Different Terminologies

The Diffusion Equation $D\nabla^2u$ for $u(x,y)$:

$$\frac{\partial u}{\partial t} = D\nabla^2u$$ (15)

$$= D\nabla\cdot\nabla u$$

$$= D\mbox{ div}(\mbox{grad }u)$$

$$= \displaystyle{D\biggl[\begin{array}{cc} \frac{\partial}{\partial ... ...frac{\partial u}{\partial x}& \frac{\partial u}{\partial y} \end{array}\biggr]}$$

$$= D\frac{\partial^2u}{\partial x^2}+D\frac{\partial^2u}{\partial y^2}$$

$$u_t = Du_{xx}+Du_{yy}$$

The Laplace-Operator $${\Delta=\sum_{i=1}^{n}\partial_i^2}$$ for $u(x,y)$:

$$\Delta u = \nabla^2 u$$ (16)

$$= \nabla\cdot\nabla u$$

$$= \mbox{div}(\mbox{grad }u)$$

$$= \displaystyle{\biggl[\begin{array}{cc} \frac{\partial}{\partial x... ...frac{\partial u}{\partial x}& \frac{\partial u}{\partial y} \end{array}\biggr]}$$

$$= \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}$$

$$= u_{xx}+u_{yy}$$

$$\equiv 0$$

The Chemotaxis-Equation for $u(x,y)$:

$$\frac{\partial n}{\partial t} = \overbrace{D\nabla^2n}^\textrm{Diffusion}- \overbrace{\nabla\cdot(n\nabla u)}^\textrm{Chemotaxis}$$ (17)

$$= D\nabla^2n- \nabla\cdot\left(n\biggl[\begin{array}{cc} \frac{\partial u}{\partial x}& \frac{\partial u}{\partial y} \end{array}\biggr]\right)$$

$$= D\nabla^2n-\biggl[\begin{array}{cc} \frac{\partial}{\partial x}& ... ...partial u}{\partial x}& \frac{\partial u}{\partial y} \end{array}\biggr]\right)$$

$$= D\nabla^2n-\Bigl(\frac{\partial n}{\partial x}\frac{\partial u}{\... ...rtial y}\frac{\partial u}{\partial y}+ n \frac{\partial^2u}{\partial y^2}\Bigr)$$

$$= D\frac{\partial^2n}{\partial x^2}+D\frac{\partial^2n}{\partial y^... ...rtial y}\frac{\partial u}{\partial y}+ n \frac{\partial^2u}{\partial y^2}\Bigr)$$