Coupled partial differential equations

[abaset]

Hello every one,
In my physics problem, i end up having two coupled second-order nonlinear differential equations where the coupling terms include, the variable, the first derivatives, and also a second derivative coupling. I appreciate any help on how to handle this system before setting it for numerical simulation: that system generally looks like the equations attached.


-For my first trial, i have eliminated the term with d^2(y) from the first equation using the second equation. similarly i eliminated the term with d^2(x) from the second equation using the first. Hence i obtained a new coupled system that looks more complicated but numerically solvable. i would like to know if this process is mathematically valid or not. i.e. does the new coupled equations typically represent or give the same results as the old one.

-For my second suggestion is just to neglect those higher coupling terms from both equations then solve them, however, I'm not sure how valid that is, anything in math or physics can validate neglecting those terms.

-What is the best (standard) way to solve such system?

-My final question is, any recommendation for a good book that discuss these coupled differential equations especially ones like the above.

Thanks in advance.

 

[haruspex]

Hello every one,
In my physics problem, i end up having two coupled second-order nonlinear differential equations where the coupling terms include, the variable, the first derivatives, and also a second derivative coupling. I appreciate any help on how to handle this system before setting it for numerical simulation: that system generally looks like the equations attached.


-For my first trial, i have eliminated the term with d^2(y) from the first equation using the second equation. similarly i eliminated the term with d^2(x) from the second equation using the first. Hence i obtained a new coupled system that looks more complicated but numerically solvable. i would like to know if this process is mathematically valid or not. i.e. does the new coupled equations typically represent or give the same results as the old one.

Sounds fine to me. In fact, you could do the same within the numerical solution. At each step, plug in the current values for all except the two second derivatives; this gives you a pair of simultaneous equations with those two derivatives as the only unknowns; solve and iterate. It comes to the same, but avoids the heavy algebra you went through.

-For my second suggestion is just to neglect those higher coupling terms from both equations then solve them, however, I'm not sure how valid that is, anything in math or physics can validate neglecting those terms.

The possible bases for neglecting them would be (a) that their coefficients in the equations are very small or (b) the values they take are very small. The first appears false, and the second would imply the first derivatives are more-or-less constant. You could certainly search for such a special case solution, but I doubt it would be right to ignore them generally.
Besides, does it help? Even a numerical solution from the resulting equation looks messy.

-What is the best (standard) way to solve such system?

Pass. I don't have much practical experience in this area.