{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:41:16Z","timestamp":1760233276145,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,25]],"date-time":"2022-12-25T00:00:00Z","timestamp":1671926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Abel\u2019s integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel\u2019s integral equation and its solution using G\u03b1-transform, which is a symmetric relation between Laplace and Sumudu transforms. G\u03b1-transform, as defined via distribution space, is employed to establish a solution to Abel\u2019s integral equation, interpreted in the sense of distributions. As an application to the given theory, certain examples are given to demonstrate the efficiency and suitability of using the G\u03b1-transform method in solving integral equations.<\/jats:p>","DOI":"10.3390\/sym15010053","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T04:58:01Z","timestamp":1672117081000},"page":"53","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Certain Solutions of Abel\u2019s Integral Equations on Distribution Spaces via Distributional G\u03b1-Transform"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4621-2503","authenticated-orcid":false,"given":"Supaknaree","family":"Sattaso","sequence":"first","affiliation":[{"name":"Faculty of Science and Engineering, Kasetsart University, Chalermphrakiat Sakon Nakhon Province Campus, Sakon Nakhon 47000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0983-164X","authenticated-orcid":false,"given":"Hwajoon","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of IT Engineering, Kyungdong University, Yangju 11458, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8955-5552","authenticated-orcid":false,"given":"Shrideh","family":"Al-Omari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Amman 11134, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,25]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Wazwaz, A.M. (2011). Linear and Nonlinear Integral Equations Methods and Applications, Higher Education Press.","key":"ref_1","DOI":"10.1007\/978-3-642-21449-3"},{"key":"ref_2","first-page":"248","article-title":"The combined Laplace-Adomian method for handling singular integral equation of heat transfer","volume":"10","author":"Wazwaz","year":"2010","journal-title":"Int. J. Nonlinear Sci."},{"doi-asserted-by":"crossref","unstructured":"Gorenflo, R., and Vessella, S. (1991). Abel Integral Equations, Springer.","key":"ref_3","DOI":"10.1007\/BFb0084665"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1090\/qam\/42596","article-title":"Heat transfer between solids and gases under nonlinear boundary conditions","volume":"9","author":"Mann","year":"1951","journal-title":"Q. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1137\/0706055","article-title":"Inversion of Abel\u2019s integral equation by means of orthogonal polynomials","volume":"6","author":"Minerbo","year":"1969","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_6","first-page":"49","article-title":"A new operational method to solve Abel\u2019s and generalized Abel\u2019s integral equations","volume":"317","author":"Sadri","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_7","first-page":"1785","article-title":"A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method","volume":"218","author":"Bougoffa","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1748","DOI":"10.1016\/j.camwa.2008.04.003","article-title":"Approximate solution of Abel integral equation","volume":"56","author":"Huang","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_9","first-page":"4827","article-title":"On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials","volume":"217","author":"Gulsu","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.asej.2011.10.002","article-title":"Comparison Chebyshev wavelets method with BPFs method for solving Abels\u2019s integral equation","volume":"2","author":"Sohrabi","year":"2011","journal-title":"Ain Shams Eng. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.jksus.2014.09.004","article-title":"Solving Abel integral equations of first kind via fractional calculus","volume":"27","author":"Jahanshahi","year":"2015","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_12","first-page":"574","article-title":"Numerical solution of Abel\u2019s integral equation by using Legendre wavelets","volume":"175","author":"Yousefi","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_13","first-page":"656","article-title":"An efficient numerical method for solving Abel integral equation","volume":"227","author":"Yang","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"480528","DOI":"10.1155\/2011\/480528","article-title":"On the solution of distributional Abel Integral equation by distributional Sumudu transform","volume":"2011","author":"Loonker","year":"2011","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1666","DOI":"10.1016\/j.aml.2012.01.034","article-title":"A reliable treatment of Abel\u2019s second kind singular integral equations","volume":"25","author":"Khan","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1080\/09720502.2020.1861787","article-title":"Solving Volterra integral equation by using a new transformation","volume":"24","author":"Jaabar","year":"2021","journal-title":"J. Interdiscip. Math."},{"doi-asserted-by":"crossref","unstructured":"Ali, S., Ullah, A., Nonlaopon, K., and Akg\u00fcl, A. (2022). Analysis of Kink behaviour of KdV-mKdV equation under Caputo fractional operator with non-singular kernel. Symmetry, 14.","key":"ref_17","DOI":"10.3390\/sym14112316"},{"doi-asserted-by":"crossref","unstructured":"Fang, J., Nadeem, M., Habib, M., and Akg\u00fcl, A. (2022). Numerical investigation of nonlinear shock wave equations with fractional order in propagating disturbance. Symmetry, 14.","key":"ref_18","DOI":"10.3390\/sym14061179"},{"doi-asserted-by":"crossref","unstructured":"Sadiq, G., Ali, A., Ahmad, S., Nonlaopon, K., and Akg\u00fcl, A. (2022). Bright soliton behaviours of fractal fractional nonlinear good Boussinesq equation with nonsingular kernels. Symmetry, 14.","key":"ref_19","DOI":"10.3390\/sym14102113"},{"key":"ref_20","first-page":"179","article-title":"Distributional and tempered distributional diffraction Fresnel transforms and their extension to Boehmian spaces","volume":"30","year":"2013","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_21","first-page":"87","article-title":"On distributional Abel integral equation for distributional Elzaki transform","volume":"81","author":"Loonker","year":"2014","journal-title":"J. Indian Math. Soc."},{"key":"ref_22","first-page":"801","article-title":"On the distributional Mellin transformation and its extension to Boehmian spaces","volume":"6","year":"2011","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_23","first-page":"139","article-title":"Generalized functions for double Sumudu transformation","volume":"6","year":"2012","journal-title":"Int. J. Algebra"},{"key":"ref_24","first-page":"729","article-title":"On the application of natural transforms","volume":"85","year":"2013","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1762729","DOI":"10.1155\/2017\/1762729","article-title":"The intrinsic structure and properties of Laplace-typed integral transforms","volume":"2017","author":"Kim","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_26","first-page":"16083","article-title":"The solution of Laguerre\u2019s equation by using G-transform","volume":"12","author":"Kim","year":"2017","journal-title":"Int. J. Appl. Eng. Res."},{"key":"ref_27","first-page":"195","article-title":"Further properties of Laplace-typed integral transforms","volume":"28","author":"Sattaso","year":"2019","journal-title":"Dyn. Syst. Appl."},{"key":"ref_28","first-page":"257","article-title":"An application of generalized Laplace transform in PDEs","volume":"14","author":"Kim","year":"2019","journal-title":"Adv. Dyn. Syst. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.29020\/nybg.ejpam.v14i4.4066","article-title":"Analytical study for certain ordinary differential equations with variable coefficients via G\u03b1-transform","volume":"14","author":"Prasertsang","year":"2021","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"17859","DOI":"10.3934\/math.2022984","article-title":"On the application of G\u03b1 integral transform to nonlinear dynamical models with non-integer order derivatives","volume":"7","author":"Nuruddeen","year":"2022","journal-title":"AIMS Math."},{"unstructured":"Estrada, R., and Kanwal, R.P. (2012). Singular Integral Equations, Birkhauser.","key":"ref_31"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:48Z","timestamp":1760147448000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,25]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010053"],"URL":"https:\/\/doi.org\/10.3390\/sym15010053","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,25]]}}}