To predict the behaviour of thermal physiology of a finite biological tissue in severe cold climatic conditions, a mathematical model has been established based on Pennes' bio-heat transfer equation with oscillatory boundary condition and time dependent heat source term. Crank-Nicholson scheme has been employed to obtain the solution of the boundary value problem to understand the change in stable temperature profiles at the peripheral tissues of human body subjected to forced convection due to cold. Thermal stress at these regions with respect to different input parameters has been computed under extreme environmental conditions using MATLAB Software. The results have shown a relative significance and provide a reasonable outcome in terms of variable metabolic heat generation and oscillatory heat source. The oscillations of the temperature profiles from the mean temperatures were computed in relation with tissue medium and other physiological parameters.