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What type of PDEs (partial differential equations) are the following:

  1. $\frac{\mu}{K}\textbf{u} + \frac{\partial \textbf{u}}{\partial t} = -\nabla p $ (Darcy's law),
  2. $\frac{\partial c} {\partial t} + \textbf{u} \cdot \nabla c= \nabla^2 c$ (convection-diffusion equation).

Next how many boundary conditions and initial conditions does a PDE require to solve $?$ for example

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This is not a complete answer to your question. So one may not be satisfied to it.

  1. Hint: Write it done in its full expansion and see.

  2. It is a combination of parabolic and hyperbolic partial differential equation.

Number of initial and boundary conditions will depend on the region where you have modeled your equations, co-ordinate system you are using, and the method of solution (Analytical, Numerical using finite difference method, Finite element method etc.) you are using.

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