Lecture Notes on Partial Differential Equations

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Partial Differential Equations Lecture Notes Erich Miersemann Department of Mathematics Leipzig University Version October, 20122Contents 1 Introduction 9 1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Equations from variational problems . . . . . . . . . . . . . . 15 1.2.1 Ordinary differential equations . . . . . . . . . . . . . 15 1.2.2 Partial differential equations . . . . . . . . . . . . . . 16 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Equations of first order 25 2.1 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Quasilinear equations . . . . . . . . . . . . . . . . . . . . . . 31 2.2.1 A linearization method . . . . . . . . . . . . . . . . . 32 2.2.2 Initial value problem of Cauchy . . . . . . . . . . . . . 33 2.3 Nonlinear equations in two variables . . . . . . . . . . . . . . 40 2.3.1 Initial value problem of Cauchy . . . . . . . . . . . . . 48 n 2.4 Nonlinear equations inR . . . . . . . . . . . . . . . . . . . . 51 2.5 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . 53 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Classification 63 3.1 Linear equations of second order . . . . . . . . . . . . . . . . 63 3.1.1 Normal form in two variables . . . . . . . . . . . . . . 69 3.2 Quasilinear equations of second order. . . . . . . . . . . . . . 73 3.2.1 Quasilinear elliptic equations . . . . . . . . . . . . . . 73 3.3 Systems of first order . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Systems of second order . . . . . . . . . . . . . . . . . . . . . 82 3.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5 Theorem of Cauchy-Kovalevskaya . . . . . . . . . . . . . . . . 84 3.5.1 Appendix: Real analytic functions . . . . . . . . . . . 90 34 CONTENTS 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Hyperbolic equations 107 4.1 One-dimensional wave equation . . . . . . . . . . . . . . . . . 107 4.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.1 Case n=3 . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2.2 Case n=2 . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3 Inhomogeneous equation . . . . . . . . . . . . . . . . . . . . . 117 4.4 A method of Riemann . . . . . . . . . . . . . . . . . . . . . . 120 4.5 Initial-boundary value problems . . . . . . . . . . . . . . . . . 125 4.5.1 Oscillation of a string . . . . . . . . . . . . . . . . . . 125 4.5.2 Oscillation of a membrane . . . . . . . . . . . . . . . . 128 4.5.3 Inhomogeneous wave equations . . . . . . . . . . . . . 131 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5 Fourier transform 141 5.1 Definition, properties . . . . . . . . . . . . . . . . . . . . . . . 141 5.1.1 Pseudodifferential operators . . . . . . . . . . . . . . . 146 5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6 Parabolic equations 151 6.1 Poisson’s formula . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.2 Inhomogeneous heat equation . . . . . . . . . . . . . . . . . . 155 6.3 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . 156 6.4 Initial-boundary value problem . . . . . . . . . . . . . . . . . 162 6.4.1 Fourier’s method . . . . . . . . . . . . . . . . . . . . . 162 6.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 164 6.5 Black-Scholes equation . . . . . . . . . . . . . . . . . . . . . . 164 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7 Elliptic equations of second order 175 7.1 Fundamental solution . . . . . . . . . . . . . . . . . . . . . . 