Unformatted text preview: ∞ < x < ∞ , and all times t starting from t = 0 until a ±nal time t = T . ²ind u ( x, t ) for the initial condition: u ( x, 0) = u n e i nx where i = √1 and n and u n are constants. Based on your result, is the initial value problem properly posed? If so, why? 3. Consider a mesh with xvalues: x , x 1 , x 2 , . . . , x j1 , x j , x j +1 , . . . , x J and corresponding mesh point values of some function u ( x ): u , u 1 , u 2 , . . . , u j1 , u j , u j +1 , . . . , u J . The xvalues are spaced a constant distance Δ x apart. Show using Taylor series expansion that u j +12 u j + u j1 (Δ x ) 2 approximates the second order derivative of u ( x ) at point x j to second order accuracy. Verify your answer using the operator form of the above expression....
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 Spring '09
 Dommelen

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