Abstract
The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, we use the CESTAC method and the CADNA library. Using this method, we can find the optimal step of the method, the optimal approximation, the optimal error, and some of numerical instabilities. They are the main novelties of the DSA in comparison with the FPA. The error analysis of the method is proved. Furthermore, the main theorem of the CESTAC method is presented. Using this theorem we can apply a new termination criterion instead of the traditional absolute error. Several examples are approximated based on the FPA and the DSA. The numerical results show the applications and advantages of the DSA than the FPA.
Highlights
In 1823, the AIE was studied by Niels Abel for solving mathematical physics problems [1,2,3]
Singular integral equations are among the important and applicable kinds of integral equations which have been solved by many authors [5,6,7,8]
We have used the CESTAC method and the CADNA library, which are based on the DSA, to validate the obtained numerical results
Summary
In 1823, the AIE was studied by Niels Abel for solving mathematical physics problems [1,2,3]. The AIEs are singular form of the Volterra IEs. Singular integral equations are among the important and applicable kinds of integral equations which have been solved by many authors [5,6,7,8]. Singular integral equations are among the important and applicable kinds of integral equations which have been solved by many authors [5,6,7,8] This problem has many applications in various areas such as simultaneous dual relations [9], stellar winds [10], water wave [11], spectroscopic data [12], and others [1,13,14].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
