in this paper, a numerical method is developed for solving a linear sixth order boundaryvalue problem (6vbp ) by using the hyperbolic uniform spline of order 3 (lower order). thereis proved to be first-order convergent. numerical results confirm the order of convergencepredicted by the analysis.

We study a boundary value problem for Caputo-type fractional differential equation subjected to periodic conditions. For an auxiliary with the simplified right-hand side, we explicitly construct its unique solution. In addition, based on theory of topological index, prove existence at least one solution original problem.