How can one discretize one-dimensional delayed reaction diffusive system: $\frac{\partial u}{partial t} = u - u(t)*u(t-\tau) +\frac{\partial^2 u}{partial x^2}$, where $\tau$ is time delay?
Discretization of partial differential equations (PDEs) usually involves transforming continuous functions into discrete approximations, which enables numerical solutions to be calculated.
We can use finite differences to approximate the spatial and temporal derivatives for the delayed reaction diffusive system you described. However, the delay differential term $u(t-\tau)$ introduces some complexity, as it depends on the function's value at a previous time.
Let's first handle the spatial derivative using a central difference approximation:
(see fig. 1 attached)
where $x_{i+1}$ and $x_{i-1}$ are the points to the right and left of $x_i$, respectively, and $\Delta x$ is the step size in the $x$-direction.
Now let's discretize the temporal derivative using a forward difference approximation:
(see fig. 2 attached)
where $t_{j+1}$ is the next point in time and $\Delta t$ is the step size in time.
So the discretized form of your PDE is:
(see fig. 3 attached)
The term $u(x_i,t_j-\tau)$ represents the value of $u$ at the same spatial point $x_i$, but at a time delay of $\tau$ earlier. If $\tau$ is a multiple of $\Delta t$, i.e., $\tau = k\Delta t$ for some integer $k$, then we can simply replace this term by $u(x_i, t_{j-k})$. However, if $\tau$ is not an exact multiple of $\Delta t$, then some interpolation would be needed to estimate the value of $u(x_i,t_j-\tau)$.
Note: The PDE you've given is a nonlinear one (due to the term $u(t)*u(t-\tau)$) and typically requires more advanced numerical methods to solve, like the Newton-Raphson method or fixed-point iteration methods. Also, the stability and convergence of the numerical scheme need to be checked and may depend on the specific values of $\Delta x$, $\Delta t$, and $\tau$. In many practical cases, this kind of problem requires sophisticated software to solve efficiently.
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