Goal. The purpose of this exercise is to write down a formula for the solution of...

 

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Goal. The purpose of this exercise is to write down a formula for the solution of the IVP xux-yuy=2x-u, u(x, 1) = 0. We propose that the characteristics are given by a collection of parametric curves (r(s), y(s)), where s is a parameter. We then consider the following system of ODES dr ds dy ds du ds = x, =-y, y(0) = 1 x(0) = { = =2x-u, u(0) = 0, where is a new parameter. Observe that the above system allows us to make, at s = 0, the function u attain the value u(x(0), y(0)) = u(s = 0) = 0 = u(,1). This means that the ODE system produced accommodates the requirement that u is = 0 along the line y = 1 (for all r). Part I. (a) Show that x = Ses and yes and conclude that du ds 25e - u (b) Show that the solution to the ODE (*) (obtained in part (a), above) is u(s) = (ese¯³). Hint. Multiply (*) by eª and note that (eu) = ..., and so the ODE (*), multiplied by e*, becomes .. Part II. You may refer to the results of Part I and use them to solve the next question: Use the method of characteristics to solve the PDE problem Turyuy=2x-u, u(1, y) = 0. (*) Goal. The purpose of this exercise is to write down a formula for the solution of the IVP xux-yuy=2x-u, u(x, 1) = 0. We propose that the characteristics are given by a collection of parametric curves (r(s), y(s)), where s is a parameter. We then consider the following system of ODES dr ds dy ds du ds = x, =-y, y(0) = 1 x(0) = { = =2x-u, u(0) = 0, where is a new parameter. Observe that the above system allows us to make, at s = 0, the function u attain the value u(x(0), y(0)) = u(s = 0) = 0 = u(,1). This means that the ODE system produced accommodates the requirement that u is = 0 along the line y = 1 (for all r). Part I. (a) Show that x = Ses and yes and conclude that du ds 25e - u (b) Show that the solution to the ODE (*) (obtained in part (a), above) is u(s) = (ese¯³). Hint. Multiply (*) by eª and note that (eu) = ..., and so the ODE (*), multiplied by e*, becomes .. Part II. You may refer to the results of Part I and use them to solve the next question: Use the method of characteristics to solve the PDE problem Turyuy=2x-u, u(1, y) = 0. (*)