There are 3 non-homogeneous second order PDE equations with 3 functions (u,v, and w) to be found.
Each function having 3 variables.
i am looking for analytical solution not numerical.
Regards
Dear Shahab Sadeghpour ,
I suggest the following method:
Step (1) Use the first PDE and
determine the quantities ∂²w/∂x∂z and ∂²w/∂x² by direct differentiation,
then substitute them in the second and the third and equations.
Step(2) we obtain two second order PDE's (*) and (**)
(You can hold y as constant, it doesn't play any rule related to the differentiation process)
u and v are considered as the dependent variables , x and z are the independent variables.
Step(3) Differentiate the first pde (*) equation with respect to (z) and the second pde(**) with respect to (x).
Step(4) Eliminate ∂²u/∂x∂z from the two equations (*) and (**).
Then we obtain a third order PDE where v is the dependent function and x,z are the independent variables.
Step(5) Let T = ∂²v/∂x∂z , and then the final PDE transforms into a first order linear ordinary differential equation in T.
Solve for T and then by backward integrations you can determine the general solution for v and u.
Finally, check the obtained solution to satisfy the original system
of the given PDE's.
Wish you good luck.
The suggestions by Issam are good. Some simplifications:
* When ∂²w/∂x∂z are eliminated from the last equation, by Issam's Step (1), all references to u are also eliminated.
* So you end up with a first order differential equation (in z) for the variable V == ∂v/∂x. Hence v can be "found" by integrations.
* With this solution inserted into the middle equation, you end up with a first order differential equation (in x) for the variable u, provided a2 - a1 is non-zero (otherwise a consistency condition emerges).
But I am not convinced that the formal solution thus obtained will be immediately useful, since the integrals will involve unknown integration "constants" that can depend on the remaining variables. The nature of the underlying problem must provide further information.
Since we have no information about what system you're trying to model, this suggestion may not apply perfectly, but I'd recommend you to do a nondimensionalization of the system, if possible. It would reduce the number of parameters and make it easier to find exact solutions.
The separate equation for v is (a1 v-z v',z)'x=k-g',z with solution v=v0(x,y,z).
If a1≠a2, there exists separate equation for u:
(a1-a2) u',x=g',x-(h+a3 v0',z+z v0",xx) with solution u=u0(x,y,z),
and for w: w',x=a1 u0-g.
In the case a1=a2, additional condition is required
g',x= h+a3 v0',z+z v0",xx
and only one equation exists for two functions u and w:
a1 u-w',x=g.
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