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 x-values: x , x 1 , x 2 , . . . , x j-1 , x j , x j +1 , . . . , x J and corresponding mesh point values of some function u ( x ): u , u 1 , u 2 , . . . , u j-1 , u j , u j +1 , . . . , u J . The x-values are spaced a constant distance Δ x apart. Show using Taylor series expansion that u j +1-2 u j + u j-1 (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