Open AccessAnalytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions
Author(s) -
Norazlina Subani,
Faizzuddin Jamaluddin,
Muhammad Arif Hannan Mohamed,
Ahmad Danial Hidayatullah Badrolhisam
Publication year - 2020
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1551/1/012002
Subject(s) - neumann boundary condition , poincaré–steklov operator , partial differential equation , boundary value problem , mathematics , heat equation , first order partial differential equation , mathematical analysis , parabolic partial differential equation , mixed boundary condition , cauchy boundary condition , separation of variables , robin boundary condition , free boundary problem , differential equation , homogeneous differential equation , elliptic partial differential equation , ordinary differential equation , differential algebraic equation
A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.