Partial Differential Equations: Graduate Level Problems And .
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 condition must be satisfied:f (s) · Fq [f, g, h, φ, ψ](s) g (s) · Fp [f, g, h, φ, ψ](s) 0.5.3.2Three Spatial DimensionsWrite a general nonlinear equation F (x1 , x2 , x3 , u, ux1 , ux2 , ux3 ) 0 asF (x1 , x2 , x3, z, p1 , p2, p3 ) 0.Γ is parameterized by Γ : f1 (s1 , s2 ), f2 (s1 , s2 ), f3 (s1 , s2 ), h(s1 , s2 ), φ1 (s1 , s2 ), φ2 (s1 , s2 ), φ3 (s1 , s2 ) x1 (s1 ,s2 ,0) x2 (s1 ,s2 ,0) x3 (s1 ,s2 ,0) z(s1 ,s2 ,0)p1 (s1 ,s2 ,0)p2 (s1 ,s2 ,0)p3 (s1 ,s2 ,0)We need to complete Γ to a strip. Find φ1 (s1 , s2 ), φ2 (s1 , s2 ), and φ3 (s1 , s2 ), the initialconditions for p1 (s1 , s2 , t), p2 (s1 , s2 , t), and p3 (s1 , s2 , t), respectively: F f1 (s1 , s2 ), f2 (s1 , s2 ), f3 (s1 , s2 ), h(s1 , s2 ), φ1 , φ2 , φ3 0 f1 f2 f3 h φ1 φ2 φ3 s1 s1 s1 s1 h f1 f2 f3 φ1 φ2 φ3 s2 s2 s2 s2The characteristic equations aredx2dx3dx1 Fp1 Fp2 Fp3dtdtdtdz p1 Fp1 p2 Fp2 p3 Fp3dtdp1dp2dp3 Fx1 p1 Fz Fx2 p2 Fz Fx3 p3 Fzdtdtdt
Partial Differential Equations66.1Igor Yanovsky, 200514Second-Order EquationsClassification by CharacteristicsConsider the second-order equation in which the derivatives of second-order all occurlinearly, with coefficients only depending on the independent variables:a(x, y)uxx b(x, y)uxy c(x, y)uyy d(x, y, u, ux, uy ).(6.1)The characteristic equation is b b2 4acdy .dx2a b2 4ac 0 two characteristics, and (6.1) is called hyperbolic; b2 4ac 0 one characteristic, and (6.1) is called parabolic; b2 4ac 0 no characteristics, and (6.1) is called elliptic.These definitions are all taken at a point x0 R2 ; unless a, b, and c are all constant,the type may change with the point x0 .6.2Canonical Forms and General Solutions➀ uxx uyy 0 is hyperbolic (one-dimensional wave equation).➁ uxx uy 0 is parabolic (one-dimensional heat equation).➂ uxx uyy 0 is elliptic (two-dimensional Laplace equation).By the introduction of new coordinates μ and η in place of x and y, the equation(6.1) may be transformed so that its principal part takes the form ➀, ➁, or ➂.If (6.1) is hyperbolic, parabolic, or elliptic, there exists a change of variables μ(x, y) andη(x, y) under which (6.1) becomes, respectively, η, u, uμ, uη )uμη d(μ, η, u, uμ, uη ),uμμ d(μ, ȳ, u, ux̄, uȳ ),ux̄x̄ uȳ ȳ d(x̄, η, u, uμ, uη ).uμμ uηη d(μ,Example 1. Reduce to canonical form and find the general solution:uxx 5uxy 6uyy 0.Proof. a 1, b 5, c 6 b2 4ac 1 0characteristics.The characteristics are found by solving 5 13dy dx22to find y 3x c1 and y 2x c2 .(6.2) hyperbolic two
Partial Differential EquationsIgor Yanovsky, 200515μ(x, y) 3x y, η(x, y) 2x y.Letμx 3,ηx 2,μy 1,ηy 1.u u(μ(x, y), η(x, y));ux uμ μx uη ηx 3uμ 2uη ,uy uμ μy uη ηy uμ uη ,uxx (3uμ 2uη )x 3(uμμ μx uμη ηx) 2(uημ μx uηη ηx) 9uμμ 12uμη 4uηη ,uxy (3uμ 2uη )y 3(uμμ μy uμη ηy ) 2(uημμy uηη ηy ) 3uμμ 5uμη 2uηη ,uyy (uμ uη )y (uμμ μy uμη ηy uημ μy uηη ηy ) uμμ 2uμη uηη .Inserting these expressions into (6.2) and simplifying, we obtainuμη 0,which is the Canonical form,uμ f (μ),u F (μ) G(η),u(x, y) F (3x y) G(2x y),General solution.Example 2. Reduce to canonical form and find the general solution:y 2 uxx 2yuxy uyy ux 6y.(6.3)Proof. a y 2 , b 2y, c 1 b2 4ac 0 parabolic one characteristic.The characteristics are found by solving 2y1dy 2dx2yyy2 c x.to find 22Let μ y2 x. We must choose a second constant function η(x, y) so that η is notparallel to μ. Choose η(x, y) y.μx 1,ηx 0,μy y,ηy 1.u u(μ(x, y), η(x, y));ux uμ μx uη ηx uμ ,uy uμ μy uη ηy yuμ uη ,uxx (uμ )x uμμ μx uμη ηx uμμ ,uxy (uμ )y uμμ μy uμη ηy yuμμ uμη ,uyy (yuμ uη )y uμ y(uμμ μy uμη ηy ) (uημ μy uηη ηy ) uμ y 2 uμμ 2yuμη uηη .
