Applying the Reproducing Kernel Method to Fractional Differential Equations with Periodic Conditions in Hilbert Space

Abstract

In this article, the reproducing kernel method is presented for the fractional differential equations with periodic conditions in the Hilbert space. This method gives an approximate solution to the problem. The approximate and exact solutions are displayed in the form of series in the reproduction kernel space. In addition, we provide an error analysis for this technique. The presented method is tested by some examples to show its precision.

Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

“…Finally, we consider the function space in Equation (11) on hB, and it is an RKHS with reproducing kernel in Equation (12).…”

“…Proof. The proof is straightforward by Theorem 1 when we constrained functions on H 2 (hB, d(ρ ⊗ µ)) with a reproducing kernel in Equation (12).…”

“…In numerical analysis, Song has proposed the WPOAFD method for the Helmholtz equation [11]. The RKHS method is applied to fractional partial differential equations [12] and shows the potential to perform well compared with the finite difference method, the finite element method, and the finite volume method. There is some other work, please see [13][14][15] for details.…”