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$$\left\{\begin{array}{ll} u_t+a u_x = f(t,x) & (t>0,\ -\inf... ...ty),\\ u(0,x) = g(x) & (-\infty<x<\infty) \end{array}\right.$$ (35)

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$$u_t(t,x)\approx \frac{u(t+\Delta t,x)-u(t,x)}{\Delta t},\ u_x(t,x)\approx \frac{u(t,x+\Delta x)-u(t,x)}{\Delta x}$$

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$$\frac{u(t+\Delta t,x)-u(t,x)}{\Delta t} + a\frac{u(t,x+\Delta x)-u(t,x)}{\Delta x} \approx f(t,x)$$ (36)

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$$u(t+\Delta t,x)\approx -a\frac{\Delta t}{\Delta x}u(t,x+\D... ...left(1+a\frac{\Delta t}{\Delta x}\right)u(t,x) + \Delta tf(t,x)$$ (37)

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$$\begin{array}{l} \left\{\begin{array}{l} \displaystyle u^... ...elta x),\ n=0,1,2,\ldots, \ j=0,\pm1,\pm2,\ldots) \end{array}$$ (38)

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