Construction Of Generalized Green’s Function For The Heat Operator

ABSTRACT

 In this work, we construct the generalized Green’s function for the initial value problem involving a partial differential operator specifically the heat operator in n , using Fourier transform method. The homogeneous and non-homogeneous heat equations are solved with specified initial condition. In particular employing the Duhamel’s principle which is a procedure for expressing the solution of a non-homogeneous problem as an integral of the solutions to the homogeneous problem in the way that the source term is interpreted as the initial condition, we obtained the solution of the non-homogeneous problem. We also study the properties of the Green’s function and for the two dimensional case, the function is plotted in three dimensions using MATLAB. Our result shows that the Green’s function constructed is unbounded in any neighbourhood of the origin. We also applied the concept of Green’s function to electrostatics. In particular we solved a generalized problem involving a unit source charge positioned at a specific point in space. We then solve the Poisson’s equation using the Green’s function solution giving us an inverse law for the associated electrostatic potential.