# Solution of a system of first order coupled PDEs

I have a system of coupled PDEs
$$\dot{x}+y\prime+xy-y=0\tag{1}$$ $$\dot{y}+x\prime+x+xy=0$$
Where $$x=x(s,t)$$, and $$y=y(s,t)$$.

$$\dot{()}$$ represents the derivative w.r.t time $$'t'$$ and $$()\prime$$ refers to derivative w.r.t. the space variable $$'s'$$.

The initial conditions are given by,
$$x(s,0)=s, y(s,0)=1$$
I need to solve these equations analytically (if possible) or numerically.

What I have done:

The system of equations can be written in matrix form as:

$$\left[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix}\right]\dot{\begin{Bmatrix} x \\ y\\ \end{Bmatrix}}+ \left[\begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix}\right]\begin{Bmatrix} x \\ y\\ \end{Bmatrix}^\prime+\begin{Bmatrix} xy-y \\ x+xy\\ \end{Bmatrix}=0$$
Which can be written as:
$$A\dot{U}+BU^\prime+f(U)=0 \tag{2}$$
Where,
$$U=\begin{Bmatrix} x \\ y\\ \end{Bmatrix}, A=\left[\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix}\right], B=\left[\begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix}\right], f(U)= \begin{Bmatrix} xy-y \\ x+xy\\ \end{Bmatrix}$$
Rewriting the system by multiplying with $${A}^{-1}$$,
$$\dot{U}+{A}^{-1}BU^\prime+{A}^{-1}f(U)=0\tag{3}$$
Diagonalizing $${A}^{-1}B$$
$${A}^{-1}B=\left[\begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix}\right]=SD{S}^{-1}$$
Where,
$$S=\left[\begin{matrix} -1 & 1 \\ 1 & 1 \\ \end{matrix}\right], D=\left[\begin{matrix} -1 & 0 \\ 0 & 1 \\ \end{matrix}\right]$$
Where $$D$$ represents the diagonal matrix with its diagonal elements as the Eigen values.
Since the Eigen values are real and distinct, the system of PDEs is hyperbolic.

Rewriting Equation (3),
$$\dot{U}+SD{S}^{-1}U^\prime+{A}^{-1}f(U)=0$$
Multiplying with $$S^{-1}$$ throughout
$$S^{-1}\dot{U}+D{S}^{-1}U^\prime+S^{-1}{A}^{-1}f(U)=0\tag{4}$$
If we introduce a new vector, $$V=\begin{Bmatrix} v_1 \\ v_2\\ \end{Bmatrix}=S^{-1}U$$, Equation(4) becomes:
$$\dot{V}+DV^\prime+S^{-1}{A}^{-1}f(SV)=0\tag{5}$$
In Equation(5), the matrix $$D$$ decouples the derivative terms. But the term $$S^{-1}{A}^{-1}f(SV)$$ is causing problem. It is not allowing the decoupling, as shown in Equation (6). The system of equations become:
$$\dot{\begin{Bmatrix} v_1 \\ v_2\\ \end{Bmatrix}}+ \left[\begin{matrix} -1 & 0 \\ 0 & 1 \\ \end{matrix}\right]\begin{Bmatrix} v_1 \\ v_2\\ \end{Bmatrix}^\prime+\begin{Bmatrix} v_2 \\ {v_2}^2-{v_1}^2-v_1\\ \end{Bmatrix}=0\tag{6}$$

Can I use the method of characteristics to solve this system? If not please suggest any other method.

How should I proceed to solve this system if $$f(U)$$ can be any nonlinear functions in $$x$$ and $$y$$? Please help...

• It is often the case that systems of balance laws don't decouple (it is even more likely here since $f(U)$ is non-linear w.r.t. $U$). In this case, the method of characteristics isn't very useful... Jul 25, 2020 at 13:30
• Thank you for answering the question. Since the method of characteristics is not useful for the problem, I should try other methods. Now I am considering method of lines to convert the system into ODEs. Can you please suggest a method which can be a better way to solve the given problem Jul 27, 2020 at 6:22
• Sure! I'd suggest the use of finite volume schemes Jul 27, 2020 at 8:29