Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem

Abstract

An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems.

A Conservative Difference Scheme for Solving the Coupled Fractional Schrödinger–Boussinesq System

“…Additionally, we set −x l = x r = 40 and T = 2. Tables 1 and 2 demonstrate that the numerical scheme ( 26)- (30) achieves second-order accuracy in time and fourth-order accuracy in space for numerical solutions U n and V n with α = 2. Table 3 presents the l 2 h -norm errors and convergence orders for different α ∈ (1, 2).…”

“…, and z = (a T , b T , c T ) T ; then, z is a 3(J − 1)-dimensional vector or a point of 3(J − 1)-dimensional Euclidean space R 3(J−1) . Now, we use the Schauder fixed point to prove the existence of the solutions for the finite difference scheme ( 26)- (30). For this purpose, we construct a mapping T λ : R 3(J−1) −→ R 3(J−1) of the 3(J − 1)-dimensional Euclidean space into itself, with a parameter λ ∈ (0, 1)…”

“…We present the error estimation of the conservative difference scheme ( 26)- (30) in the following theorem. Readers can refer to [23,24,32,33] for the same style of analysis method regarding the proof of convergence.…”

“…Algorithm 1: The conservative scheme ( 26)- (30) of the FCSBS 1 Given: U n , V n and Φ n . 2 Step 1: Solve V n+1 and Φ n+1 from ( 27) and (28).…”

“…It is widely used in addressing various nonlinear problems, including the fractional wave equations [22], fractional Schrödinger equation [23], fractional Ginzburg-Landau equation [24], fractional coupled Schrödinger-Boussinesq equation [25], fractional Korteweg-de Vries equation [26][27][28], and fractional Fornberg-Whitham equation [29]. Additional notable results on this subject are available in [30][31][32][33].…”