Difference between revisions of "Beltrami method"
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
(gather refs) |
||
| Line 13: | Line 13: | ||
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) $, | 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} ) = \ | 4 \pi t _ {0} ^ {2} u (x _ {0} , t _ {0} ) = \ | ||
{\int\limits \int\limits } _ \Omega | {\int\limits \int\limits } _ \Omega | ||
| Line 21: | Line 22: | ||
\right ) d \sigma , | \right ) d \sigma , | ||
| − | + | \end{equation} | |
where $ \Omega $ | where $ \Omega $ | ||
| Line 29: | Line 30: | ||
denotes differentiation in the direction of the outward normal on the cone $ | x - x _ {0} | ^ {2} = (t - t _ {0} ) ^ {2} $. | 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. | The method is applicable to the cases of inhomogeneous equations and to equations with any odd number of spatial variables. | ||
| − | |||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The formula above is called Beltrami's formula. | + | The formula \eqref{f:1} above is called Beltrami's formula. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. John, "Partial differential equations" , Springer (1978)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience (1965) (Translated from German)</TD></TR> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> F. John, "Partial differential equations" , Springer (1978)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 05:53, 30 May 2023
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) |
Beltrami method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beltrami_method&oldid=46009