Abel integral equation

The integral equation

\begin{equation}\int\limits_0^x\frac{\phi(s)}{\sqrt{x-s}}ds=f(x),\label{1}\end{equation}

to which the solution of the Abel problem is reduced. The generalized Abel integral equation is the equation

\begin{equation}\int\limits_a^x\frac{\phi(s)}{(x-s)^\alpha}ds=f(x),\quad a\leq x\leq b,\label{2}\end{equation}

where $a>0$ and $0<\alpha<1$ are known constants, $f(x)$ is a known function and $\phi(x)$ is the unknown function. The expression $(x-s)^{-\alpha}$ is called the kernel of the Abel integral equation, or Abel kernel. An Abel integral equation belongs to the class of Volterra equations of the first kind (cf. Volterra equation). The equation

\begin{equation}\int\limits_a^b\frac{\phi(s)}{|x-s|^\alpha}ds=f(x),\quad a\leq x\leq b,\label{3}\end{equation}

is called Abel's integral equation with fixed limits.

If $f(x)$ is a continuously-differentiable function, then the Abel integral equation \eqref{2} has a unique continuous solution given by the formula

\begin{equation}\phi(x)=\frac{\sin\alpha\pi}\pi\frac d{dx}\int\limits_a^x\frac{f(t)dt}{(x-t)^{1-\alpha}},\label{4}\end{equation}

or

\begin{equation}\phi(x)=\frac{\sin\alpha\pi}\pi\left[\frac{f(a)}{(x-a)^{1-\alpha}}+\int\limits_a^x\frac{f'(t)dt}{(x-t)^{1-\alpha}}\right],\label{5}\end{equation}

which is the same thing. Formula \eqref{5} also gives the solution of an Abel integral equation \eqref{2} under more general assumptions [3], [4]. Thus, it has been shown [3] that if $f(x)$ is absolutely continuous on the interval $[a,b]$, the Abel integral equation \eqref{2} has the unique solution in the class of Lebesgue-integrable functions given by formula \eqref{5}. For the solution of the Abel integral equation \eqref{3} see [2]; see also [6].

References

[1]	M. Bôcher, "On the regions of convergence of power-series which represent two-dimensional harmonic functions" Trans. Amer. Math. Soc. , 10 (1909) pp. 271–278
[2]	T. Carleman, "Ueber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen" Math. Z. , 15 (1922) pp. 111–120
[3]	L. Tonelli, "Su un problema di Abel" Math. Ann. , 99 (1928) pp. 183–199
[4]	J.D. Tamarkin, "On integrable solutions of Abel's integral equation" Ann. of Math. (2) , 31 (1930) pp. 219–229
[5]	S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian)
[6]	F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)

Comments

The left-hand side of \eqref{2} is also known as a Riemann–Liouville fractional integral, where $\operatorname{Re}\alpha<1$, cf. [a1]. If one integrates in \eqref{1} and \eqref{2} from $x$ to $\infty$ and replaces $x-s$ by $s-x$, then one obtains left-hand sides which are known as Abel transform and Weyl fractional integral (cf. [a1]), respectively. This Abel transform is the case $\SL(2,\mathbf R)$ of the Abel transform on a real semi-simple Lie group, cf. [a2].

A general source for integral equations is [a3].

References

[a1]	A. Erdélyi, W. Magnus, F. Oberhetinger, F.G. Tricomi, "Tables of integral transforms" , II , McGraw-Hill (1954) pp. Chapt. 13
[a2]	R. Godement, "Introduction aux travaux de A. Selberg" , Sem. Bourbaki , 144 (1957)
[a3]	S. Fenyö, H.W. Stolle, "Theorie und Praxis der linearen Integralgleichungen" , 3 , Birkhäuser (1984) pp. Sect. 13.2.4