# Using MATLAB to solve a system of two PDEs encountered in Mathematical Biology

I am in an undergraduate summer project and am studying a paper, I hope I could solve the following two PDEs with two boundary conditions for $$y_1(t,a)$$ and $$y_2(t,a)$$ using MATLAB. $$\frac{\partial y_1}{\partial t}+\frac{\partial y_1}{\partial a}=f_1(t)y_1, \qquad t\geqslant0, \quad a\in[0,\tau_1],$$ with boundary condition $$y_1(t,0)=c_1$$. $$\frac{\partial y_2}{\partial t}+\frac{\partial y_2}{\partial a}=f_2(t)y_2, \qquad t\geqslant0,\quad a\in[0,\tau_2],$$ with boundary condition $$y_2(t,0)=c_2y_1(t,\tau_1)$$.

Here, $$y_1(t,a)$$ and $$y_2(t,a)$$ are functions of $$t$$ and $$a$$. The functions $$f_1(t)$$ and $$f_2(t)$$ are known functions of $$t$$ but are rather complicated, and $$\tau_1,\tau_2,c_1,c_2$$ are known constants.

Since $$y_1(t,a)$$ is involved in the boundary condition of the second PDE for $$y_2(t,a)$$, I aim to solve the first PDE for $$y_1(t,a)$$ first, and then use the results in solving the second PDE for $$y_2(t,a)$$.

I am new in both PDEs and MATLAB, although I have spent several days searching for possible ways in doing it, I still have got no hints. It would be greatly appreciated if some hints or possible directions I could look into could be provided. Thanks in advance.

• Please edit your post to add context. *What do the variables $y_{1,2}$, $t$ and $a$ represent? *What are the initial conditions? *What are the bibliographical references of the studied paper? + Note that cross-posting the same question on various SE websites is not recommended. Aug 8 '19 at 15:24
• You can use pdepe in MATLAB to solve this system. The information page on pdepe has several examples. It is quite easy to follow. Aug 15 '19 at 22:15

First order PDEs can generally be solved by the method of characteristics. To solve your first PDE using this method, put $$\phi_z(x) = y_1(z + x, x)\ .$$ Then $$\begin{eqnarray} \frac{d\phi_z}{dx}(x)&=&\frac{\partial y_1}{\partial t}(z+x,x)+\frac{\partial y_1}{\partial a}(z+x,x)\\ &=& f_1(z+x)\,\phi_z(x)\ . \end{eqnarray}$$ This is a first-order ODE for $$\ \phi_z(x)\$$, which has the solution: $$\begin{eqnarray} \frac{\phi_z(x)}{\phi_z(0)}&=&e^{\int_0^xf_1(z +u)\,du}\\ &=& e^{\int_z^{x+z}f_1(u)\,du}\ . \end{eqnarray}$$
Now, $$\ \phi_z(0)=y_1(z, 0) = c_1\$$, and $$\ y_1(t,a)=\phi_{t-a}(a)\$$, so $$y_1(t,a)=c_1 e^{\int_{t-a}^tf_1(u)\,du}\ .$$ Your second PDE can now be solved similarly. When solving the corresponding ODE, however, you will need to integrate over the interval from $$\ x\$$ to $$\ \tau_1\$$, rather than from $$\ 0\$$ to $$\ x\$$ as I did above.
The method of characteristics provides the solution in the domain $$t\geq a \geq 0$$. The characteristic curves along which the information $$\frac{\text d}{\text d t} y_k = f_k y_k$$ propagates are the straight parallel lines $$x = t-t_0$$ with $$t_0 \geq 0$$. Hence, we have $$y_1(t,a) = c_1 \exp\left(\int_{t-a}^t f_1(s)\, \text d s\right) = c_1 \exp\left(\int_{0}^a f_1(t-\tau)\, \text d \tau\right) .$$ Similarly, we have for $$t\geq a \geq 0$$ $$y_2(t,a) = c_2\, y_1(t,\tau_1) \exp\left(\int_{0}^a f_2(t-\tau)\, \text d \tau\right)$$ where $$y_1(t,\tau_1)$$ with $$t\geq \tau_1$$ can be deduced from the previous step. Of course, this solution can then be implemented in Matlab, e.g. by using the Matlab function integral for the numerical integration of $$f_k$$. Alternatively, an iterative numerical resolution by finite-volume methods can be implemented.