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In this paper heat conduction equation for an eccentric spherical annulus with the inner surface kept at a constant temperature and the outer surface subjected to convection is solved analytically. Eccentric problem domain is first transformed into a concentric domain via formulating the problem in bispherical co-ordinate system. Since an analytical Green?s function for the heat conduction equation in bispherical co-ordinate for an eccentric sphere subject to boundary condition of third type can not be found, an analytical Green's function obtained for Dirichlet boundary condition is employed in the solution. Utilizing this Green's function yields a boundary integral equation for the unknown normal derivative of the surface temperature distribution. The resulting boundary integral equation is solved analytically using method of moments. The method has been applied to heat generating eccentric spherical annuli and results are compared to the simulation results of FLUENT CFD code. A very good agreement was observed in temperature distribution computations for various geometrical configurations and a wide range of Biot number. Variation of heat dissipation with radii and eccentricity ratios are studied and a very good agreement with FLUENT has been observed

Ultraspherical polynomial P(λ)n approximations are used in reflected slab criticality calculations for strongly anisotropic scattering. A unique formulation is developed for reflected slab criticality conditions whose special cases are spherical harmonics approximations, Chebyshev polynomial approximations of first and second kind. It is shown in the calculations that all ultraspherical polynomial approximations are equivalent regarding the convergence of the solution of one-speed neutron transport equation for various degrees of anisotropy and cross section parameters. Although some ultraspherical polynomials approximations converge in the mean as the order of approximation is increased and some converge uniformly, they all converge to the same critical thickness which implies their equi-convergence. It is also demonstrated that the first order approximation P(λ)1 would give the exact critical thickness for a certain value of the parameter λ.

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