RANDOM WALK AND DISCRETE HEAT EQUATION

• Main Author: RATNAWATY (NIM : 10110101); Pembimbing : Dr. Yudi Soeharyadi, NITA
• Format: Final Project
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/19066
• Institution: Institut Teknologi Bandung

Heat flow is a diffusion, that is a local averaging process. Solution to the heat flow problem is usually obtained from solving Boundary Value Problems of the heat <br /> <br /> <br /> <br /> <br /> (also known as diffusion) partial differential equation. Random walk provides a probabilistic interpretation to the diffusion process. In this interpretation, the solution to the diffusion problem in discrete structures A of Zd is obtained. Prior to solving (discrete) diffusion problem, solution to the discrete Laplace equation is explored. Solution to the Laplace equation is the equilibrium solution to the related (discrete) heat equation. The solution u(x) to the Laplace equation at the point x is the expectation of the random walk starting at x, reaching the boundary @A at the first time. In comparison, the methods used to solve discrete heat equation and the one used to solve "continuous" version, are similar. In the continuous version, the Laplace operator is decomposed into eigenfunctions. In the discrete version, the matrix Q representing (probabilistic) state transition is decomposed into eigenvectors. Solution to the heat equation (discrete or continuous version) is then expressed as a linear combination of eigenfunctions or <br /> <br /> <br /> <br /> <br /> eigenvectors.