homework1 - u ( x, t ), deFned for ( x, t ) R 2 , is said...

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Homework Set 1: Math 716, Due: Friday, January 23rd 1. Calculate and plot the caracteristic curves for u x 1 + 2 u x 2 + (2 x 1 x 2 ) u = x 1 x 2 , ( x 1 , x 2 ) R 2 Afterwards, derive the general solution u ( x 1 , x 2 ) and in particular Fnd solution satisfying u (0 , x 2 ) = e x 2 . 2. Solve u x u y + u = e x - 2 y with u ( x, 0) = 0. 3. Use the method of characteristics to Fnd representation for solution to u t + uu x = u for x R , t R + , u ( x, 0) = e - x What is the restriction on t , if any, for solution to be smooth in x ? 4. By direct veriFcation show that for any integer n , u ( x, t ) = e - n e κn 2 t sin n ( x t ) is a solution to the backwards heat equation with advection: u t + u x = κu xx for x R , t R + with u ( x, 0) = e - n sin nx Use this to prove that the solution is unstable to variations in initial conditions and there- fore the problem is not well-posed in the b . b (sup norm in x ). 5. A function
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Unformatted text preview: u ( x, t ), deFned for ( x, t ) R 2 , is said to be a weak solution of the linear second order wave equation: u tt c 2 u xx = 0 if i - i - u ( x, t ) b tt ( x, t ) c 2 xx ( x, t ) B dxdt = 0 for all test functions ( x, t ) with compact support, i.e. for any smooth function ( x, t ) that is deFned for ( x, t ) R 2 and that vanishes outside a bounded region of R 2 . Verify that u ( x, t ) = f ( x ct ) + g ( x + ct ) is a weak solution for any continuous functions f and g . Hint: You may think of transforming independent variables from ( x, t ) ( , ), where = x ct , = x + ct . 1...
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