
doi: 10.1007/bf01057517
The author proposes a scheme for solving the problem \[ u''+2u'/x+\beta ^2u^\nu =0,\;u(0)=0, \quad u'(0)=0. \] After rather cumbersome manipulations he obtains the integral equation \[ u(x)=1+\lambda ^{-1}\int_{0}^{x}\Biggl[(c\lambda +\xi)\psi [a](\xi)-\varepsilon ^{-\mu }(c\xi)^{-1}\int_{0}^{\xi }\zeta ^2a(\zeta) d\zeta \Biggr]d\xi \] with \(a=-\varepsilon ^{\mu }\beta ^2u^{\nu-1 },\;\mu =1/(2-\nu),\;c,\lambda \) are parameters, and \(\psi[a](\xi)\) is a linear Volterra operator with complicated kernel. The author finds a ``natural'' Volterra equation \(u(x)=1-\int_{0}^{x}s^{-2}\int_{0}^{s}\beta ^2\xi ^2u^\nu (\xi) d\xi ds\) to be inefficient for applications since its kernel contains the singularity \(s^{-2}\). But the same singularity appears in \(\psi\) for the presence of the term \((\varepsilon ^\mu cx^2)^{-1}\int_{0}^{x}\xi ^2a(\xi) d\xi \). So, the advantages of the author's approach are not obvious.
Volterra integral equations, Fredholm equation, Volterra equation, Emden equation, Fredholm integral equations, Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations, Nonlinear ordinary differential equations and systems, Poisson integral, Theoretical approximation of solutions to ordinary differential equations
Volterra integral equations, Fredholm equation, Volterra equation, Emden equation, Fredholm integral equations, Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations, Nonlinear ordinary differential equations and systems, Poisson integral, Theoretical approximation of solutions to ordinary differential equations
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