Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems.

Solution of Abel Integral Equation Using Differential Transform Method

The application of fractional differential transform method, developed for differential equations of fractional order, are extended to derive exact analytical solutions of fractional order Abel integral equations. The fractional integrations are described in the Riemann-Liouville sense and fractional derivatives are described in the Caputo sense. Abel integral equation occurs in the mathematical modeling of various problems in physics, astrophysics, solid mechanics and applied sciences. An analytic technique for solving Abel integral equation of first kind by the proposed method is introduced here. Also illustrative examples with exact solutions are considered to show the validity and applicability of the proposed method. Abel integral equation, Differential transform method, Fractional differential transform method.

Hybrid Rational Haar Wavelet and Block Pulse Functions Method for Solving Population Growth Model and Abel Integral Equations

We use a computational method based on rational Haar wavelet for solving nonlinear fractional integro-differential equations. To this end, we apply the operational matrix of fractional integration for rational Haar wavelet. Also, to show the efficiency of the proposed method, we solve particularly population growth model and Abel integral equations and compare the numerical results with the exact solutions.