The method of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences

A method for solving free boundary problems for journal bearings by means of finite differences has been proposed by Christopherson. We analyse Christopherson’s method in detail for the case of an infinite journal bearing where the free boundary problem is as follows: Given T > 0 T > 0 and h ( t ) h(t) find τ ∈ ( 0 , T ] τ ∈ (0,T] and p ( t ) p(t) such that (i) [ h 3 p ′ ] ′ = h ′ [{h^3}p’]’ = h’ for t ∈ ( 0 , τ ) t ∈ (0,τ ) , (ii) p ( 0 ) = 0 p(0) = 0 , (iii) p ( t ) = 0 p(t) = 0 for t ∈ [ τ , T ] t ∈ [τ ,T] , and (iv) p ′ ( τ − 0