = X (L)=0 L cos L xn=0, 1, 2,... 2nπ 2 2nπ X(0) = X(L),X(0) = X (L) L sin L xn=1, 2,... 2nπ cos L xn=0, 1, 2,... − − nπ 2 nπ X( L)=X(L),X( L)=X (L) L sin L xn=1, 2,... nπ cos L xn=0, 1, 2,... X − λX =0 Boundary conditions Eigenvalues λ Eigenfunctions X n n nπ 4 nπ X(0) = X(L)=0,X(0) = X (L)=0 L sin L xn=1, 2,... nπ 4 nπ X (0) = X (L)=0,X (0) = X (L)=0 L cos L xn=0, 1, 2,... Partial Differential 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 + λu =0. X (x) −Y (y) − X Y = = λ. + + λ =0. (λ = μ2 + ν2) X(x) Y (y) X Y X + λX =0. X + μ2X =0,Y + ν2Y =0. X (t) Y (θ) = − = λ. X(t) Y (θ) Y (θ)+λY (θ)=0. 2 uxx + uyy + k u =0. Wave Equation: utt − uxx =0. X Y 2 2 − = + k = c . X (x) T (t) X Y = = −λ. X(x) T (t) X + c2X =0, X + λX =0. Y +(k2 − c2)Y =0. utt +3ut + u = uxx. T T X − +3 +1 = = λ. 2 T T X uxx + uyy + k u =0. X + λX =0. Y X − − = + k2 = c2. utt uxx + u =0. Y X T X Y + c2Y =0, +1 = = −λ. T X X +(k2 − c2)X =0. X + λX =0. 2 utt + μut = c uxx + βuxxt, (β>0) X = −λ, X 1 T μ T β T X + = 1+ . c2 T c2 T c2 T X 4th Order: utt = −kuxxxx. X 1 T − = = −λ. X k T X − λX =0. Heat Equation: ut = kuxx. T X = k = −λ. T X λ X + X =0. k 4th Order: ut = −uxxxx. T X = − = −λ. T X X − λX =0. Partial Differential Equations Igor Yanovsky, 2005 10 5First-OrderEquations 5.1 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 Γ parameterized by (f(s),g(s),h(s)). The characteristic equations are dx dy dz = a(x, y, z), = b(x, y, z), = c(x, y, z), dt dt dt with initial conditions x(s, 0) = f(s),y(s, 0) = g(s),z(s, 0) = h(s). dx dy In a quasilinear case, the characteristic equations for dt and dt need not decouple from dz the dt equation; this means that we must take the z values into account even to find the projected characteristic curves in the xy-plane. In particular, this allows for the possibility that the projected characteristics may cross each other. The condition for solving for s and t in terms of x and y requires that the Jacobian matrix be nonsingular: xs ys J ≡ = xsyt − ysxt =0 . xt yt In particular, at t = 0 we obtain the condition f (s) · b(f(s),g(s),h(s)) − g(s) · a(f(s),g(s),h(s)) =0 .