{"title":"Exact analytical solution of the parabolic two-step model for nanoscale heat conduction","authors":"A.S.N. Chairuca , A.J.A. Ramos , J.A.R. Nascimento , L.G.R. Miranda","doi":"10.1016/j.aml.2025.109693","DOIUrl":null,"url":null,"abstract":"<div><div>This work presents an analytical solution to the parabolic two-step model, widely used in modeling nanoscale heat conduction, especially in metallic materials subjected to ultrashort laser pulses. The model describes the thermal coupling between the electron gas and the crystal lattice through a system of coupled differential equations. The main contribution of the article is the derivation of an explicit solution using the technique of separation of variables and Fourier series, without the need to assume the <em>null initial velocity</em> (<strong>NIV</strong>) condition, which is often imposed in the literature. The analytical solution provides a rigorous description of the transient behavior of the temperatures in the electronic and lattice subsystems, revealing the presence of exponential decay modes. Additionally, computational simulations are carried out to illustrate the rapid thermal dissipation in the electronic subsystem, followed by a slower redistribution in the crystal lattice, behavior that is characteristic of electron–phonon coupling at the nanoscale.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"171 ","pages":"Article 109693"},"PeriodicalIF":2.8000,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965925002435","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This work presents an analytical solution to the parabolic two-step model, widely used in modeling nanoscale heat conduction, especially in metallic materials subjected to ultrashort laser pulses. The model describes the thermal coupling between the electron gas and the crystal lattice through a system of coupled differential equations. The main contribution of the article is the derivation of an explicit solution using the technique of separation of variables and Fourier series, without the need to assume the null initial velocity (NIV) condition, which is often imposed in the literature. The analytical solution provides a rigorous description of the transient behavior of the temperatures in the electronic and lattice subsystems, revealing the presence of exponential decay modes. Additionally, computational simulations are carried out to illustrate the rapid thermal dissipation in the electronic subsystem, followed by a slower redistribution in the crystal lattice, behavior that is characteristic of electron–phonon coupling at the nanoscale.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.