Visual Analysis of the Exact Solution of a Heat Equation

Abstract

The solution of the two-dimensional heat equation is represented as images. The considered heat equation is 2 2 2 2 . U U U t x y                  , 0  x 1, 0  y  1 . The Boundary conditions are U(0, y,t)  0, U(1, y,t)  0, U(x,0,t)  0, and U(x,1,t)  0 and the initial condition is U(x, y,0)  sin4 xcos4 y . The images provide quick but approximate insights to the solution. Approximation by image visualization depends on many factors such as the display device, human eye factors, image resolution (sampling rate and bit depth of the image), and image resize methods. We have provided images for different time t and different resolutions We have studied the impact of different sampling rate and different image resize methods on the quality of the images. User study is performed for qualitative inspection and peak signal to noise ratio (PSNR) is used as a quantitative measure of the relative similarity in images with respect to different sampling rate and time t . Qualitative inspection and PSNR follow the same trend. Further, Qualitative inspection confirms that the quality of approximation of solution by image visualization improves with increase in sampling rate. Moreover, lanczos3 is the best interpolation method to resize the images.