Beltrami method

A method for solving wave equations with three spatial variables, proposed by E. Beltrami in 1864. The method is based on the fact that the wave equation on the surface of the characteristic cone can be converted with the aid of interior differential operators to an especially simple form which can be used for the solution $ u (x, t) $,

\begin{equation} \label{f:1} 4 \pi t _ {0} ^ {2} u (x _ {0} , t _ {0} ) = \ {\int\limits \int\limits } _ \Omega \left ( u + t _ {0} \frac{\partial u }{\partial n } \right ) d \sigma , \end{equation}

where $ \Omega $ is the sphere $ \{ {x } : {| x - x _ {0} | = t _ {0} } \} $, $ d \sigma $ is its surface element and $ \partial / \partial n $ denotes differentiation in the direction of the outward normal on the cone $ | x - x _ {0} | ^ {2} = (t - t _ {0} ) ^ {2} $. The method is applicable to the cases of inhomogeneous equations and to equations with any odd number of spatial variables.

Comments

The formula \eqref{f:1} above is called Beltrami's formula.

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[a1]	F. John, "Partial differential equations" , Springer (1978)