Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients

Abstract

Applying Babenko’s approach, we construct solutions for the generalized Abel’s integral equations of the second kind with variable coefficients on R and R n , and show their convergence and stability in the spaces of Lebesgue integrable functions, with several illustrative examples.

Uniqueness of the Hadamard-type integral equations

“…Podlubny [ 16 ] also provided interesting applications to solving certain partial differential equations for heat and mass transfer by Babenko’s method. Recently, Li and Plowman [ 18 ] and Li [ 19 ] studied the generalized Abel’s integral equations of the second kind with variable coefficients by Babenko’s technique.…”

On the nonlinear Hadamard-type integro-differential equation

“…Clearly, it is always necessary to show convergence of the series obtained as solutions. Recently, Li studied the generalized Abel’s integral equations of the first [ 7 ] and second kind with variable coefficients by Babenko’s technique [ 8 – 10 ].…”

On the Boundary Value Problem of Nonlinear Fractional Integro-Differential Equations

“…BA [20] is a useful instrument in solving differential and integral equations with initial conditions by treating bounded integral operators as normal variables. The method itself is close to the Laplace transform while dealing with differential equations with constant coefficients, but it can be applied to differential and integral equations with variable coefficients [21,22]. Evidently, it is always necessary to prove the convergence of solution series, otherwise the solution is not well-defined.…”