Eigenvalue & Eigenvectors An analysis of the displacement of an object produces the following set of equations for the components of the displacement x, y and z: dx dt = 13x ?? 4y ?? 11z dy dt = ??2x + 2z dz dt = 14x ?? 4y ?? 12z Write a MAPLE worksheet to determine the General Solutions for x, y and z. [10 marks] Examine the behaviour of the solutions you have obtained for varying initial con- ditions x(0), y(0) and z(0). You should describe and include plots of the solutions obtained to illustrate all possible behaviour giving reasons for your choices of initial conditions. [20 marks] Total: 30 marks 1
Fourier Series The temperature (x; t) at any position x along a solid bar (0 x 1) and at any time t satises the partial dierential equation k @@t = @2@x2 and given that the bar is kept at zero temperature at the end points (so that (0; t) = (1; t) = 0) the solution is given by (x; t) = 1Xn=1 bn sin(nx) exp(??n22t=k) where the initial temperature has been expanded in a half-range Fourier series so that (x; 0) = 1Xn=1 bn sin(nx) 1. Given that k = 100 and that the initial temperature distribution is (x; 0) = x(1 ?? x) XN n=1 bn sin(nx) (a) nd a suitable value of N so that the series is an consistent representation of (x; t) and calculate the coecients bn; (b) produce a suitable 3-dimensional graph of the temperature for all x and 0 t 10. Comment on the shape and how changes in time. [15 marks] 2. Repeat the calculation above for a range of values of , where again k = 100 but (x; 0) = u(x ?? ) ?? u(x) Comment on the shapes and evolution in time for all cases. Full marks will be obtained for a good descriptive analysis rather than just a set of graphs. [15 marks] Total: 30 marks 2
Fourier transforms Make use of the MAPLE syntax given to nd inverse Fourier transforms of the general form F(!) = a + b! a2 + c!2 + d!4 Choose a range of parameters to show the possible behaviours including values such that F(!) (!) and where FT??1fF(!)g (t) [15 Marks] Repeat the analysis…