175 7.2 Representation formula . . . . . . . . . . . . . . . . . . . . . 177 7.2.1 Conclusions from the representation formula . . . . . 179 7.3 Boundary value problems . . . . . . . . . . . . . . . . . . . . 181 7.3.1 Dirichlet problem . . . . . . . . . . . . . . . . . . . . . 181 7.3.2 Neumann problem . . . . . . . . . . . . . . . . . . . . 182 7.3.3 Mixed boundary value problem . . . . . . . . . . . . . 183 7.4 Green’s function for4 . . . . . . . . . . . . . . . . . . . . . . 183 7.4.1 Green’s function for a ball . . . . . . . . . . . . . . . . 186CONTENTS 5 7.4.2 Green’s function and conformal mapping . . . . . . . 190 7.5 Inhomogeneous equation . . . . . . . . . . . . . . . . . . . . . 190 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956 CONTENTSPreface These lecture notes are intented as a straightforward introduction to partial differential equations which can serve as a textbook for undergraduate and beginning graduate students. Foradditionalreadingwerecommendfollowingbooks: W.I.Smirnov21, I. G. Petrowski 17, P. R. Garabedian 8, W. A. Strauss 23, F. John 10, L. C. Evans 5 and R. Courant and D. Hilbert4 and D. Gilbarg and N. S. Trudinger 9. Some material of these lecture notes was taken from some of these books. 78 CONTENTSChapter 1 Introduction Ordinary and partial differential equations occur in many applications. An ordinary differential equation is a special case of a partial differential equa- tion but the behaviour of solutions is quite different in general. It is much more complicated in the case of partial differential equations caused by the fact that the functions for which we are looking at are functions of more than one independent variable. Equation 0 (n) F(x,y(x),y (x),...,y )=0 is an ordinary differential equation of n-th order for the unknown function y(x), where F is given. An important problem for ordinary differential equations is the initial value problem 0 y (x) = f(x,y(x)) y(x ) = y , 0 0 where f is a given real function of two variables x, y and x , y are given 0 0 real numbers. Picard-Lindelof ¨ Theorem. Suppose (i) f(x,y) is continuous in a rectangle 2 Q=(x,y)∈R : x−x a, y−y b. 0 0 (ii) There is a constant K such that f(x,y)≤K for all (x,y)∈Q. (ii) Lipschitz condition: There is a constant L such that f(x,y )−f(x,y )≤Ly −y 2 1 2 1 910 CHAPTER 1. INTRODUCTION y y 0 x x 0 Figure 1.1: Initial value problem for all (x,y ),(x,y ). 1 2 1 Then there exists a unique solutiony∈C (x −α,x +α) of the above initial 0 0 value problem, where α =min(b/K,a). The linear ordinary differential equation (n) (n−1) 0 y +a (x)y +...a (x)y +a (x)y =0, n−1 1 0 where a are continuous functions, has exactly n linearly independent solu- j 2 tions. Incontrasttothispropertythepartialdifferentialu +u =0inR xx yy 2 2 hasinfinitelymanylinearlyindependentsolutionsinthelinearspaceC (R ). The ordinary differential equation of second order 00 0 y (x)=f(x,y(x),y (x)) has in general a family of solutions with two free parameters. Thus, it is naturally to consider the associated initial value problem 00 0 y (x) = f(x,y(x),y (x)) 0 y(x ) = y , y (x )=y , 0 0 0 1 where y and y are given, or to consider the boundary value problem 0 1 00 0 y (x) = f(x,y(x),y (x)) y(x ) = y , y(x )=y . 