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Biharmonic Functions 📂Functions

Biharmonic Functions

Definition1

Let’s call Δ=∇2\Delta = \nabla^{2} the Laplacian. Δ2\Delta ^{2} is called the biharmonic operator or bilaplacian. The following equation is called the biharmonic equation.

Δ2f=0 \Delta^{2} f = 0

The solutions to the biharmonic equation are called biharmonic functions.

Explanation

Let’s say ∂i=∂∂xi\partial_{i} = \dfrac{\partial}{\partial x_{i}}. In the Cartesian coordinate system, since Δ=∑i∂i∂i\Delta = \sum\limits_{i} \partial_{i}\partial_{i},

Δ2f=∑j∑i∂j∂j∂i∂if \Delta^{2} f = \sum\limits_{j} \sum\limits_{i} \partial_{j}\partial_{j} \partial_{i}\partial_{i} f

Especially in 3 dimensions,

Δ2g=∑j=13∂j∂j(∂2f∂x2+∂2f∂y2+∂2f∂z2)=∂4f∂x4+∂4f∂y4+∂4f∂z4+2∂4f∂x2∂y2+2∂4f∂y2∂z2+2∂4f∂z2∂x2 \begin{align*} \Delta^{2}g &= \sum\limits_{j=1}^{3} \partial_{j}\partial_{j} \left( \dfrac{\partial^{2} f}{\partial x^{2}} + \dfrac{\partial^{2} f}{\partial y^{2}} + \dfrac{\partial^{2} f}{\partial z^{2}}\right) \\ &= \dfrac{\partial^{4} f}{\partial x^{4}} + \dfrac{\partial^{4} f}{\partial y^{4}} + \dfrac{\partial^{4} f}{\partial z^{4}} + 2\dfrac{\partial^{4} f}{\partial x^{2} \partial y^{2}} + 2\dfrac{\partial^{4} f}{\partial y^{2} \partial z^{2}} + 2\dfrac{\partial^{4} f}{\partial z^{2} \partial x^{2}} \\ \end{align*}

See Also

  1. https://en.wikipedia.org/wiki/Biharmonic_equation ↩︎