The biharmonic equation is the equation produced by applying the Euler-Lagrange equation to the simplified thin plate energy functional X _ { uu } ^ 2 + 2X _ { uv } ^ 2 + X _ { vv } ^ 2.
12.
Where \ nabla ^ 4 is the fourth power of the del operator and the square of the Laplacian operator \ nabla ^ 2 ( or \ Delta ), and it is known as the "'biharmonic operator "'or the "'bilaplacian operator " '.
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Letting \ rho ^ 2 = x ^ 2 + y ^ 2 + z ^ 2 + h ^ 2 and computing \ Delta ( 1 / \ rho ) =-3h ^ 2 / \ rho ^ 5 again indicates that the right hand side of the PDEs for the biharmonic and triharmonic RBFs are Dirac delta functions.
