Abelian integral equation

The Abelian integral equation is a special volterra integral equation of the first kind. It has the form:

${\displaystyle f(h)=\int _{0}^{h}{\frac {u(y)}{\sqrt {h-y}}}\,\mathrm {d} y}$.

where is given and the function sought is. The volterra integral equation of the first kind is more general than ${\displaystyle f}$ ${\displaystyle u}$

${\displaystyle f(h)=\int _{0}^{h}u(y)K(h,y)\mathrm {d} y}$

defined with a core function . Especially for core functions of the form ${\displaystyle K}$ ${\displaystyle K}$

${\displaystyle K(h,y)={\frac {1}{{(h-y)}^{a}}}}$

with there is a general solution method by tracing back to the formula for Euler's beta function . It results: ${\displaystyle 0<a<1}$

${\displaystyle u(y)={\frac {\sin(\pi a)}{\pi }}{\frac {\mathrm {d} }{\mathrm {d} y}}\int _{0}^{y}{(y-x)}^{(a-1)}f(x)\mathrm {d} x}$

Is in the Abelian integral equation . ${\displaystyle a={\tfrac {1}{2}}}$

The relationship between the functions and expressed by the generalization of the Abel integral equation for is also referred to as the Abel transformation , that is , the Abel transform of . The formula for provided by the above-mentioned solution method for gives the inverse formula of the Abel transformation. ${\displaystyle 0<a<1}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle 0<a<1}$ ${\displaystyle u}$

Application and history

In 1823 Niels Henrik Abel was one of the first to investigate integral equations in connection with a mechanical problem. Until then, mechanics was mainly determined by differential equations . Abel considered a body moving under the influence of gravity along a curve in a vertical plane from to (0,0). ${\displaystyle P1(x_{0},y_{0})}$

Based on the classic formula for speed

${\displaystyle v={\frac {\mathrm {d} s}{\mathrm {d} t}}={\sqrt {2g(y_{0}-y)}}}$

the fall time is obtained through integration over the route

${\displaystyle T(y_{0})=\int _{0}^{l}{\frac {\mathrm {d} s}{\sqrt {2g(y_{0}-y)}}}}$.

By substituting to the final form: ${\displaystyle s=f(y)\,\Rightarrow \,f'(y)=\mathrm {d} s/\mathrm {d} y\,\Rightarrow \,\mathrm {d} s=f'(y)\cdot \mathrm {d} y}$

${\displaystyle T(y_{0})={\frac {1}{\sqrt {2g}}}\int _{0}^{y_{0}}{\frac {f'(y)}{\sqrt {y_{0}-y}}}\,\mathrm {d} y}$.

If you know the curve , you get the fall time. Abel also considered the opposite problem: if the fall time is given, one obtains an Abelian integral equation for the unknown function . ${\displaystyle f(y)}$ ${\displaystyle f'(y)}$

There are further applications of the Abel integral equation or the Abel transformation in astrophysics, in geophysics ( Herglotz - Wiechert method of determining the speed distribution from arrival times of seismic waves) and, for example, in determining the atmospheric data of planets by radio Occultation . As in the original application, these are typical inverse problems .

literature

• Rudolf Gorenflo, Sergio Vessella Abel Integral Equations- Analysis and Applications , Springer, Lecture Notes in Mathematics, Vol. 1461, 1991
• Flügge methods of mathematical physics , vol. 1, Springer Verlag, p. 130
• Tricomi Integral Equations , Interscience, 1957, pp. 39f
• Rudolf Rothe On Abel's integral equation , Mathematische Zeitschrift, Volume 33, 1931, online

Web links

• Eric W. Weisstein : Abel Transform . In: MathWorld (English).