MIT OpenCourseWarehttp://ocw.mit.edu 18.306 Advanced Partial Differential Equations with ApplicationsFall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 04 2009 09 21 MON TOPICS: First order scalar pde. Examples of solutions by characteristics. Domain of influence. Review characteristics. Examples in detail: 1) x*u_x + y*u_y = 0, for y >= 1, with u(x, 1) = F(x) 2) x*u_x + y*u_y = 1+y^2, for y >= 1, with u(x, 1) = F(x) Domain of dependence and domain of influence. Where is the solution defined and where it is not. Examples showing solution not unique outside domain of influence: For case (1), with F(x) = exp(-x^2), consider (in the plane without the origin = P0) u1 = exp(-x^2/y^2) ................. for x^2+y^2 > 0. u = exp(-x^2/y^2) .................. for y >= 0 and x^2+y^2 > 0. = exp(-3*x^2/y^2) ................ for y <=0 and x^2+y^2 > 0. Both u1 and u2 are smooth and solve the equation and given data, but they are not equal outside y >= 0 and x^2+y^2 > 0. Can construct infinitely many such
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