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4. Derive an explicit finite difference scheme for solving the equation \frac{\partial U}{\partial t}=\frac{\partial U}{\partial x}+U \frac{\partial^{2} U}{\partial x^{2}} in the region 0 <= x <= 1 based on a

forward difference in time t and central differences in x. Find the principal part of the local truncation error at x = ih, t = jk where hand k are the step lengths in x and t respectively and hence confirm that your scheme is consistent. Use two time steps of the scheme to find a numerical solution for U at intervals of 1/4 in x at the final time t = 1/16, given that: U=0 \quad \text { at } \quad x=0 \text { for } t \geq 0 U=1.3 \text { at } x=1 \text { for } t \geq 0 U=1.3 x \text { at } t=0 \text { for } 0 \leq x \leq 1

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