0 0 1 1 Initial and boundary value problems play an important role also in the theory of partial differential equations. A partial differential equation for1.1. EXAMPLES 11 y y 1 y 0 x x x 0 1 Figure 1.2: Boundary value problem the unknown function u(x,y) is for example F(x,y,u,u ,u ,u ,u ,u )=0, x y xx xy yy where the function F is given. This equation is of second order. An equation is said to be of n-th order if the highest derivative which occurs is of order n. An equation is said to be linear if the unknown function and its deriva- tives are linear in F. For example, a(x,y)u +b(x,y)u +c(x,y)u=f(x,y), x y where the functions a, b, c and f are given, is a linear equation of first order. An equation is said to be quasilinear if it is linear in the highest deriva- tives. For example, a(x,y,u,u ,u )u +b(x,y,u,u ,u )u +c(x,y,u,u ,u )u =0 x y xx x y xy x y yy is a quasilinear equation of second order. 1.1 Examples 1. u =0, where u=u(x,y). All functions u=w(x) are solutions. y 2. u = u , where u = u(x,y). A change of coordinates transforms this x y equation into an equation of the first example. Set ξ = x+y, η = x−y, then µ ¶ ξ+η ξ−η u(x,y)=u , =:v(ξ,η). 2 212 CHAPTER 1. INTRODUCTION 1 Assume u∈C , then 1 v = (u −u ). η x y 2 If u = u , then v = 0 and vice versa, thus v = w(ξ) are solutions for x y η 1 arbitraryC -functionsw(ξ). Consequently,wehavealargeclassofsolutions of the original partial differential equation: u = w(x+y) with an arbitrary 1 C -function w. 1 3. A necessary and sufficient condition such that for given C -functions M, N the integral Z P 1 M(x,y)dx+N(x,y)dy P 0 isindependentofthecurvewhichconnectsthepointsP withP inasimply 0 1 2 connected domain Ω ⊂R is the partial differential equation (condition of integrability) M =N y x in Ω. Ω y P 1 P 0 x Figure 1.3: Independence of the path This is one equation for two functions. A large class of solutions is given 2 by M = Φ , N = Φ , where Φ(x,y) is an arbitrary C -function. It follows x y 1 from Gauss theorem that these are all C -solutions of the above differential equation. 4. Method of an integrating multiplier for an ordinary differential equation. Consider the ordinary differential equation M(x,y)dx+N(x,y)dy =01.1. EXAMPLES 13 1 1 for given C -functions M, N. Then we seek a C -function μ(x,y) such that μMdx+μNdy is a total differential, i. e., that (μM) =(μN) is satisfied. y x This is a linear partial differential equation of first order for μ: Mμ −Nμ =μ(N −M ). y x x y 1 5. TwoC -functionsu(x,y)andv(x,y)aresaidtobefunctionallydependent if µ ¶ u u x y det =0, v v x y whichisalinearpartialdifferentialequationoffirstorderforuifv isagiven 1 C -function. A large class of solutions is given by u=H(v(x,y)), 1 where H is an arbitrary C -function. 6. Cauchy-Riemannequations. Setf(z)=u(x,y)+iv(x,y),wherez =x+iy 1 2 andu, varegivenC (Ω)-functions. HereisΩadomaininR . Ifthefunction f(z)isdifferentiablewithrespecttothecomplexvariablez thenu, v satisfy the Cauchy-Riemann equations u =v , u =−v . x y y x It is known from the theory of functions of one complex variable that the real part u and the imaginary part v of a differentiable function f(z) are solutions of the Laplace equation 4u=0, 4v =0, where4u=u +u . xx yy 7. The Newton potential 1 u=p 2 2 2 x +y +z 3 is a solution of the Laplace equation inR \(0,0,0), i. e., of u +u +u =0. xx yy zz14 CHAPTER 1. INTRODUCTION 8. Heat equation. Let u(x,t) be the temperature of a point x∈ Ω at time 3 t, where Ω ⊂R is a domain. Then u(x,t) satisfies in Ω×0,∞) the heat equation u =k4u, t where4u=u +u +u andk isapositiveconstant. Thecondition x x x x x x 1 1 2 2 3 3 u(x,0)=u (x), x∈Ω, 0 where u (x) is given, is an initial condition associated to the above heat 0 equation. The condition u(x,t)=h(x,t), x∈∂Ω, t≥0, where h(x,t) is given is a boundary condition for the heat equation. Ifh(x,t)=g(x), that is, h is independent oft, then one expects that the solution u(x,t) tends to a function v(x) if t → ∞. Moreover, it turns out thatv isthesolutionoftheboundary value problemfortheLaplaceequation 4v = 0 in Ω v = g(x) on ∂Ω. 9. Wave equation. The wave equation y u(x,t ) 1 u(x,t ) 2 x l Figure 1.4: Oscillating string 2 u =c 4u, tt where u = u(x,t), c is a positive constant, describes oscillations of mem- branesorofthreedimensionaldomains,forexample. Intheone-dimensional case 2 u =c u tt xx describes oscillations of a string.1.2. EQUATIONS FROM VARIATIONAL PROBLEMS 15 Associated initial conditions are u(x,0)=u (x), u (x,0)=u (x), 0 t 1 where u , u are given functions. Thus the initial position and the initial 0 1 velocity are prescribed. If the string is finite one describes additionally boundary conditions, for example u(0,t)=0, u(l,t)=0 for all t≥0. 1.2 Equations from variational problems A large class of ordinary and partial differential equations arise from varia- tional problems. 1.2.1 Ordinary differential equations Set Z b 0 E(v)= f(x,v(x),v (x)) dx a and for given u , u ∈R a b 2 V =v∈C a,b: v(a)=u , v(b)=u , a b where −∞ a b ∞ and f is sufficiently regular. One of the basic problems in the calculus of variation is (P) min E(v). v∈V Euler equation. Let u∈V be a solution of (P), then d 0 0 f 0(x,u(x),u(x))=f (x,u(x),u(x)) u u dx in (a,b). 2 Proof. Exercise. Hints: For fixed φ ∈ C a,b with φ(a) = φ(b) = 0 and 0 real ², ² ² , set g(²) = E(u+²φ). Since g(0)≤ g(²) it follows g (0) = 0. 0 0 Integration by parts in the formula for g (0) and the following basic lemma in the calculus of variations imply Euler’s equation.16 CHAPTER 1. INTRODUCTION y y 1 y 0 a x b Figure 1.5: Admissible variations Basic lemma in the calculus of variations. Let h∈C(a,b) and Z b h(x)φ(x) dx=0 a 1 for all φ∈C (a,b). Then h(x)≡0 on (a,b). 0 Proof. Assume h(x ) 0 for an x ∈ (a,b), then there is a δ 0 such that 0 0 (x −δ,x +δ)⊂(a,b) and h(x)≥h(x )/2 on (x −δ,x +δ). Set 0 0 0 0 0 ½ ¡ ¢ 2 2 2 δ −x−x if x∈(x −δ,x +δ) 0 0 0 φ(x)= . 0 if x∈(a,b)\x −δ,x +δ 0 0 1 Thus φ∈C (a,b) and 0 Z Z b x +δ 0 h(x ) 0 h(x)φ(x) dx≥ φ(x) dx0, 2 a x −δ 0 which is a contradiction to the assumption of the lemma. 2 1.2.2 Partial differential equations Thesameprocedureasaboveappliedtothefollowingmultipleintegralleads to a second-order quasilinear partial differential equation. Set Z E(v)= F(x,v,∇v) dx, Ω1.2. EQUATIONS FROM VARIATIONAL PROBLEMS 17 n where Ω ⊂R is a domain, x = (x ,...,x ), v = v(x) : Ω 7→R, and 1 n ∇v = (v ,...,v ). Assume that the function F is sufficiently regular in x x n 1 its arguments. For a given function h, defined on ∂Ω, set 2 V =v∈C (Ω): v =h on ∂Ω. Euler equation. Let u∈V be a solution of (P), then n X ∂ F −F =0 u u x i ∂x i i=1 in Ω. Proof. Exercise. Hint: Extendtheabovefundamentallemmaofthecalculus of variations to the case of multiple integrals. The interval (x −δ,x +δ) in 0 0 the definition of φ must be replaced by a ball with center at x and radius 0 δ. Example: Dirichlet integral In two dimensions the Dirichlet integral is given by Z ¡ ¢ 2 2 D(v)= v +v dxdy x y Ω and the associated Euler equation is the Laplace equation4u=0 in Ω. Thus, there is natural relationship between the boundary value problem 4u=0 