hw1solns.pdf - APM346 HOMEWORK 1 Remarks \u2022 See the \u201cMultivariable calculus review\u201d notes \u2022 R denotes the set of real numbers Pn 2 \u2022 If u(x y

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APM346 HOMEWORK 1 Remarks See the “Multivariable calculus review” notes. R denotes the set of real numbers. If u ( x, y ) is a function of x R m and y R n , then Δ y = n i =1 2 y i denotes the Laplacian with respect to y . A naked “Δ” with no subscript means the Laplacian with respect to all the variables. A similar convention holds for the gradient . Problems 1. Each of the following PDE has a scaling symmetry in the sense that there exists a > 0 such that if u is a solution, then so is u λ ( t, x ) := u ( λ a t, λx ) for any λ > 0. Determine the exponent a for each of the following equations: (a) u t + u x = 0, t, x R . (b) t u - Δ x u = 0, ( t, x ) R × R 3 Solution. (a) a = 1 (b) a = 2 2. For each of the following classical equations from physics or finance, determine its order with respect to each variable and whether it is linear or nonlinear. (a) u t + u xxx = 0, ( t, x ) R × R . (b) |∇ φ | = 1, where φ is a function on R 2 . (c) t V + 1 2 x 2 2 x V + x∂ x V - V = 0, where t, x R . (d) div u (1 + |∇ u | 2 ) 1 2 = 0 , where u is a function on R 2 . (e) 2 t u - Δ x u = u 5 , ( t, x ) R × R 3 . Solution. (a) Order 1 in t , 3 in x , linear (b) Order 1, nonlinear.

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