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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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- Spring '08