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PDE 2012 Problem Set 4 Hand in no later than October 24 Consider the IBVP on domain 0≤x≤1, 0≤t for ! ! + ! ! = 0, ! ! − ! ! = 0, with initial conditions ! ! , 0 = ! ! ! , ! ! , 0 = ! ! ! , and boundary conditions (a) or (b): ! ! 0, ! − ! ! 0, ! = ! ! ! , ! 1, ! − ! ! 1, ! = ! ! ! , ! ! 0, ! − ! ! 0, ! = ! ! ! , ! 1, ! − ! ! 1, ! = ! ! ! . 1. Investigate illposedness by looking for solutions e st with s=x+iy, x=Re(s)>0 of the corresponding problem with homogeneous boundary conditions. Show that one of the sets, (a) or (b), results in an illposed problem. Hint: you may investigate the determinant condition numerically. 2. Consider homogeneous initial data. For the other set of boundary conditions, show an estimate of the Laplace transformed solution in terms of the inhomogeneaus boundary functions. For which s is the estimate is valid? 3. Transform the estimate to physical space using ! !!!" !(∙, ! ) ! ! ! !" = ! !! ! (∙, ! + !" ) ! ! !! !"
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