Landau Legendre Wavelet Galerkin Method Applied to Study Two Phase Moving Boundary Problem of Heat Transfer in Finite Region

Abstract

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase MBP of heat transfer in finite region and represents as MBP of system of parabolic non-linear second order PDEs. We developed a Landau Legendre Wavelet Galerkin Method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of non-linear ODEs by the use of dimensionless variables and the Landau transform. The problem is converted into system of algebraic equations with the application of Legendre Wavelet Galerkin Method. In particular case, we compared present solution with Laplace transform solution and found approximately same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially varies in temperatures, the effect of dimensionless parameters: Thermal diffusivity $(\alpha_{12})$, ratio of Thermal conductivity $(k_{12})$, dimensionless temperature $(\theta_{f})$, Fourier number $(F_0)$,Stefan number $(Ste)$ and ratio of densities $\left(\rho_{1}/\rho_{2} \right)$ are discussed in detail.