This paper introduces an approach to decoupling singularly perturbed boundary
value problems for fourth-order ordinary differential equations that feature a
small positive parameter ϵ multiplying the highest derivative. We
specifically examine Lidstone boundary conditions and demonstrate how to break
down fourth-order differential equations into a system of second-order
problems, with one lacking the parameter and the other featuring ϵ
multiplying the highest derivative. To solve this system, we propose a mixed
finite element algorithm and incorporate the Shishkin mesh scheme to capture
the solution near boundary layers. Our solver is both direct and of high
accuracy, with computation time that scales linearly with the number of grid
points. We present numerical results to validate the theoretical results and
the accuracy of our method.Comment: 15 pages, 7 figure