0

I have a system of PDE which I would like to solve it by Matlab(numerically or analytically). How can I do this? Are there any known analytical approaches to problems of this kind?

$$\frac{\partial f_1(x,t)}{\partial x}=2f_3(x,t)+3f_1(x,t)-f_2(x,t),\\ \frac{\partial f_2(x,t)}{\partial t}=-2f_4(x,t)-3.2f_1(x,t)+f_2(x,t),\\ \frac{\partial f_3(x,t)}{\partial x}=-3f_3(x,t)+3.2f_4(x,t)-0.045f_1(x,t),\\ \frac{\partial f_4(x,t)}{\partial t}=f_3(x,t)-f_4(x,t),\\$$ with following conditions: $$f_3(0,t)=-e^{-2t},\\ f_4(x,0)=e^{-3x}.cos(2\pi x),\\ f_1(3,3)=0,\\ f_2(3,3)=0.$$

Thanks in advance for any help.

Jean Marie
  • 81,803
Mahmood Dadkhah
  • 11
  • Related : https://math.stackexchange.com/q/3316781 with the method of characteristics. – Jean Marie Feb 16 '20 at 11:25
  • Another track : use Laplace Transform. References :http://www.robots.ox.ac.uk/~jmb/lectures/pdelecture4.pdf, http://people.3sr-grenoble.fr/users/bloret/enseee/maths/enseee-maths-IBVPs-1.pdf, https://people.math.gatech.edu/~herod/conted/M40.pdfhttps://math.stackexchange.com/q/2540373, etc. – Jean Marie Feb 16 '20 at 18:11
  • Please add context to the question (where is this problem from?) – EditPiAf Feb 23 '20 at 20:41
  • This question is obtained from an optimal control problem. – Mahmood Dadkhah Feb 28 '20 at 13:55

1 Answers1

1

Note that $\partial_t f_4 = f_3 - f_4$ with $f_4(x,0)= \text{e}^{-3x}\cos(2\pi x)$ can be integrated as $$ f_4(x,t) = \text{e}^{-3x-t}\cos(2\pi x) + \int_0^t \text{e}^{-(t-\tau)} f_3(x,\tau)\, \text d \tau \, . $$ We can proceed in a similar manner with $\partial_x f_3 = -3 f_3 +3.2 f_4 - .045 f_1$ for which $$ f_3(x,t) = -\text{e}^{-2t-3x} + \int_0^x \text{e}^{-3(x-\xi)} \left(3.2 f_4 - .045 f_1\right)(\xi,t)\, \text d \xi \, . $$ However, injecting one of those into another, one observes that $f_4$, $f_3$ are solutions of an integral equation. I don't even know if the above problem is well posed.

For numerical resolution, I'd suggest to implement finite difference methods.

EditPiAf
  • 20,898