in Ω, u=h on ∂Ω and the variational problem min D(v). v∈V But these problems are not equivalent in general. It can happen that the boundary value problem has a solution but the variational problem has no solution, see for an example Courant and Hilbert 4, Vol. 1, p. 155, where h is a continuous function and the associated solution u of the boundary value problem has no finite Dirichlet integral. Theproblemsareequivalent,providedthegivenboundaryvaluefunction 1/2 h is in the class H (∂Ω), see Lions and Magenes 14.18 CHAPTER 1. INTRODUCTION Example: Minimal surface equation The non-parametric minimal surface problem in two dimensions is to find a minimizer u=u(x ,x ) of the problem 1 2 Z q 2 2 min 1+v +v dx, x x 1 2 v∈V Ω where for a given function h defined on the boundary of the domain Ω 1 V =v∈C (Ω): v =h on ∂Ω. S Ω Figure 1.6: Comparison surface 2 Supposethattheminimizersatisfiestheregularityassumptionu∈C (Ω), then u is a solution of the minimal surface equation (Euler equation) in Ω Ã Ã ∂ u ∂ u x x 1 2 p + p =0. (1.1) 2 2 ∂x ∂x 1 1+∇u 2 1+∇u 2 In fact, the additional assumption u∈ C (Ω) is superfluous since it follows from regularity considerations for quasilinear elliptic equations of second order, see for example Gilbarg and Trudinger 9. 2 Let Ω =R . Each linear function is a solution of the minimal surface equation (1.1). It was shown by Bernstein 2 that there are no other solu- tions of the minimal surface quation. This is true also for higher dimensions1.2. EQUATIONS FROM VARIATIONAL PROBLEMS 19 n≤7, see Simons 19. Ifn≥8, then there exists also other solutions which define cones, see Bombieri, De Giorgi and Giusti 3. The linearized minimal surface equation over u≡0 is the Laplace equa- 2 tion 4u = 0. InR linear functions are solutions but also many other functions in contrast to the minimal surface equation. This striking differ- ence is caused by the strong nonlinearity of the minimal surface equation. More general minimal surfaces are described by using parametric rep- 1 resentations. An example is shown in Figure 1.7 . See 18, pp. 62, for example, for rotationally symmetric minimal surfaces. Figure 1.7: Rotationally symmetric minimal surface Neumann type boundary value problems 1 Set V =C (Ω) and Z Z E(v)= F(x,v,∇v) dx− g(x,v) ds, Ω ∂Ω n where F and g are given sufficiently regular functions and Ω ⊂R is a bounded and sufficiently regular domain. Assume u is a minimizer of E(v) in V, that is u∈V : E(u)≤E(v) for all v∈V, 1 An experiment from Beutelspacher’s Mathematikum, Wissenschaftsjahr 2008, Leipzig20 CHAPTER 1. INTRODUCTION then Z n X ¡ ¢ F (x,u,∇u)φ + F (x,u,∇u)φ dx u x u x i i Ω i=1 Z − g (x,u)φ ds=0 u ∂Ω 1 2 for all φ ∈ C (Ω). Assume additionally u ∈ C (Ω), then u is a solution of the Neumann type boundary value problem n X ∂ F −F = 0 in Ω u u x i ∂x i i=1 n X F ν −g = 0 on ∂Ω, u i u x i i=1 whereν =(ν ,...,ν ) is the exteriorunit normal at the boundary∂Ω. This 1 n follows after integration by parts from the basic lemma of the calculus of variations. Example: Laplace equation Set Z Z 1 2 E(v)= ∇v dx− h(x)v ds, 2 Ω ∂Ω then the associated boundary value problem is 4u = 0 in Ω ∂u = h on ∂Ω. ∂ν Example: Capillary equation 2 Let Ω⊂R and set Z Z Z p κ 2 2 E(v)= 1+∇v dx+ v dx−cosγ v ds. 2 Ω Ω ∂Ω Here κ is a positive constant (capillarity constant) and γ is the (constant) boundary contact angle, i. e., the angle between the container wall and

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