The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes

On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

“…Then, we can prove the following theorem (where for notational simplicity, we restrict to m = 1), the statement of which is analogous to that on the convergence of the spline extension associated with BS methods [18]. In the proof of the theorem, we relate to the quasi-interpolation approach for function approximation, the peculiarity of which consists of being a local approach.…”

“…Given a non-decreasing set of abscissas Θ := {θ i } M i=0 , we say that a function g 1 agrees with another function g 2 at Θ if g Then, we can formulate the following proposition, (18) and associated with the B-spline basis of S 2R is nonsingular.…”

“…Here, we are interested in deriving its more compact B-spline representation. Besides being more compact, this also allows us to clarify the connection between BSHO and BS methods previously introduced in [16][17][18]. For this aim, let us introduce some necessary notation.…”

“…We recall that the order two BSHO method corresponds to the well-known trapezoidal rule, the property of conjugate symplecticity of which is well known (see for example [12]) and the continuous extension by the B-spline of which has been already developed in [18]. The order four BSHO belongs also to the EMHO class, and it has been analyzed in detail in [10].…”

“…Here, we follow the local strategy used in [18]. Taking into account the banded structure of A i , i = 1, .…”