Difference between revisions of "Fundamental solution"
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$ x \in \mathbf R ^ {n} $, | $ x \in \mathbf R ^ {n} $, | ||
with coefficients of class $ C ^ \infty $, | with coefficients of class $ C ^ \infty $, | ||
− | in the form of a function $ | + | in the form of a function $ E ( x, y) $ |
that satisfies, for fixed $ y \in \mathbf R ^ {n} $, | that satisfies, for fixed $ y \in \mathbf R ^ {n} $, | ||
the equation | the equation | ||
$$ | $$ | ||
− | L | + | L E ( x, y) = \delta ( x - y),\ \ |
x \neq y, | x \neq y, | ||
$$ | $$ | ||
Line 40: | Line 40: | ||
$$ | $$ | ||
− | + | E ( x, y) = | |
− | + | \begin{cases} \displaystyle | |
− | \begin{ | ||
\left [ \sum _ {i, j = 1 } ^ { n } | \left [ \sum _ {i, j = 1 } ^ { n } | ||
A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) | A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) | ||
− | \right ] ^ {( 2 - n)/2 } , | + | \right ] ^ {( 2 - n)/2 } , & n > 2 , |
\\ | \\ | ||
− | + | \displaystyle \log \left [ \sum _ {i, j = 1 } ^ { n } | |
− | A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] , | + | A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] , & n = 2 , |
− | \end{ | + | \end{cases} |
− | |||
$$ | $$ | ||
Line 64: | Line 62: | ||
====Comments==== | ====Comments==== | ||
− | Fundamental solutions are also used in the study of Cauchy problems (cf. [[Cauchy problem|Cauchy problem]]) for hyperbolic and parabolic equations. The name "elementary solution of a linear partial differential | + | Fundamental solutions are also used in the study of Cauchy problems (cf. [[Cauchy problem|Cauchy problem]]) for hyperbolic and parabolic equations. The name "elementary solution of a linear partial differential equation" is also used. |
See also [[Green function|Green function]]. | See also [[Green function|Green function]]. |
Latest revision as of 08:49, 13 May 2022
of a linear partial differential equation
A solution of a partial differential equation $ Lu ( x) = 0 $, $ x \in \mathbf R ^ {n} $, with coefficients of class $ C ^ \infty $, in the form of a function $ E ( x, y) $ that satisfies, for fixed $ y \in \mathbf R ^ {n} $, the equation
$$ L E ( x, y) = \delta ( x - y),\ \ x \neq y, $$
which is interpreted in the sense of the theory of generalized functions, where $ \delta $ is the delta-function. There is a fundamental solution for every partial differential equation with constant coefficients, and also for arbitrary elliptic equations. For example, for the elliptic equation
$$ \sum _ {i, j = 1 } ^ { n } a _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } = 0 $$
with constant coefficients $ a _ {ij} $ forming a positive-definite matrix $ a $, a fundamental solution is provided by the function
$$ E ( x, y) = \begin{cases} \displaystyle \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] ^ {( 2 - n)/2 } , & n > 2 , \\ \displaystyle \log \left [ \sum _ {i, j = 1 } ^ { n } A _ {ij} ( x _ {i} - y _ {i} ) ( x _ {j} - y _ {j} ) \right ] , & n = 2 , \end{cases} $$
where $ A _ {ij} $ is the cofactor of $ a _ {ij} $ in the matrix $ a $.
Fundamental solutions are widely used in the study of boundary value problems for elliptic equations.
References
[1] | V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) |
[2] | L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) |
[3] | F. John, "Plane waves and spherical means: applied to partial differential equations" , Interscience (1955) |
Comments
Fundamental solutions are also used in the study of Cauchy problems (cf. Cauchy problem) for hyperbolic and parabolic equations. The name "elementary solution of a linear partial differential equation" is also used.
See also Green function.
References
[a1] | A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) |
[a2] | O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian) |
[a3] | O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, "Linear and quasilinear parabolic equations" , Amer. Math. Soc. (1968) (Translated from Russian) |
[a4] | L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1957–1959) |
[a5] | I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian) |
Fundamental solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_solution&oldid=47027