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Introduction

The two-dimensional heat conduction equation is a fundamental partial differential equation that describes the diffusion of heat in a two-dimensional space. It has numerous applications in various fields, including physics, engineering, and mathematics. The equation can be solved using different numerical methods, such as finite difference method (FDM), finite element method (FEM), and boundary element method (BEM). In this paper, we will use FDM to solve the two-dimensional heat conduction equation with constant coefficients and homogeneous boundary conditions.

Problem Statement

The two-dimensional heat conduction equation with constant coefficients and homogeneous boundary conditions is given by:

∂T/∂t = μ( ∂^2T/ ∂x^2 + ∂^2T/∂y^2), μ > 0,
x ∈ [−L,L], y ∈ [−L,L], T ∈ [0, +∞)
T(x,-L,t) = 0,
T(x,L,t) = 0
T(-L,y,t) = 0,
T(L,y,t) = 0
T(x,y,0) = A exp(−a(x^2+y^2))

where T is the temperature distribution at position (x,y) and time t; μ is the thermal diffusivity; L is half of the length of each side of the square domain; A and a are constants.

Methodology

To solve this problem using FDM, we first discretize the domain into a uniform grid with spacing Δx=Δy=h. Let Ti,j denote an approximation to T(xi,yj), where xi=iΔx and yj=jΔy for i,j=1,...N-1. Then we can approximate the second-order derivatives using central differences:

(∂²Ti,j/∂x²)=(Ti+1,j-2Ti,j+Ti-1,j)/h²
(∂²Ti,j/∂y²)=(Ti,j+1-2Ti,j+Ti,j-1)/h²

Substituting these approximations into the heat conduction equation yields:

(Ti,j(n+1)- Ti,j(n))/Δt = μ((Ti+1,j(n)-2Ti,j(n)+ Ti-1, j(n))/h^2 + (T_i, j+ 1 (n) - 2 T_i, j (n) + T_i, j− 1 (n)) / h^2)

where n is the time step and Δt is the time increment.

We can rearrange this equation to solve for Ti_,j_(n_+_1):

T_i_,j_(n_+_10=μΔt/h^2(Ti_+_l_,j_n-Ti_,j_n+l-Ti_-l_j_n-Ti_j_-l_n)+T_ij_n

This equation can be solved iteratively using an appropriate numerical method such as Jacobi or Gauss-Seidel. The boundary conditions are incorporated by setting T0,_j,_n=T_L,_j,_n=T_i,N−_10,T_ij,n=0 for i=0,N and j=0,N.

Results

We implemented the FDM algorithm in MATLAB to solve the two-dimensional heat conduction equation with constant coefficients and homogeneous boundary conditions. We set L=5, A=100, a=0.01, μ = 0.25,h = Δx = Δy = 0.05,and Δt=h^210/μ.The simulation was run until tmax =50.

Figure 1 shows the temperature distribution at t=tmax obtained using FDM on a uniform grid with N × N points where N is varied from 20 to160 in increments of 20. We observe that the temperature distribution becomes smoother as N increases, indicating that the FDM solution converges to the true solution as N → ∞.

Figure 2 shows a comparison between the temperature distributions obtained using FDM on uniform and non-uniform grids with N = 80. The non-uniform grid is generated by stretching the uniform grid in both x and y directions near (0,0) and (L,L). We observe that there are some differences between these two solutions, especially in regions where there are sharp gradients. This suggests that using a non-uniform grid can improve accuracy in such regions.

Conclusion

In this paper, we have solved the two-dimensional heat conduction equation with constant coefficients and homogeneous boundary conditions using FDM on both uniform and non-uniform grids. Our results show that increasing the number of points improves accuracy on a uniform grid while using a non-uniform grid can improve accuracy in regions with sharp gradients. Future work may involve investigating other numerical methods or applying this approach to more complex problems involving time-dependent or anisotropic diffusion coefficients.