Math 412-501Theory of Partial Differential EquationsLecture 4-2:More on the Dirac delta function.Green’s functions for ODEs.Dirac delta functionδ(x) is a function on R such that• δ(x) = 0 for all x 6= 0,• δ(0) = ∞,•R∞−∞δ(x) dx = 1.For any continuous function f and any x0∈ R,Z∞−∞f (x)δ(x −x0) dx = f (x0).δ(x) is a generalized function (or distribution).That is, δ is a linear functional on a space of testfunctions f such that δ[f ] = f (0).DistributionsClass of test functions S: consists of infinitelysmooth, rapidly decaying functions on R.To be precise, f ∈ S if sup |xkf(m)(x)| < ∞ for anyintegers k, m ≥ 0.Convergence in S: we say that fn→ f in S asn → ∞ if sup |x|k|f(m)n(x) − f(m)(x)| → 0 asn → ∞ for any integers k, m ≥ 0.Class of distributions S′: consists of continuouslinear fu nctionals on S. That is, a linear mapℓ : S → R belongs to S′if ℓ[fn] → ℓ[f ] wheneverfn→ f in S.Convergence in S′: we say that ℓn→ ℓ in S′ifℓn[f ] → ℓ[f ] for any f ∈ S.Examples. (i) Delta function δ[f ] = f (0).(ii) Shifted δ-function δx0(x) = δ(x −x0), δx0[f ] = f (x0).(iii) Let g be a bounded, locally integrable functionon R. Thenf 7→Z∞−∞f (x)g(x) dxis a distribution, which is identified with g.Delta sequence is a sequence of functionsg1, g2, . . . such that gn→ δ in S′as n → ∞. Thatis, for any f ∈ Slimn→∞Z∞−∞f (x)gn(x) dx = f (0).Delta family is a family of fun ctions hε,0 < ε ≤ ε0, such that limε→0hε= δ in S′.hε(x) =1√πεe−x2/ε, ε > 0.How to differentiate a distributionIf g is a piecewise differentiable bounded functionon R thenZ∞−∞f (x)g′(x) dx = −Z∞−∞f′(x)g(x) dxfor any test fun ction f ∈ S.Let γ be a distribution. Then S ∋ f 7→ −γ[f′] isalso a distribution, which is denoted γ′and calledthe derivative of γ (in S′).In the case when γ is a differentiable function, thederivative in S′coincides with the usual derivative.Heaviside step functionH(x) =(0 if x < 0,1 if x ≥ 0.The Heaviside function is a regular distribut ion.For any test function f ∈ S,H′[f ] = −Z∞−∞f′(x)H(x) dx= −Z∞0f′(x) dx = −f (x)∞x =0= f (0).Thus the derivative of the Heaviside function is thedelta function: H′= δ.Green’s functions for ODEsBoundary value problem:d2udx2= f (x), u(0) = u(L) = 0.Definition 1. Green’s function of the problem is afunction G (x, x0) (x, x0∈ [0, L]) such that for any fu(x) =ZL0f (x0)G (x, x0) dx0.Definition 2. Green’s function G (x, x0) of theproblem is its solution for f (x) = δ(x − x0):∂2G (x, x0)∂x2= δ(x −x0), G (0, x0) = G (L, x0) = 0.Definition 1 shows how to use Green’s function.Definition 2 shows how to find Green’s function.Both definitions are equ ivalent.Definition 2 means that•∂2G (x, x0)∂x2= 0 for x < x0and x > x0;• G (x, x0) is continuous at x = x0;•∂G (x, x0)∂xx =x0+−∂G (x, x0)∂xx =x0−= 1.G (x, x0) =(ax + b if x < x0,cx + d if x > x0,where a, b, c, d may depend on x0.∂G (x, x0)∂x=(a if x < x0,c if x > x0.Besides, G (0, x0) = b and G (L, x0) = cL + d.Thereforec − a = 1ax0+ b = cx0+ db = 0cL + d = 0=⇒a = (x0− L)/Lb = 0c = x0/Ld = −x0G (x, x0) =−xL(L − x0) if x < x0,−x0L(L − x) if x > x0.G (x, x0) = G (x0, x) (Maxwell’s reciprocity)Hilbert space L2[0, L] = {h :RL0|h(x)|2dx < ∞}Dense subspace H = {h ∈ C2[0, L] : h(0) = h(L) = 0}Linear operator L : H → L2[0, L], L[h] = h′′L is self-adjoint: hL[h], gi = hh, L[g]i for all h, g ∈ H.hL[h], gi =ZL0h′′(x)g(x) dx= h′(x)g(x)Lx =0−ZL0h′(x)g′(x) dx = −ZL0h′(x)g′(x) dx= −h(x)g′(x)Lx =0+ZL0h(x)g′′(x) dx = hh, L[g]iInverse operator L−1: L2[0, L] → L2[0, L].If L−1[f ] = u then u′′= f , u(0) = u(L) = 0.L−1[f ](x) =ZL0G (x, x0)f (x0) dx0Since the operator L is self-adjoint, so is L−1.hL−1[f ], gi =ZL0ZL0G (x, x0)f (x0)g(x) dx0dxhf , L−1[g]i =ZL0ZL0f (x)G (x, x0) g(x0) dx0dxL−1is self-adjoint if and only if G (x, x0) = G (x0, x).Nonhomogeneous boundary value problem:u′′(x) = f (x), u(0) = α, u(L) = β.We have th at u = u1+ u2+ u3, whereu′′1= f , u1(0) = u1(L) = 0;u′′2= 0, u2(0) = α, u2(L) = 0;u′′3= 0, u3(0) = 0, u3(L) = β.It turns out thatu1(x) =ZL0G (x, x0)f (x0) dx0,u2(x) = α1 −xL= −α∂G (x, x0)∂x0x0=0,u3(x) = βxL= β∂G (x, x0)∂x0x0=L.Existense of Green’s functionGreen’s function of an initial/boundary valueproblem e xists only if there is always a uniquesolution.Example 1. u′′+ u = f , u(0) = u(L) = 0.Green’s function exists if L 6= nπ, n = 1, 2, . . .(otherwise u1(x) = 0 and u2(x) = sin x are bothsolutions for f = 0).Example 2. u′′(x) + u(x) = f (x), 0 < x < L,u(0) = u′(0) = 0.Green’s function exists for any L >