Transcription

Partial Differential Equations: Graduate Level Problemsand SolutionsIgor Yanovsky1

Partial Differential EquationsIgor Yanovsky, 20052Disclaimer: This handbook is intended to assist graduate students with qualifyingexamination preparation. Please be aware, however, that the handbook might contain,and almost certainly contains, typos as well as incorrect or inaccurate solutions. I cannot be made responsible for any inaccuracies contained in this handbook.

Partial Differential EquationsIgor Yanovsky, 20053Contents1 Trigonometric Identities62 Simple Eigenvalue Problem83 Separation of Variables:Quick Guide94 Eigenvalues of the Laplacian:Quick Guide95 First-Order Equations5.1 Quasilinear Equations . . . . . . . . . . . . . . .5.2 Weak Solutions for Quasilinear Equations . . . .5.2.1 Conservation Laws and Jump Conditions5.2.2 Fans and Rarefaction Waves . . . . . . . .5.3 General Nonlinear Equations . . . . . . . . . . .5.3.1 Two Spatial Dimensions . . . . . . . . . .5.3.2 Three Spatial Dimensions . . . . . . . . .10101212121313136 Second-Order Equations146.1 Classification by Characteristics . . . . . . . . . . . . . . . . . . . . . . . 146.2 Canonical Forms and General Solutions . . . . . . . . . . . . . . . . . . 146.3 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Wave Equation7.1 The Initial Value Problem . . . . . . . . . .7.2 Weak Solutions . . . . . . . . . . . . . . . .7.3 Initial/Boundary Value Problem . . . . . .7.4 Duhamel’s Principle . . . . . . . . . . . . .7.5 The Nonhomogeneous Equation . . . . . . .7.6 Higher Dimensions . . . . . . . . . . . . . .7.6.1 Spherical Means . . . . . . . . . . .7.6.2 Application to the Cauchy Problem7.6.3 Three-Dimensional Wave Equation .7.6.4 Two-Dimensional Wave Equation . .7.6.5 Huygen’s Principle . . . . . . . . . .7.7 Energy Methods . . . . . . . . . . . . . . .7.8 Contraction Mapping Principle . . . . . . .8 Laplace Equation8.1 Green’s Formulas . . . . . . . . . . . . . . . . . . . . .8.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . .8.3 Polar Laplacian in R2 for Radial Functions . . . . . .8.4 Spherical Laplacian in R3 and Rn for Radial Functions8.5 Cylindrical Laplacian in R3 for Radial Functions . . .8.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . .8.7 Maximum Principle . . . . . . . . . . . . . . . . . . .8.8 The Fundamental Solution . . . . . . . . . . . . . . . .8.9 Representation Theorem . . . . . . . . . . . . . . . . .8.10 Green’s Function and the Poisson Kernel . . . . . . . 42

Partial Differential EquationsIgor Yanovsky, 200548.11 Properties of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . .8.12 Eigenvalues of the Laplacian . . . . . . . . . . . . . . . . . . . . . . . . .44449 Heat Equation459.1 The Pure Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . 459.1.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 459.1.2 Multi-Index Notation . . . . . . . . . . . . . . . . . . . . . . . . 459.1.3 Solution of the Pure Initial Value Problem . . . . . . . . . . . . . 499.1.4 Nonhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . 509.1.5 Nonhomogeneous Equation with Nonhomogeneous Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.1.6 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . 5010 Schrödinger Equation5211 Problems: Quasilinear Equations5412 Problems: Shocks7513 Problems: General Nonlinear Equations8613.1 Two Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8613.2 Three Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 9314 Problems: First-Order Systems10215 Problems: Gas Dynamics Systems15.1 Perturbation . . . . . . . . . . . .15.2 Stationary Solutions . . . . . . . .15.3 Periodic Solutions . . . . . . . . .15.4 Energy Estimates . . . . . . . . . .12712712813013616 Problems: Wave Equation16.1 The Initial Value Problem . . . .16.2 Initial/Boundary Value Problem16.3 Similarity Solutions . . . . . . . .16.4 Traveling Wave Solutions . . . .16.5 Dispersion . . . . . . . . . . . . .16.6 Energy Methods . . . . . . . . .16.7 Wave Equation in 2D and 3D . 249.17 Problems: Laplace Equation17.1 Green’s Function and the Poisson Kernel . . .17.2 The Fundamental Solution . . . . . . . . . . .17.3 Radial Variables . . . . . . . . . . . . . . . .17.4 Weak Solutions . . . . . . . . . . . . . . . . .17.5 Uniqueness . . . . . . . . . . . . . . . . . . .17.6 Self-Adjoint Operators . . . . . . . . . . . . .17.7 Spherical Means . . . . . . . . . . . . . . . .17.8 Harmonic Extensions, Subharmonic Functions.

