System of second order linear coupled pde with constant coefficient

[galuoises]

Someone know how to uncouple this system of pde?

Δu$_{1}$(x) + a u$_{1}$(x) + b u$_{2}$(x) =f(x)
Δu$_{2}$(x) + c u$_{1}$(x) + d u$_{2}$(x) =g(x)

a,b,c,d are constant.

I would like to find a solution in one, two, three dimension, possibily in terms of Green function...someone could help me, please?

 

[Ben Niehoff]

Let

$$y_1 = u_1 + \lambda_1 u_2, \qquad y_2 = u_1 + \lambda_2 u_2$$
where $\lambda_1, \lambda_2$ are constants. For the right choice of constants, the equations will separate when written in terms of $y_1, y_2$.

$\lambda_{1,2}$ will have to solve a quadratic equation that involves a, b, c, d, hence generically you will get two roots.

 

[HallsofIvy]

Why do you refer to this as a "PDE" when you have only the single independent variable, x?

 

[galuoises]

Thank you so much Ben Niehoff!

Sorry for the notation for the variable x, HallsofIvy, I intended it is a vector
x$\equiv$(x,y,z)
and
Δ$\equiv$$\partial_{xx}+\partial_{yy}+\partial_{zz}$

 

[blue_raver22]

I am familiar with systems of second order linear coupled partial differential equations (PDEs) with constant coefficients. These types of systems are commonly encountered in many areas of science and engineering, and they can be challenging to solve.

To uncouple this system of PDEs, one approach could be to use the method of separation of variables. This involves assuming that the solution can be written as a product of functions, each depending on only one of the variables. By substituting this assumption into the system of PDEs and solving for each individual function, the system can be uncoupled into separate equations for each variable.

Another approach could be to use the method of eigenfunctions, where the solution is expressed as a linear combination of eigenfunctions of the differential operator. This method can also help to uncouple the system of PDEs into separate equations for each variable.

In terms of finding a solution in one, two, or three dimensions, the method of separation of variables and eigenfunctions can be applied in each case. However, the complexity of the solution may increase with the number of dimensions.

Using Green's functions can also be a useful tool for solving these types of systems of PDEs. Green's functions are solutions to the homogeneous version of the PDE with a delta function as a source term. By using these functions, the solution to the inhomogeneous system of PDEs can be expressed as an integral involving the Green's functions.

In summary, there are various mathematical methods that can be used to uncouple and solve systems of second order linear coupled PDEs with constant coefficients. I recommend consulting with a mathematician or using specialized software to find a solution that meets your specific needs and parameters.