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Unformatted text preview: Introduction to PDE’s: (Strauss, 1.1) (Haberman, 1.1) • PDE’s vs. ODE’s ← methods of solution const coeff( e λt ) undetermined coefficients variation of parameters series solutions reduction of order Laplace transform PDE’s have the complication of more than 1 independent variable For PDE’s: more difficult to “guess” the form of solution as in ODE • Definitions: linear vs. nonlinear u t = u x u t = uu x order of the equation u t = u xx u tt = u xx const. coefficient nonconst coefficient u t = cu xx u t = c ( x, t ) u xx • Analogies for linear ode’s to linear pde’s : REVIEW ODE’s: Superposition: if e λ,t , e λ 2 t are solutions of au 00 + bu + u = 0 c 1 e λ 1 + c 2 e λ 2 t is also a solution uses concepts of linear algebra for vector spaces: For a vector space V , there is associative and commutative addition, a multi plicative identity (1), and additive(0) identity Basis = smallest # of nonzero linear independnet elements from which we gen erate the rest e.g. if solutions are 1 , sin 2 x, cos 2 x for a linear pde then c 1 + c 2 sin 2 x + c 3 cos 2 x is a vector space but the basis is (sin 2 x, cos 2 x ) so we can write any solution as d 1 sin 2 x + d 2 cos 2 x Complication of PDE  more variation in additional independent variables makes solution methods more complicated 1 x U ODE f ( x , U ) = U x x x y U PDE U x = h ( x, y, U y , U, U x ) U x , U y different at different points Additional material: (from Weinberger) Classification of 2 nd order PDE’s: Examples Elliptic  u xx + u yy = 0 ∇ 2 u = 0 Laplace’s equation Parabolic  u t = u xx Heat Equation Hyperbolic  u tt = u xx Wave Equation General 2 nd order pde: a 11 u xx + 2 a 12 u xy + a 22 u yy + a 1 u x + a 2 u y + a u = 0 Classification: a 2 12 < a 11 a 22 Elliptic a 2 12 = a 11 a 22 parabolic a 2 12 > a 11 a 22 hyperbolic Ex (from Weinberger:) Lu = A ∂ 2 ∂t 2 + 2 B ∂ 2 ∂x∂t + C ∂ 2 ∂x 2 u = 0 (2 nd order part only) Make a real change of variables: ξ = αx + βt η = γx + δt ∂ t = β∂ ξ + δ∂ η ∂ x = α∂ ξ + γ∂ η ⇒ [ Aβ 2 + 2 Bαβ + Cα 2 ] u ξξ + u ξη [2 Aβδ + 2 B [ αδ + βγ ] + 2 Cαγ ] + u ηη [ Aδ 2 + 2 Bγδ + Cγ 2 ] = 0 if 4 B 2 4 AC > 0, we can find α, β, δ, γ to set coefficient of u ξξ , u ηη = 0 ⇒ u ξη = 0. Then we can find α, β, δ, γ to write pde as u ξξ + u ηη = 0 Similarly for parabolic, hyperbolic ⇒ can find change of variables to put 2 nd order part in form of wave, heat equation Derivation of heat equation: 1 D (Strauss 1.3, Haberman 1.2) Conservation of heat energy in a 1 D rod a b (lateral sides insulated) 2 d dt Z b a e dx  {z } change in heat energy = φ ( a, t ) φ ( b, t )  {z } flux at x = a flux at x = b (flowing to the right)  {z } R b a ∂φ ∂x dx + Z b a Q dx  {z } internal sources ⇒ R b a h ∂e ∂t + ∂φ ∂x Q i dx = 0 for arbitrary a, b, Q ∂e ∂t = ∂φ ∂x...
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 Fall '09
 DeVille

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