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The series expansions and blow-up time estimation for the solutions of convolution Volterra-Hammerstein integral equations

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Abstract

The blow-up phenomenon may occur in nonlinear integral equations and differential equations, which has important significance for the simulation of some practical problems. This paper is devoted to predicting the blow-up time for convolution Volterra-Hammerstein integral equations using the series expansion of the solution about the origin. First, the finite-term series expansion for the solution about the origin is obtained via Picard iteration and symbolic computation, which has high accuracy near the origin. Second, the Padé approximation of the series expansion is made to extend the convergence region of the series. The blow-up time of the solution is estimated by calculating the smallest positive root of the denominator of the rational function. Third, an integral transform is performed to the series expansion to convert the branch point to a pole of order one at the blow-up time, so the predicted accuracy of the blow-up time is improved remarkably. Numerical examples illustrate that the proposed method is very efficient for estimating the blow-up time of Volterra integral equations and some kinds of differential equations.

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Acknowledgements

The authors are very grateful to Editors and Referees for their valuable suggestions, which improve the quality of the paper significantly.

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 11971241 and the Program for Innovative Research Team in Universities of Tianjin under Grant No. TD13-5078.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Yuxuan Wang, Tongke Wang & Huan Lian

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  1. Yuxuan Wang
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  2. Tongke Wang
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  3. Huan Lian
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All authors contributed to the study conception and design. The manuscript was written by Yuxuan Wang and Tongke Wang. All authors read and approved the final manuscript.

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Correspondence to Tongke Wang.

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Wang, Y., Wang, T. & Lian, H. The series expansions and blow-up time estimation for the solutions of convolution Volterra-Hammerstein integral equations. Numer Algor 95, 637–663 (2024). https://doi.org/10.1007/s11075-023-01584-z

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