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Exercise 8: Derivation of the diffusion-convection conservation law PDE.

In some physical problems, the flux of material M has both a diffusive (down-gradient) component and a convective component (due to a velocity v) in which case the one dimensional flux can be expressed as

with v and allowed to depend on x, t, and u in general. Use the conservation law, , r = cu + b, and the above flux expression to derive the diffusion-convection conservation law PDE . In the case that c, v, and are constants, classify this PDE as hyperbolic, parabolic, or elliptic.