Partial Differential EquationsIgor Yanovsky, 2005518 Problems: Heat Equation25518.1 Heat Equation with Lower Order Terms . . . . . . . . . . . . . . . . . . 26318.1.1 Heat Equation Energy Estimates . . . . . . . . . . . . . . . . . . 26419 Contraction Mapping and Uniqueness - Wave27120 Contraction Mapping and Uniqueness - Heat27321 Problems: Maximum Principle - Laplace and Heat27921.1 Heat Equation - Maximum Principle and Uniqueness . . . . . . . . . . . 27921.2 Laplace Equation - Maximum Principle . . . . . . . . . . . . . . . . . . 28122 Problems: Separation of Variables - Laplace Equation28223 Problems: Separation of Variables - Poisson Equation30224 Problems: Separation of Variables - Wave Equation30525 Problems: Separation of Variables - Heat Equation30926 Problems: Eigenvalues of the Laplacian - Laplace32327 Problems: Eigenvalues of the Laplacian - Poisson33328 Problems: Eigenvalues of the Laplacian - Wave33829 Problems: Eigenvalues of the Laplacian - Heat34629.1 Heat Equation with Periodic Boundary Conditions in 2D(with extra terms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36030 Problems: Fourier Transform36531 Laplace Transform38532 Linear Functional Analysis32.1 Norms . . . . . . . . . . . .32.2 Banach and Hilbert Spaces32.3 Cauchy-Schwarz Inequality32.4 Hölder Inequality . . . . . .32.5 Minkowski Inequality . . . .32.6 Sobolev Spaces . . . . . . .393393393393393394394

Partial Differential Equations1Trigonometric Identitiescos(a b) cos a cos b sin a sin bcos(a b) cos a cos b sin a sin bsin(a b) sin a cos b cos a sin bsin(a b) sin a cos b cos a sin bcos a cos b sin a cos b sin a sin b cos(a b) cos(a b)2sin(a b) sin(a b)2cos(a b) cos(a b)2cos 2t cos2 t sin2 tIgor Yanovsky, 2005 0mπxnπxcosdx cosLLL L L0mπxnπxsindx sinLLL L Lmπxnπxcosdx 0sinLL L L L0 nπxmπxcoscosdx LLL0mπxnπxsindx sinLLsin 2t 2 sin t cos t 1 cos t21 cos t21cos2 t 21sin2 t 2 Leinx eimx dx 0 0 cot t 1 csc tsin x eix e ix2eix e ix2icosh x sinh x ex e x2ex e x2dcosh x sinh(x)dxdsinh x cosh(x)dxcosh2 x sinh2 x 1 du u2 du a2 u2a21utan 1 Caau sin 1 Ca 0n mn mn mn mL2n mn mL2n mn m 00Ln mn m0Ln 0n 0x sin x cos x 22 x sin x cos xcos2 x dx 22 tan2 x dx tan x x cos2 xsin x cos x dx 22cos x Leinx dx 1 tan2 t sec2 t26sin2 x dx ln(xy) ln(x) ln(y)xln ln(x) ln(y)yln xr r lnx ln x dx x ln x x x ln x dx 2 z2 e z dz RRx2x2ln x 24e 2 dz π2π

Partial Differential Equations A a bc d , 1A1 det(A)Igor Yanovsky, 2005 d b c a 7

Partial Differential Equations2Igor Yanovsky, 20058Simple Eigenvalue ProblemX λX 0Boundary conditionsEigenvalues λn nπ 2 L1 2X(0) X(L) 0Eigenfunctions Xn(n 2 )πL (n 12 )π 2L nπ 2L 2nπ2LX(0) X (L) 0X (0) X(L) 0X (0) X (L) 0X(0) X(L), X (0) X (L)X( L) X(L), X ( L) X (L) nπ 2Lsin nπL x(n 1 )πsin L2 x(n 1 )πcos L2 xcos nπL x2nπsin L xcos 2nπL xnπsin L xcos nπL xn 1, 2, . . .n 1, 2, . . .n 1, 2, . . .n 0, 1, 2, . . .n 1, 2, . . .n 0, 1, 2, . . .n 1, 2, . . .n 0, 1, 2, . . .X λX 0Boundary conditionsX(0) X(L) 0, X (0) X (L) 0X (0) X (L) 0, X (0) X (L) 0Eigenvalues λn nπ 4L nπ4LEigenfunctions Xnsin nπL xcos nπL xn 1, 2, . . .n 0, 1, 2, . . .

Partial Differential Equations3Igor Yanovsky, 2005Separation of Variables:Quick Guide u 0.Laplace Equation:X (x)Eigenvalues of the Laplacian: Quick GuideLaplace Equation: (y)Y λ.X(x)Y (y)X λX 0. 4uxx uyy λu 0.X Y λ 0. (λ μ2 ν 2 )XYY ν 2 Y 0.X μ2 X 0,X (t)Y (θ) λ.X(t)Y (θ)Y (θ) λY (θ) 0.uxx uyy k2 u 0.utt uxx 0.Wave Equation:T (t)X (x) λ.X(x)T (t)X λX 0.Y X k 2 c2 .XYX c2 X 0, Y (k2 c2 )Y 0.utt 3ut u uxx .T X T 3 1 λ.TTXX λX 0.X T 1 λ.TXX λX 0.utt μut c2 uxx βuxxt,(β 0)X λ,Xμ T β T X 1 T 1 .c2 Tc2 Tc2 T X4th Order: utt k uxxxx.1 T X λ.Xk TX λX 0. Heat Equation:T T kut kuxx .X λ.XλX 0.kut uxxxx .X 4th Order:uxx uyy k2 u 0.X Y k 2 c2 .YXY c2 Y 0, utt uxx u 0.X T λ.TXX λX 0.9X (k2 c2 )X 0.

