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Numerical Schemes for the Generalized Abel’s Integral Equations

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Abstract

This paper presents and studies two numerical approximations for the generalized Abel’s integral equations (GAIEs). Two numerical schemes such as linear scheme and quadratic scheme are proposed to solve GAIEs numerically. The error convergence of the presented schemes is also established where it is observed that the quadratic scheme achieves the convergence order up to 3. Some examples of GAIEs from literature are considered to perform the numerical investigations and the obtained numerical results are shown in tabular form. We analyze that the presented schemes work well and provide good numerical results. It is also observed that the accuracy in the numerical solutions can be achieved with smaller value of the step size. As the GAIEs reduces to the Abel’s integral equation of the first kind in special case, therefore a similar scheme could be developed to solve such equations.

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Acknowledgements

Authors are thankful to the reviewers for the comments to improve the manuscript. The first and third author acknowledge the financial supports from the Indian Institute of Technology (BHU) Varanasi, India and the University Grant Commission, New Delhi, India under the SRF schemes, respectively. The second author acknowledges the National Board of Higher Mathematics for providing the financial support for the research project (Ref. No. NBHM/R.P. 70/2015/Fresh/163).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, 221005, India

    Kamlesh Kumar, Rajesh K. Pandey & Shiva Sharma

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  1. Kamlesh Kumar
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  2. Rajesh K. Pandey
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  3. Shiva Sharma
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Correspondence to Rajesh K. Pandey.

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Kumar, K., Pandey, R.K. & Sharma, S. Numerical Schemes for the Generalized Abel’s Integral Equations. Int. J. Appl. Comput. Math 4, 68 (2018). https://doi.org/10.1007/s40819-018-0501-2

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  • DOI: https://doi.org/10.1007/s40819-018-0501-2

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