{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T02:04:44Z","timestamp":1767924284881,"version":"3.49.0"},"reference-count":55,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T00:00:00Z","timestamp":1723161600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this research article, we demonstrate the generalized expansion method to investigate nonlinear integro-partial differential equations via an efficient mathematical method for generating abundant exact solutions for two types of applicable nonlinear models. Moreover, stability analysis and modulation instability are also studied for two types of nonlinear models in this present investigation. These analyses have several applications including analyzing control systems, engineering, biomedical engineering, neural networks, optical fiber communications, signal processing, nonlinear imaging techniques, oceanography, and astrophysical phenomena. To study nonlinear PDEs analytically, exact traveling wave solutions are in high demand. In this paper, the (1 + 1)-dimensional integro-differential Ito equation (IDIE), relevant in various branches of physics, statistical mechanics, condensed matter physics, quantum field theory, the dynamics of complex systems, etc., and also the (2 + 1)-dimensional integro-differential Sawda\u2013Kotera equation (IDSKE), providing insights into the several physical fields, especially quantum gravity field theory, conformal field theory, neural networks, signal processing, control systems, etc., are investigated to obtain a variety of wave solutions in modern physics by using the mentioned method. Since abundant exact wave solutions give us vast information about the physical phenomena of the mentioned models, our analysis aims to determine various types of traveling wave solutions via a different integrable ordinary differential equation. Furthermore, the characteristics of the obtained new exact solutions have been illustrated by some figures. The method used here is candid, convenient, proficient, and overwhelming compared to other existing computational techniques in solving other current world physical problems. This article provides an exemplary practice of finding new types of analytical equations.<\/jats:p>","DOI":"10.3390\/computation12080161","type":"journal-article","created":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T05:19:38Z","timestamp":1723439978000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Exploring New Traveling Wave Solutions to the Nonlinear Integro-Partial Differential Equations with Stability and Modulation Instability in Industrial Engineering"],"prefix":"10.3390","volume":"12","author":[{"given":"J. R. M.","family":"Borhan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Jashore University of Science and Technology, Jashore 7408, Bangladesh"}]},{"given":"I.","family":"Abouelfarag","sequence":"additional","affiliation":[{"name":"Mathematics Department, Khurmah University College, Taif University, Taif 21944, Saudi Arabia"}]},{"given":"K.","family":"El-Rashidy","sequence":"additional","affiliation":[{"name":"Technology and Science Department, Ranyah University College, Taif University, Taif 21944, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6277-8334","authenticated-orcid":false,"given":"M. Mamun","family":"Miah","sequence":"additional","affiliation":[{"name":"Division of Mathematical and Physical Sciences, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan"},{"name":"Department of Mathematics, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh"}]},{"given":"M. Ashik","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Khulna Agricultural University, Khulna 9100, Bangladesh"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0526-0015","authenticated-orcid":false,"given":"Mohammad","family":"Kanan","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, College of Engineering, University of Business and Technology, Jeddah 21448, Saudi Arabia"},{"name":"Department of Mechanical Engineering, College of Engineering, Zarqa University, Zarqa 13110, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.7498\/aps.42.1731","article-title":"Solitary wave solutions to a generalized KdV equation","volume":"42","author":"Ma","year":"1993","journal-title":"Acta Phys. 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