# Parabolic differential equations with time delay

Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$

How to solve:

$$\left\{ \begin{array}{lc} \dfrac{\partial u(t,x)}{\partial t}=d_1\triangle u(t,x)+u(t,x)\left(r_1-a_{11}u(t-\tau,x)-a_{12}v(t,x)\right),& t>0,x\in\Omega \\ \dfrac{\partial v(t,x)}{\partial t}=d_2\triangle v(t,x)+v(t,x)\left(-r_2+a_{21}u(t,x)-a_{22}v(t,x)\right),& t>0,x\in\Omega\\ \dfrac{\partial u}{\partial n}=\dfrac{\partial v}{\partial n}=0,\quad t\ge0,x\in\partial\Omega \quad(\text{Neumann conditions})\\ u(t,x)=\phi(t,x)\ge 0,\qquad v(t,x)=\psi(t,x)\ge 0, &(t,x)\in[-\tau,0]\times\Omega \end{array} \right.$$

• What's the particular problem here? Is it the time-delay? For this, interpolate the solution you already have, or try a constant stepsize which is a fraction of $\tau$ (so that the solution at $t-\tau$ is directly available). May 9, 2017 at 18:30
• Can you please include some kind of context here, like what you have already tried, and how it worked or didn't? Otherwise it's just a naked problem statement. May 9, 2017 at 20:20
• There have been much work for the equations which are listed above. eg, $\bf {monograph [Changyou Wang]}.$ do you use which numerical method for simulation? Just you can easily develop your programme by means of your formulation scheme of the equations. just the problem which you consider is the case of constant delay. May 11, 2017 at 11:08

Do the same discretization that you normally do for the nonlinear Heat Equation to turn $$\Delta$$ into $$A$$, the Strang matrix second order discretization (the [1 -2 1] tridiagonal matrix). Now you have a system of DDEs. Use a DDE solver on this. MATLAB's DDE23, Julia's DDE solvers, etc.