Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The numerical solution is needed to obtain the 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 numerically by using Implicit Crank Nicolson method. Our main objective is to determine the flow characteristics of heat equation with Dirichlet boundary condition on homogeneous heat equation. The method of Implicit Crank Nicolson has been chosen because of the stability of the method. The results have been compared with the exact analytical solution. The validated results show that the numerical results remains same as the exact analytical solutions. The results show that the changes of the temperature profile depends on the types of boundary conditions. The boundary conditions will be affected the flow characteristics of the heat equation.