Study material for Mathematical and Numerical Methods in Engineering, shared by the Studwiz community and reviewed by moderators.
Study material for Mathematical and Numerical Methods in Engineering, shared by the Studwiz community and reviewed by moderators.
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MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING EXAM OF THE MATHEMATICAL PART (Prof. F. Gazzola) 18.01.21 1. (10 points) Solve the following conservation law with the initial conditions given: 𝑢!+𝑒"𝑢#=0 𝑥∈ℝ,𝑡>0 𝑎) 𝑢(𝑥,0)=0−log2 𝑓𝑜𝑟 𝑥<0 0 𝑓𝑜𝑟 0<𝑥<1log𝑥 𝑓𝑜𝑟 𝑥>1 𝑏) 𝑢(𝑥,0)=<log2 𝑓𝑜𝑟 𝑥<00 𝑓𝑜𝑟 𝑥>0 2. (10 points) i) Solve the following problem: ⎩⎪⎨⎪⎧∆𝑢=0 (𝑥,𝑦)∈(0,1)×(0,2)𝑢(𝑥,0)=0 𝑥∈(0,1) 𝑢$(𝑥,2)=𝑥 𝑥∈(0,1) 𝑢#(0,𝑦)=0 𝑦∈(0,2) 𝑢$(1,𝑦)=0 𝑦∈(0,2) ii) Solve the following problem in view of the solution of the previous point: ⎩⎪⎨⎪⎧∆𝑣=2 (𝑥,𝑦)∈(0,1)×(0,2)𝑣(𝑥,0)=0 𝑥∈(0,1) 𝑣$(𝑥,2)=𝑥+4 𝑥∈(0,1) 𝑣#(0,𝑦)=0 𝑦∈(0,2) 𝑣$(1,𝑦)=0 𝑦∈(0,2) 3. Prove the D’Alambert formula for the wave equation in 1D. (6 points) 4. Derive the heat equation from a model of heat conduction. (6 points)
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