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Mathematical and Numerical Methods in Engineering - 16jan18 1. (10 points) Consider the following Cauchy problem: ut +u2ux = 0, x∈ R, t∈ (0, 3] u(x, 0) =g(x) :=    −1 x≤ 0 0 x∈ (0, 1) −1 x> 1 x∈ R, Find the solution for (x,t )∈ R×[0, 3] by the method of characteristics, specifying where it is a continuous function. Moreover, write the differential equation satisfied by the shock line starting at t = 3, and find the slope of that line at t = 3. 2. (10 points) Solve the following problem:    ut−uxx = 0 x∈ (0, 1),t> 0 u(x, 0) =x2, x ∈ [0, 1] u(0,t ) = 1, u(1,t ) = 0 t> 0. 3. (6 points) Write and discuss the Rankine-Hugoniot condition. 4. (6 points) Consider a string moving according to the following system:    ρ0utt−τ0muxx = 0 x∈ (0, 1),t> 0 u(x, 0) =g(x), x ∈ [0, 1] u(0,t ) = 0, u(1,t ) = 0 t> 0. Define its energy and show that it is constant. Mathematical and Numerical Methods in Engineering - 06feb18 1. (10 points) Consider the following Cauchy problem: ut + 2uux = 0, x∈ R,t> 0 u(x, 0) =g(x) :=    0 x< 0 2x x ∈ (0, 2) −2 x> 2 x∈ R, Find the solution by the method of characteristics, specifying whether it is a continuous function. 2. (10 points) Solve the following problem:    ut−uxx−u = 0 x∈ (0, 1),t> 0 u(x, 0) =x− 1 2, x ∈ (0, 1) ux(0,t ) = 0, ux(1,t ) = 0 t> 0. Finally establish whether, for any given x∈ (0, 1), lim t→+∞ u(t,x ) exists and is finite, or not. 3. (6 points) Give a derivation of the heat equation, specifying the physical assumptions made. 4. (6 points) State and prove the d’Alembert formula. 1 Mathematical and Numerical Methods in Engineering - 26jun18 1. (10 points) Consider the following Cauchy problem: ut + 3u2ux = 0, x∈ R,t> 0 u(x, 0) =g(x) :=    0 x< 0 −1 x∈ (0, 2) 0 x> 2 Find the solution by the method of characteristics,…