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For nonlinear challenges, the Quasi-Newton linearization formula is applied to successfully eliminate non-linearity from the model. To evaluate the technique\u2019s performance, we analyze key metrics such as maximum absolute errors, root mean square errors, and computational convergence rates with varying numbers of collocation points. The proposed approach consistently outperforms existing methods, particularly in situations involving abrupt changes in the solution space or discontinuities between boundary and initial conditions, delivering stable solutions in these critical scenarios. The combination of strong theoretical foundations and computational stability, along with excellent convergence rates and comprehensive numerical studies, firmly validates the accuracy and versatility of this method, confirming its wide range of applications.<\/jats:p>","DOI":"10.1007\/s12190-025-02551-8","type":"journal-article","created":{"date-parts":[[2025,6,22]],"date-time":"2025-06-22T16:11:11Z","timestamp":1750608671000},"page":"6585-6620","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Robust numerical techniques for modeling telegraph equations in multi-scale and heterogeneous environments"],"prefix":"10.1007","volume":"71","author":[{"given":"Muhammad","family":"Asif","sequence":"first","affiliation":[]},{"given":"Faisal","family":"Bilal","sequence":"additional","affiliation":[]},{"given":"Nadeem","family":"Haider","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3303-0623","authenticated-orcid":false,"given":"Fahd","family":"Jarad","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,22]]},"reference":[{"key":"2551_CR1","first-page":"100773","volume":"11","author":"M. 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