Mixed-type functional differential equations: A numerical approach

The equations considered in this paper are mixed-type functional equations (sometimes known as forward–backward differential equations) that take the form u ′ ( t ) = a u ( t ) + b u ( t − 1 ) + c u ( t + 1 ) , t ∈ I ⊂ R . We consider basic existence and uniqueness properties when I = [ t 1 , t 2 ] and we seek solutions u ∈ C [ t 1 − 1 , t 2 + 1 ] that satisfy u ( t ) = w 1 ( t ) , t ∈ [ t 1 − 1 , t 1 ] , u ( t ) = w 2 ( t ) , t ∈ [ t 2 , t 2 + 1 ] , for prescribed functions w 1 , w 2 absolutely continuous, respectively, on [ t 1 − 1 , t 1 ] , [ t 2 , t 2 + 1 ] . With arbitrary boundary conditions specified in this way, the problem turns out to be ill-posed and so existence and uniqueness questions have an important role to play in developing numerical schemes. sider, with t 1 , t 2 ∈ N , numerical approximations of a solution when it exists. The numerical methods that we consider are linear θ -methods and we investigate computationally their effectiveness through some illustrative examples.