{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:23:23Z","timestamp":1760228603775,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,17]],"date-time":"2022-05-17T00:00:00Z","timestamp":1652745600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100016152","name":"Yayasan Universiti Teknologi PETRONAS under YUTP-Fundamental Research","doi-asserted-by":"publisher","award":["015LC0-083","015MC0-033"],"award-info":[{"award-number":["015LC0-083","015MC0-033"]}],"id":[{"id":"10.13039\/501100016152","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Universiti Teknologi PETRONAS\u2014Universiti Malaysia Pahang Matching","award":["015LC0-083","015MC0-033"],"award-info":[{"award-number":["015LC0-083","015MC0-033"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Initially, the concept of the complexity reduction approach was applied to solve symmetry algebraic systems that were generated from the discretization of the partial differential equations. Consequently, in this paper, the effectiveness of a complexity reduction approach based on half- and quarter-sweep iteration concepts for solving linear Fredholm integral equations of the second kind is investigated. Half- and quarter-sweep iterative methods are applied to solve dense linear systems generated from the discretization of the second kind of linear Fredholm integral equations using a repeated modified trapezoidal (RMT) scheme. The formulation and implementation of the proposed methods are presented. In addition, computational complexity analysis and numerical results of test examples are also included to verify the performance of the proposed methods.<\/jats:p>","DOI":"10.3390\/sym14051017","type":"journal-article","created":{"date-parts":[[2022,5,17]],"date-time":"2022-05-17T04:04:19Z","timestamp":1652760259000},"page":"1017","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Complexity Reduction Approach for Solving Second Kind of Fredholm Integral Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Mohana Sundaram","family":"Muthuvalu","sequence":"first","affiliation":[{"name":"Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Seri Iskandar 32610, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4629-0483","authenticated-orcid":false,"given":"Elayaraja","family":"Aruchunan","sequence":"additional","affiliation":[{"name":"Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia"}]},{"given":"Majid Khan Majahar","family":"Ali","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, Gelugor 11800, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1195-0955","authenticated-orcid":false,"given":"Jackel Vui Lung","family":"Chew","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah Labuan International Campus, Labuan 87000, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4556-602X","authenticated-orcid":false,"given":"Andang","family":"Sunarto","sequence":"additional","affiliation":[{"name":"Tadris Matematika, IAIN Bengkulu, Bengkulu 65144, Indonesia"}]},{"given":"Ramoshweu","family":"Lebelo","sequence":"additional","affiliation":[{"name":"Department of Education, Vaal University of Technology, Private Bag X021, Vanderbijlpark 1911, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9538-6588","authenticated-orcid":false,"given":"Jumat","family":"Sulaiman","sequence":"additional","affiliation":[{"name":"Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu 88400, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,17]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Atkinson, K.E. 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Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1602","DOI":"10.1080\/00207160802406523","article-title":"The method of moments for solution of second kind Fredholm integral equations based on B-spline wavelets","volume":"87","author":"Maleknejad","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_6","first-page":"508","article-title":"Solving second kind Fredholm integral equations by periodic wavelet Galerkin method","volume":"175","author":"Xiao","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1016\/j.jco.2007.03.004","article-title":"Lattice-Nystr\u00f6m method for Fredholm integral equations of the second kind with convolution type kernels","volume":"23","author":"Dick","year":"2007","journal-title":"J. Complex."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1090\/S0025-5718-1994-1218345-X","article-title":"Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind","volume":"62","author":"Kaneko","year":"1994","journal-title":"Math. Comput."},{"key":"ref_9","first-page":"2943","article-title":"Quarter-Sweep Arithmetic Mean (QSAM) iterative method for second kind linear Fredholm integral equations","volume":"4","author":"Muthuvalu","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"174","DOI":"10.22436\/jmcs.001.03.06","article-title":"Comparisons of quadrature schemes with Arithmetic Mean iterative method for second kind linear Fredholm integral equations","volume":"1","author":"Muthuvalu","year":"2010","journal-title":"J. Math. Comput. Sci."},{"key":"ref_11","first-page":"5442","article-title":"Half-Sweep Arithmetic Mean method with composite trapezoidal scheme for solving linear Fredholm integral equations","volume":"217","author":"Muthuvalu","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_12","first-page":"85","article-title":"Numerical solution of second kind linear Fredholm integral equations using QSGS iterative method with high-order Newton-Cotes quadrature schemes","volume":"5","author":"Muthuvalu","year":"2011","journal-title":"Malays. J. Math. Sci."},{"key":"ref_13","first-page":"980","article-title":"Solving linear integral equations of the second kind with repeated modified trapezoid quadrature method","volume":"189","author":"Heidari","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1080\/00207169108803958","article-title":"The four point Explicit Decoupled Group (EDG) method: A fast Poisson solver","volume":"38","author":"Abdullah","year":"1991","journal-title":"Int. J. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1080\/00207160008805020","article-title":"An efficient four points Modified Explicit Group Poisson solver","volume":"76","author":"Othman","year":"2000","journal-title":"Int. J. Comput. Math."},{"key":"ref_16","first-page":"425","article-title":"Rotated Krylov preconditioned iterative schemes in the solution of convection-diffusion equations","volume":"206","author":"Ali","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_17","first-page":"75","article-title":"Half-Sweep Geometric Mean method for solution of linear Fredholm equations","volume":"24","author":"Muthuvalu","year":"2008","journal-title":"Matematika"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/j.parco.2007.10.004","article-title":"Performance analysis of explicit group parallel algorithms for distributed memory multicomputer","volume":"34","author":"Ng","year":"2008","journal-title":"Parallel Comput."},{"key":"ref_19","first-page":"45","article-title":"Half- and quarter-sweeps implementation of Finite-Difference Time-Domain method","volume":"3","author":"Nusi","year":"2009","journal-title":"Malays. J. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/00207169808804726","article-title":"The halfsweeps multigrid method as a fast multigrid Poisson solver","volume":"69","author":"Othman","year":"1998","journal-title":"Int. J. Comput. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.procs.2010.04.041","article-title":"MEGSOR iterative method for the triangle element solution of 2D Poisson equations","volume":"1","author":"Sulaiman","year":"2010","journal-title":"Procedia Comput. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.cam.2007.07.002","article-title":"The preconditioned Gauss-Seidel method faster than the SOR method","volume":"219","author":"Niki","year":"2008","journal-title":"J. Comput. Appl. Math."},{"unstructured":"Isaacson, E., and Keller, H.B. (1994). Analysis of Numerical Methods, Dover Publications.","key":"ref_23"},{"key":"ref_24","first-page":"946","article-title":"A new mechanical algorithm for solving the second kind of Fredholm integral equation","volume":"172","author":"Wang","year":"2006","journal-title":"Appl. Math. Comput."},{"doi-asserted-by":"crossref","unstructured":"Ahmad, B., Alruwaily, Y., Alsaedi, A., and Ntouyas, S.K. (2021). Riemann-Stieltjes integral boundary value problems involving mixed Riemann-Liouville and Caputo fractional derivatives. J. Nonlinear Funct. Anal., 11.","key":"ref_25","DOI":"10.23952\/jnfa.2021.11"},{"key":"ref_26","first-page":"1","article-title":"Positive solutions of a fractional integro-differential equation with integral boundary conditions","volume":"2020","author":"Lachouri","year":"2020","journal-title":"Commun. Optim. Theory"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1017\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:11:36Z","timestamp":1760137896000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/1017"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,17]]},"references-count":26,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14051017"],"URL":"https:\/\/doi.org\/10.3390\/sym14051017","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,17]]}}}