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# pb1_09 - MPO 662 Problem Set 1 Feel free to use symbolic...

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MPO 662 – Problem Set 1 Feel free to use symbolic computation software such as mathematica or the symbolic toolbox in matlab to carry out the algebra 1. Find the characteristics of the following PDEs (a) u xx + 3 u xy + u yy = 0 (b) u xx - 2 u xy + u yy = 0 2. Consider the following PDE: ∂t + U ∂x 2 ψ - c 2 2 ψ ∂x 2 = 0 where U and c are positive constants, and the first term is to be interpreted as: 2 ψ ∂t 2 + 2 U 2 ψ ∂x∂t + U 2 2 ψ ∂x 2 Classify this PDE sketch the domain of dependence and the domain of influence in the x - t plane for the point ( x 0 , t 0 ). Consider the case U > c and U < c . 3. You can use a symbolic manipulator here to ease the algebra. Derive the following approximations to the first derivative at x = i Δ x : 3 rd order upstream using the i - 2 , i - 1 , i and i + 1. Derive a 4 th order centered approximations to the second derivative. 4. Write a FORTRAN program to implement the numerical differentiation formula devel- oped previously for the second order derivative. Confirm the convergence rates of the

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pb1_09 - MPO 662 Problem Set 1 Feel free to use symbolic...

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