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)=au(t)+bu(t−1)+cu(t+1),t∈I⊂R. We consider basic existence and uniqueness properties when I=[t1,t2]I=[t1,t2] and we seek solutions u∈C[t1−1,t2+1]u∈C[t1−1,t2+1] that satisfy u(t)=w1(t),t∈[t1−1,t1],u(t)=w2(t),t∈[t2,t2+1], for prescribed functions w1,w2w1,w2 absolutely continuous, respectively, on [t1−1,t1],[t2,t2+1][t1−1,t1],[t2,t2+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.We consider, with t1,t2∈Nt1,t2∈N, numerical approximations of a solution when it exists. The numerical methods that we consider are linear θθ-methods and we investigate computationally their effectiveness through some illustrative examples.