Partial Differential EquationsIgor Yanovsky, 2005Inserting these expressions into (6.3) and simplifying, we obtainuηη 6y,uηη 6η,which is the Canonical form,2uη 3η f (μ),u η 3 ηf (μ) g(μ),y2y2 x g x ,u(x, y) y 3 y · f22General solution.16
Partial Differential EquationsIgor Yanovsky, 200517Problem (F’03, #4). Find the characteristics of the partial differential equationxuxx (x y)uxy yuyy 0,x 0, y 0,(6.4)and then show that it can be transformed into the canonical form(ξ 2 4η)uξη ξuη 0whence ξ and η are suitably chosen canonical coordinates. Use this to obtain the generalsolution in the form ηg(η ) dη u(ξ, η) f (ξ) 1(ξ 2 4η ) 2where f and g are arbitrary functions of ξ and η.Proof. a x, b x y, c y b2 4ac (x y)2 4xy 0 for x 0,y 0 hyperbolic two characteristics.➀ The characteristics are found by solving 2xdyb b2 4acx y (x y)2 4xyx y (x y)2x 1 2ydx2a2x2x xy 2x dxdy ,yxln y ln x 1 c 2 ,c2y .xy x c1 ,➁ Let μ x y and η xyμx 1,ηx y,μy 1,ηy x.u u(μ(x, y), η(x, y));ux uμ μx uη ηx uμ yuη ,uy uμ μy uη ηy uμ xuη ,uxx (uμ yuη )x uμμ μx uμη ηx y(uημμx uηη ηx ) uμμ 2yuμη y 2 uηη ,uxy (uμ yuη )y uμμ μy uμη ηy uη y(uημμy uηη ηy ) uμμ xuμη uη yuημ xyuηη ,uyy ( uμ xuη )y uμμ μy uμη ηy x(uημ μy uηη ηy ) uμμ 2xuμη x2 uηη ,Inserting these expressions into (6.4), we obtainx(uμμ 2yuμη y 2 uηη ) (x y)( uμμ xuμη uη yuημ xyuηη ) y(uμμ 2xuμη x2 uηη ) 0,(x2 2xy y 2 )uμη (x y)uη 0, (x y)2 4xy uμη (x y)uη 0,(μ2 4η)uμη μuη 0,which is the Canonical form.
Partial Differential EquationsIgor Yanovsky, 2005➂ We need to integrate twice to get the general solution:(μ2 4η)(uη)μ μuη 0, μ(uη )μdμ,dμ uημ2 4η1ln uη ln (μ2 4η) g̃(η),21ln uη ln (μ2 4η) 2 g̃(η),g(η)uη 1,(μ2 4η) 2 g(η) dηu(μ, η) f (μ) 1 ,(μ2 4η) 2General solution.18
Partial Differential Equations6.3Igor Yanovsky,
Partial Differential Equations Igor Yanovsky, 2005 12 5.2 Weak Solutions for Quasilinear Equations 5.2.1 Conservation Laws and Jump Conditions Consider shocks for an equation u t f(u) x 0, (5.3) where f is a smooth function ofu. If we integrate (5.3) with respect to x for a x b,
1.2. Rewrite the linear 2 2 system of di erential equations (x0 y y0 3x y 4et as a linear second-order di erential equation. 1.3. Using the change of variables x u 2v; y 3u 4v show that the linear 2 2 system of di erential equations 8 : du dt 5u 8v dv dt u 2v can be rewritten as a linear second-order di erential equation. 5
69 of this semi-mechanistic approach, which we denote as Universal Di erential 70 Equations (UDEs) for universal approximators in di erential equations, are dif- 71 ferential equation models where part of the di erential equation contains an 72 embedded UA, such as a neural network, Chebyshev expansion, or a random 73 forest. 74 As a motivating example, the universal ordinary di erential .
Numerical Recipes in Fortran (2nd Ed.), W. H. Press et al. Introduction to Partial Di erential Equations with Matlab, J. M. Cooper. Numerical solution of partial di erential equations, K. W. Morton and D. F. Mayers. Spectral methods in Matlab, L. N. Trefethen 8
Shearer and Levy: Partial Di erential Equations { Solutions Hunter Stu ebeam 2017 Forward The following is a collection of my solutions for Michael Shearer and Rachel Levy’s text Partial Di erential Equations: An Introduction to Theory and Applications. These solutions were worked out over the summer of 2017, and will almost certainly contain .
General theory of di erential equations of rst order 45 4.1. Slope elds (or direction elds) 45 4.1.1. Autonomous rst order di erential equations. 49 4.2. Existence and uniqueness of solutions for initial value problems 53 4.3. The method of successive approximations 59 4.4. Numerical methods for Di erential equations 62
Unit #15 - Di erential Equations Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Basic Di erential Equations 1.Show that y x sin(x) ˇsatis es the initial value problem dy dx 1 cosx To verify anything is a solution to an equation, we sub it in and verify that the left and right hand sides are equal after
Ordinary Di erential Equations by Morris Tenenbaum and Harry Pollard. Elementary Di erential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima. Introduction to Ordinary Di erential Equations by Shepley L. Ross. Di
First Order Linear Di erential Equations A First Order Linear Di erential Equation is a rst order di erential equation which can be put in the form dy dx P(x)y Q(x) where P(x);Q(x) are continuous functions of x on a given interval. The above form of the equation is called the Standard Form of the equation.
