Difference between revisions of "Beltrami method"
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| − | + | 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) $, | |
| + | |||
| + | $$ | ||
| + | 4 \pi t _ {0} ^ {2} u (x _ {0} , t _ {0} ) = \ | ||
| + | {\int\limits \int\limits } _ \Omega | ||
| + | \left ( | ||
| + | u + t _ {0} | ||
| + | \frac{\partial u }{\partial n } | ||
| + | |||
| + | \right ) d \sigma , | ||
| + | $$ | ||
| + | |||
| + | 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. | ||
====References==== | ====References==== | ||
<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></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></table> | ||
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====Comments==== | ====Comments==== | ||
Revision as of 10:58, 29 May 2020
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) $,
$$ 4 \pi t _ {0} ^ {2} u (x _ {0} , t _ {0} ) = \ {\int\limits \int\limits } _ \Omega \left ( u + t _ {0} \frac{\partial u }{\partial n } \right ) d \sigma , $$
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.
References
| [1] | R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) |
Comments
The formula above is called Beltrami's formula.
References
| [a1] | F. John, "Partial differential equations" , Springer (1978) |
How to Cite This Entry:
Beltrami method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beltrami_method&oldid=46009
Beltrami method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beltrami_method&oldid=46009
This article was adapted from an original article by Sh.A. AlimovV.A. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article