Can anyone help me to implement the solving of partial differential equations in Mathematica? The specific system of partial differential equations is:
du(x,y,z)/dx = dv(x,y,z)/dy - dq(x,y,z)/dz
3du(x,y,z)/dy = 2 dv(x,y,z)/dz + dq(x,y,z)/dx
1/2 du(x,y,z)/dz = 6 dv(x,y,z)/dx - 2 dq(x,y,z)/dy
I would like to solve for 3 functions u(x,y,z), v(x,y,z), and q(x,y,z)
Admittably, the problem you brought up appeared to be not too hard at first glance. After trying to solve it with Mathematica, to me it seems that the standard procedures in Mathematica cannot solve this particular system of coupled PDEs. I can safely say that I have a lot of experience with Mathematica.
Anyway, I kept hanging on to your problem and I managed to solve with pencil and paper for at least one of the three unknowns, of course the analysis should be checked further for errors. If you are interested, feel free to contact me via Email:
Regards, Markus
To my understanding there is no other way to tackle this problem other than to do it by hands.
Mathematica is a powerful package that is capable of solving coupled differential equations symbolically. However, finding a general solution to a set of coupled linear PDEs is probably too hard to do for a software package. Instead, you may obtain a general result by pencil and paper using cross-differentiation of equations to eliminate unknowns. For instance, you may differentiate equation 1 for y and equation 2 for x to eliminate u. I proceeded further in that direction and obtained a simple PDE solely for q, from which a general solution for q readily follows.
On the other hand, if it suffices to find a particular solution that satisfies some BCs, you may first determine the character of the system (classification as hyperbolic, elliptic, parabolic). I recommend to have a look at the classical book of Courant and Hilbert. To solve the problem in Mathematica (the question you orginally posted), you can use DSolve and then simply add the boundary conditions to the set of equations, e.g like in the following ODE:
DSolve[{y''[x] + 4 y[x] == 0, y[0] == 1, y'[0] == 4}, y, x]
The set-up is accordingly for PDEs. I have mailed you a more detailed answer.
Regards
Dear my friend Phuong Hoang,
It seems your PDEs need at least 9 independent equations to solve them. Therefore Mathematica cannot solve your system of PDEs. You should find 6 more equations to complete an integrable system of PDEs from your physics problem.
The initial conditions or boundary conditions are also necessary to obtain the solution.
Best wishes,
Dai-Nam Le
I would kindly disagree with Dai-Nam Le's comment that you need additional equations in order to solve this problem. It is three linear (independent) equations for three unknown functions and that's just fine.
Mathematica may not return a result because there is simply too much posibility for solutions in this particular (quite simple) case. Sometimes Mathematica returns a symbolic solution to a PDE, being more likely if one provides boundary conditions to the field equation.
I would like to extend my initial answer where I stated that solutions may be obtained working with pencil and paper. I've played around a bit and found a solution. Here is one for demonstration:
u=x+(2/3)*y+(2/3)*x*y+(4/3)*z^2;
v=y+z;
q=-(2/3)*y*z+x^2;
I would like to know what the syntax is to solve coupled partial differential equations for multivariable in mathematica. In the Mathematica documentation, it is mentioned separately on solving coupled partial differential equations and solving the multivariable equation. I have not found so far what the syntax should be if I want to do both as in your case.
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