Biharmonic Equation with Coefficients

[skrieger]

I'm working on a problem and I've run into a differential equation that very strongly resembles the biharmonic equation but is fundamentally different:

0 = a(∂^4 ψ/∂x^4) + b(∂^4 ψ/∂y^4) + c(∂^4 ψ/∂x^2 ∂y^2).

where a,b,c are scalar coefficients.

Any ideas? I think these were originally solved using eigenfunction expansions so that's my first plan, but if anybody knows anything easier...

I tried it in Mathematica with poor results, but DSolve tends to struggle with the biharmonic equation in Cartesian coordinates anyway.

 

[gtoguy97]

One approach that you could take is to use the method of separation of variables. This involves making an ansatz of the form ψ(x,y) = X(x)Y(y) and substituting this into the differential equation. This will lead to two separate equations in x and y, which can be solved separately. The general solution will then be a linear combination of the individual solutions. Another approach that may work is to transform the coordinates to polar coordinates. This can sometimes simplify the equation and make it easier to solve.