NumericalAnalysis-684-2010jan - 1 MAT 684 Qualifying Exam...

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Unformatted text preview: 1 MAT 684 Qualifying Exam Syracuse University January, Spring 2010 Ex 1. Consider the boundary value problem od determining u ( t ), 0 < t < 1 which satisfies u 00 ( t ) = 6 u ( t )- tu ( t ) + u 2 ( t ) for 0 < t < 1 u (0) = 20 u (1) = 10 (a) Formulate a finite difference method to compute u ,...,u n where u j is an approximation to u ( jh ), where j = 0 ,...,n , h = 1 /n . (b) Write a system of nonlinear algebraic equations of the form Au = b + f ( u ), u = [ u 1 ,...,u n- 1 ] T resulting from the finite difference method (c) Describe one step of Newtons method for computing the vector u . Ex 2. Let f : [0 , 1] R be a given continuous function. Consider the following boundary- value problem with non-homogeneous Dirichlet boundary conditions- u 00 ( x ) = f ( x ) , < x < 1 u (0) = , u (1) = (a) Let N be a positive integer h = 1 / ( N + 1), x j = jh , j = 0 , 1 ,...,N + 1. Define a finite element space V h of continuous, piecewise affine functions V h = { v h C ([0 , 1]) :...
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NumericalAnalysis-684-2010jan - 1 MAT 684 Qualifying Exam...

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