Approximate solution of three-point boundary value problems for second-order ordinary differential equations with variable coefficients

•Three-point boundary value problem is investigated and solved numerically.•A novel, simple and effective numerical method is proposed.•The convergence and error estimate of approximate solution are given.•Numerical results are carried out to verify the proposed methods.

Some three-point boundary value problems for a second-order ordinary differential equation with variable coefficients are investigated in the present paper. By using the integration method, the second-order three-point boundary value problems are transformed into a Fredholm integral equation of the second kind. The solutions and Green's functions for some special cases of the second-order three-point boundary value problems can be determined easily. The existence and uniqueness of the solutions of the given Fredholm integral equations are considered by using the fixed point theorem in Banach spaces. A new numerical method is further proposed to solve the second-kind Fredholm integral equation and an approximate solution is made. The convergence and error estimate of the obtained approximate solution are further analyzed. Numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures.