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John's equation

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(Redirected from John equation) Ultrahyperbolic partial differential equation

John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after German-American mathematician Fritz John.

Given a function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } with compact support the X-ray transform is the integral over all lines in R n . {\displaystyle \mathbb {R} ^{n}.} We will parameterise the lines by pairs of points x , y ∈ R n , {\displaystyle x,y\in \mathbb {R} ^{n},} x ≠ y {\displaystyle x\neq y} on each line and define u {\displaystyle u} as the ray transform where

u ( x , y ) = ∫ − ∞ ∞ f ( x + t ( y − x ) ) d t . {\displaystyle u(x,y)=\int \limits _{-\infty }^{\infty }f(x+t(y-x))dt.}

Such functions u {\displaystyle u} are characterized by John's equations

∂ 2 u ∂ x i ∂ y j − ∂ 2 u ∂ y i ∂ x j = 0 {\displaystyle {\frac {\partial ^{2}u}{\partial x_{i}\partial y_{j}}}-{\frac {\partial ^{2}u}{\partial y_{i}\partial x_{j}}}=0}

which is proved by Fritz John for dimension three and by Kurusa for higher dimensions.

In three-dimensional x-ray computerized tomography John's equation can be solved to fill in missing data, for example where the data is obtained from a point source traversing a curve, typically a helix.

More generally an ultrahyperbolic partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form

∑ i , j = 1 2 n a i j ∂ 2 u ∂ x i ∂ x j + ∑ i = 1 2 n b i ∂ u ∂ x i + c u = 0 {\displaystyle \sum \limits _{i,j=1}^{2n}a_{ij}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum \limits _{i=1}^{2n}b_{i}{\frac {\partial u}{\partial x_{i}}}+cu=0}

where n ≥ 2 {\displaystyle n\geq 2} , such that the quadratic form

∑ i , j = 1 2 n a i j ξ i ξ j {\displaystyle \sum \limits _{i,j=1}^{2n}a_{ij}\xi _{i}\xi _{j}}

can be reduced by a linear change of variables to the form

∑ i = 1 n ξ i 2 − ∑ i = n + 1 2 n ξ i 2 . {\displaystyle \sum \limits _{i=1}^{n}\xi _{i}^{2}-\sum \limits _{i=n+1}^{2n}\xi _{i}^{2}.}

It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of u can be extended to a solution.

References

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