Biharmonic equation explained

In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.

Notation

It is written as$$\backslash nabla^4\; \backslash varphi\; =\; 0$$or$$\backslash nabla^2\; \backslash nabla^2\; \backslash varphi\; =\; 0$$or$$\backslash Delta^2\; \backslash varphi\; =\; 0$$where

\nabla ^{4} , which is the fourth power of the del operator and the square of the Laplacian operator \nabla ^{2} (or \Delta ), is known as the biharmonic operator or the bilaplacian operator. In Cartesian coordinates, it can be written in n dimensions as:$$\backslash nabla^4\; \backslash varphi\; =\; \backslash sum\_^n\backslash sum\_^n\; \backslash partial\_i\backslash partial\_i\backslash partial\_j\backslash partial\_j\; \backslash varphi=\; \backslash left(\backslash sum\_^n\; \backslash partial\_i\backslash partial\_i\backslash right)\; \backslash left(\backslash sum\_^n\; \backslash partial\_j\backslash partial\_j\backslash right)\; \backslash varphi.$$Because the formula here contains a summation of indices, many mathematicians prefer the notation \Delta ^{2} over \nabla ^{4} because the former makes clear which of the indices of the four nabla operators are contracted over.

For example, in three dimensional Cartesian coordinates the biharmonic equation has the form$$+\; +\; +\; 2\; +2\; +2\; =\; 0.$$As another example, in n-dimensional real coordinate space without the origin

\left( R ^{n} \setminus 0 \right) ,$$\backslash nabla^4\; \backslash left(\backslash right)\; =$$where$$r\; =\; \backslash sqrt.$$which shows, for n=3 and n=5 only, (1)/(r) is a solution to the biharmonic equation.

A solution to the biharmonic equation is called a biharmonic function. Any harmonic function is biharmonic, but the converse is not always true.

In two-dimensional polar coordinates, the biharmonic equation is$$\backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\; \backslash left(\backslash frac\; \backslash frac\; \backslash left(r\; \backslash frac\backslash right)\; \backslash right)\backslash right)\; +\; \backslash frac\; \backslash frac\; +\; \backslash frac\; \backslash frac\; -\; \backslash frac\; \backslash frac\; +\; \backslash frac\; \backslash frac\; =\; 0$$which can be solved by separation of variables. The result is the Michell solution.

2-dimensional space

The general solution to the 2-dimensional case is$$x\; v(x,y)\; -\; y\; u(x,y)\; +\; w(x,y)$$where

u(x,y) , v(x,y) and w(x,y) are harmonic functions and v(x,y) is a harmonic conjugate of u(x,y) .

Just as harmonic functions in 2 variables are closely related to complex analytic functions, so are biharmonic functions in 2 variables. The general form of a biharmonic function in 2 variables can also be written as$$\backslash operatorname(\backslash barf(z)\; +\; g(z))$$where

f(z) and g(z) are analytic functions.

See also

• Harmonic function

References

• Eric W Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002. .
• S I Hayek, Advanced Mathematical Methods in Science and Engineering, Marcel Dekker, 2000. .
• Book: J P Den Hartog . Advanced Strength of Materials . Courier Dover Publications . Jul 1, 1987 . 0-486-65407-9.