Partial Differential Equations5Igor Yanovsky, 200510First-Order Equations5.1Quasilinear EquationsConsider the Cauchy problem for the quasilinear equation in two variablesa(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 aredy b(x, y, z),dtdx a(x, y, z),dtdz c(x, y, z),dtwith initial conditionsx(s, 0) f (s),y(s, 0) g(s),z(s, 0) h(s).dyIn a quasilinear case, the characteristic equations for dxdt and dt need not decouple fromthe dzdt equation; this means that we must take the z values into account even to findthe projected characteristic curves in the xy-plane. In particular, this allows for thepossibility 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 Jacobianmatrix be nonsingular: xs ys xs yt ys xt 0.J xt ytIn particular, at t 0 we obtain the conditionf (s) · b(f (s), g(s), h(s)) g (s) · a(f (s), g(s), h(s)) 0.Burger’s Equation. Solve the Cauchy problem ut uux 0,u(x, 0) h(x).(5.1)The characteristic equations aredydzdx z, 1, 0,dtdtdtand Γ may be parametrized by (s, 0, h(s)).x h(s)t s, y t, z h(s).u(x, y) h(x uy)(5.2)The characteristic projection in the xt-plane1 passing through the point (s, 0) is thelinex h(s)t salong which u has the constant value u h(s). Two characteristics x h(s1 )t s1and x h(s2 )t s2 intersect at a point (x, t) witht 1s2 s1.h(s2 ) h(s1 )y and t are interchanged here

Partial Differential EquationsIgor Yanovsky, 200511From (5.2), we haveux h (s)(1 ux t) ux h (s)1 h (s)tHence for h (s) 0, ux becomes infinite at the positive timet 1.h (s)The smallest t for which this happens corresponds to the value s s0 at which h (s)has a minimum (i.e. h (s) has a maximum). At time T 1/h (s0 ) the solution uexperiences a “gradient catastrophe”.

Partial Differential Equations5.25.2.1Igor Yanovsky, 200512Weak Solutions for Quasilinear EquationsConservation Laws and Jump ConditionsConsider shocks for an equationut f (u)x 0,(5.3)where f is a smooth function of u. If we integrate (5.3) with respect to x for a x b,we obtain d bu(x, t) dx f (u(b, t)) f (u(a, t)) 0.(5.4)dt aThis is an example of a conservation law. Notice that (5.4) implies (5.3) if u is C 1 , but(5.4) makes sense for more general u.Consider a solution of (5.4) that, for fixed t, has a jump discontinuity at x ξ(t).We assume that u, ux , and ut are continuous up to ξ. Also, we assume that ξ(t) is C 1in t.Taking a ξ(t) b in (5.4), we obtain ξ bdu dx u dx f (u(b, t)) f (u(a, t))dt aξ ξ but (x, t) dx ut(x, t) dx ξ (t)ul (ξ(t), t) ξ (t)ur (ξ(t), t) aξ f (u(b, t)) f (u(a, t)) 0,where ul and ur denote the limiting values of u from the left and right sides of the shock.Letting a ξ(t) and b ξ(t), we get the Rankine-Hugoniot jump condition:ξ (t)(ul ur ) f (ur ) f (ul ) 0,ξ (t) 5.2.2f (ur ) f (ul ).ur ulFans and Rarefaction WavesFor Burgers’ equationut 1 2u2x 0,xxxx ũ .ttttFor a rarefaction fan emanating from (s, 0) on xt-plane, we have: x s ul ,t f (ul ) ul ,u(x, t) x sul x st ,t ur , x s ur ,t f (ur ) ur .we have f (u) u, f ũ

Partial Differential Equations5.3Igor Yanovsky, 200513General Nonlinear Equations5.3.1Two Spatial DimensionsWrite a general nonlinear equation F (x, y, u, ux, uy ) 0 asF (x, y, z, p, q) 0.Γ is parameterized by Γ : f (s) , g(s) , h(s) , φ(s) , ψ(s) x(s,0) y(s,0) z(s,0) p(s,0) q(s,0)We need to complete Γ to a strip. Find φ(s) and ψ(s), the initial conditions for p(s, t)and q(s, t), respectively: F (f (s), g(s), h(s), φ(s), ψ(s)) 0 h (s) φ(s)f (s) ψ(s)g (s)The characteristic equations aredydx Fp Fqdtdtdz pFp qFqdtdqdp Fx Fz p Fy Fz qdtdtWe need to have the Jacobian condition. That is, in order to solve the Cauchy problemin a neighborhood of Γ, the following con