0

I'm trying to solve a system of 8 coupled partial differential equations, I have been told that it can be done using the method of characteristics, but I am struggling to see how. The equations are as follows.

$$a\frac{\delta u_1}{\delta t}+\frac{\delta u_1}{\delta z} = -bu_1(u_3+u_4)-xu_1$$ $$a\frac{\delta u_2}{\delta t}-\frac{\delta u_2}{\delta z} = -bu_2(u_3+u_4)-xu_2$$ $$a\frac{\delta u_3}{\delta t}+\frac{\delta u_3}{\delta z} = cu_3[(u_1+u_2)-(u_5+u_6)]-xu_3+y(u_1+u_2)$$ $$a\frac{\delta u_4}{\delta t}-\frac{\delta u_4}{\delta z} = cu_4[(u_1+u_2)-(u_5+u_6)]-xu_4+y(u_1+u_2)$$ $$a\frac{\delta u_5}{\delta t}+\frac{\delta u_5}{\delta z} = du_5[(u_3+u_4)-(u_7+u_8)]-xu_5+y(u_3+u_4)$$ $$a\frac{\delta u_6}{\delta t}-\frac{\delta u_6}{\delta z} = du_6[(u_3+u_4)-(u_7+u_8)]-xu_6+y(u_3+u_4)$$ $$a\frac{\delta u_7}{\delta t}+\frac{\delta u_7}{\delta z} = eu_7(u_5+u_6)-xu_7+y(u_5+u_6)$$ $$a\frac{\delta u_8}{\delta t}-\frac{\delta u_8}{\delta z} = eu_8(u_5+u_6)-xu_8+y(u_5+u_6)$$

a, b, c, d, e, x and y are all constants.

I have tried using the method of characteristics to separate the equations into odes to be solved on matlab, but I am struggling with the fact that half the equations have a positive d/dz term and half the equations have a negative d/dz term, making substitution difficult.

Any help would be appreciated, otherwise I will simply resort to an approximation using the method of lines.

HHall
  • 1
  • Use \partial for $\partial$. Also, are $x, y$ variables of the system or constants? Finally, I gave an answer here where I applied the method of characteristics for a system of first order PDEs. Try following it and see if you can apply it to your problem. – Matthew Cassell Jun 17 '20 at 15:31
  • The x and y values are constant. I'll give the method a go and see how far I get. – HHall Jun 17 '20 at 15:55

0 Answers0