Reproducing kernels of Sobolev–Slobodeckij̆ spaces via Green’s kernel approach: Theory and applications

Abstract

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains [Formula: see text] with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by [Formula: see text], and Sobolev–Slobodeckij̆ spaces, denoted by [Formula: see text], where [Formula: see text]. Our goal is accomplished by obtaining the Green’s solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green’s kernels satisfying these problems are symmetric and positive definite reproducing kernels of [Formula: see text] and [Formula: see text], respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains.