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Solving a nonlinear Volterra-Fredholm integro-differential equation with weakly singular kernels

Touati Sami, Lemita Samir, Ghiat Mourad, Aissaoui Mohamed-Zine

Fasciculi Mathematici

2019

nr 62

155--168

The aim of the current work is to investigate the numerical study of an integro-differential nonlinear Volterra-Fredholm equation with a weakly singular kernels. Our approximation technique is based on the product integration method in conjunction with an iterative scheme. The existence and uniqueness of the solution have been proved. We conclude the paper with numerical examples to illustrate the effectiveness of our method.

Bounded, asymptotically stable, and L1 solutions of caputo fractional differential equations

Islam M.N.

Opuscula Mathematica

2015

Vol. 35, no. 2

181--190

The existence of bounded solutions, asymptotically stable solutions, and L1 solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. The existence of bounded solutions are obtained employing Schauder's theorem, and then it is shown that these solutions are asymptotically stable by a definition found in [C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483]. Finally, the L1 properties of solutions are obtained using Liapunov's method