Mellin inversion method for the solution of a class of convolution integral equations. (English) Zbl 1080.45001
The author solves the well-known Mellin convolution equation
\[
\int_0^\infty u\left(\frac xy\right)f(y)\frac{dy}y=g(x) , \qquad x>0\tag{1}
\]
where \(g\) is a prescribed function, \(f\) is unknown and the kernel \(u\) is a particular function
\[
u(x)= [x^l(k+x\partial_x)]^n \{(ax^p+b)^{\alpha +\gamma n}(cx^q+d)^{\beta+\delta n}e^{-px^\eta}\}.
\]
Actually, equation (1) is solved simply: the Mellin transform
\[
H(s)=M\big[h(x);s\big]:=\int_0^\infty x^sh(x)\frac{dx}x,
\]
transforms the equation into the trivial equality \(U(s)F(s)=G(s)\), which is solved by the inverse Mellin transform. The author finds more or less explicit expression for the kernel transform \(U(s)=M\big[u(x);s\big]\) and considers several particular cases of the kernel, treated earlier by other authors.
Reviewer: Roland Duduchava (Tbilisi)
MSC:
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |