International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/3765 2020 2020 Application of Neutrosophic Pentagonal Controlled Metric Space via Orthogonality in Traffic Flow Network Using Integral Equation Research Scholar, P.G. and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivagangai, Affiliated to Alagappa University, Karaikudi, Tamilnadu, India M. M. P.G. and Associate Professor, Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivagangai, Affiliated to Alagappa University, Karaikudi, Tamilnadu, India M. Jeyaraman Faculty of Natural Sciences, Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa Rahul Shukla In this paper, we researched and confirmed some of the axioms of NOPCMS (Neutrosophic orthogonal pentagonal controlled metric space). We used NOPCMS to translate the Banach contraction principle in the formerly defined spaces. Several cases were numerically evaluated, and certain findings were supported, in or- der to review what we found. Furthermore, by demonstrating their existence with a unique and comprehensive solution, we deliver proof of usage and implementation. 2025 2025 105 131 10.54216/IJNS.260308 https://www.americaspg.com/articleinfo/21/show/3765