Numerical solution of singularly perturbed boundary value problems of the 4th order

This paper deals with a 4th-order differential equation with a small parameter at the higher derivative. A solution algorithm is proposed based on the application of a special non-uniform difference grid, with the differential equation operator approximated in two ways: 1) the 4th order operator is replaced by a more convenient operator, which is split into two operators, i.e. instead of one equation, a system of two equations of the second order is considered; 2) by the method of integral identities, the operator is approximated on a five-point pattern. In the first approach, theorems on uniform convergence on the non-uniform difference grid proposed in the work have been proved. In the second approach, the order of uniform convergence was shown by numerical experiment. The solution of the system of difference equations was carried out by a non-monotonic run. The described numerical algorithm was used to solve the linearized problem of longitudinal-transverse bending of an elastic beam with embedded ends under the action of a distributed load.