Additional Components in Standard Equations

If problems have to be modeled, the standard parabolic, hyperbolic or elliptic equations don't describe the real world problems. Additional factors and components have to be involved into the equation. For example, as already seen in the chemotaxis equation (17):

$$\frac{\partial n}{\partial t} = D\nabla^2n-\nabla\cdot(n\nabla u),$$

a diffusion of cells can be forced towards a chemical signal by introducing an extra term.

In general, the other terms can be functions of the seeked function $u$:

$$u_t=u_{xx}+g(u).$$

In the case $g(u)=\mbox{exp}(u)$ the approximation for $u_l^{m+1}$ (32) becomes

$$\frac{u_l^{m+1}-u_l^m}{k} = D\frac{u_{l-1}^m-2u_l^m+u_{l+1}^m}{h^2}+\mbox{exp}(u_l^m).$$ (37)

If additional terms are functions of different variables, the model becomes a system of partial differential equations.
A model with three PDEs which was published by (1) is the following:

$$\frac{\partial n}{\partial t} = d_n\nabla^2n\; -\; \gamma\nabla\cdot(n\nabla f),$$

$$\frac{\partial f}{\partial t} = -\; \eta\; mf,$$

$$\frac{\partial m}{\partial t} = d_m\nabla^2m\; +\; \alpha n\; -\; \beta m.$$

With the approximations for $u_t$, $u_x$ and $u_{xx}$:

$$u_t\approx\frac{u_l^{m+1}-u_l^m}{k}\; ,\hspace{1cm} u_x\approx\fr... ...l-1}^m}{2h}\; ,\hspace{1cm} u_{xx}\approx\frac{u_{l-1}^m-2u_l^m+u_{l+1}^m}{h^2}$$

we obtain the following numerical approximations:

$$\Bigg(\; \frac{\partial n}{\partial t} = d_n\frac{\partial^2 n}{\partial x^2}- \gamma\Big[\frac{\partial n... ...\frac{\partial f}{\partial x}+ n\frac{\partial^2 f}{\partial x^2}\Big]\; \Bigg)$$

$$\begin{eqnarray*} \frac{n_l^{m+1}-n_l^m}{k}&=& d_n\frac{n_{l-1}^m-2n_l^m+n_{l+1}... ...1}^m-2m_l^m+m_{l+1}^m\big)+ k\alpha\; n_l^m-k\beta\; m_l^m+m_l^m \end{eqnarray*}$$