[Paper Review] Numerical solution of one-dimensional Sine--Gordon equation using Reproducing Kernel Hilbert Space Method

Ali Akgül, Inc, Mustafa|arXiv (Cornell University)|Apr 2, 2013

Numerical methods in engineering28 references3 citations

TL;DR

This paper proposes a Reproducing Kernel Hilbert Space Method (RKHSM) for solving one-dimensional sine-Gordon equations with initial and boundary conditions. The method represents the exact solution as a convergent series in a reproducing kernel Hilbert space, avoiding discretization and enabling high-accuracy numerical solutions with minimal computation, as validated by numerical examples showing uniform convergence and errors below 10^{-6}.

ABSTRACT

In this paper, we propose a reproducing kernel Hilbert space method (RKHSM) for solving the sine--Gordon (SG) equation with initial and boundary conditions based on the reproducing kernel theory. Its exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have been studied to demonstrate the accuracy of the present method. The results obtained from the method are compared with the exact solutions and the earlier works. Results of numerical examples show that the presented method is simple and effective.

Motivation & Objective

• To develop a mesh-free numerical method for solving one-dimensional sine-Gordon equations with initial and boundary conditions.
• To provide a convergent series representation of the exact solution using reproducing kernel Hilbert space theory.
• To demonstrate the method's high accuracy and computational efficiency through numerical examples.
• To compare the RKHSM results with exact solutions and existing numerical methods, showing superior or competitive performance.

Proposed method

• The method employs reproducing kernel Hilbert space theory to construct a solution space where the exact solution is represented as an infinite series.
• A bounded linear operator is defined in the reproducing kernel space to transform the original PDE into a solvable operator equation.
• The solution is approximated iteratively using a truncated series of kernel functions, ensuring uniform convergence to the exact solution.
• The method avoids spatial and temporal discretization, eliminating the need to solve large linear systems or update Jacobian matrices.
• The algorithm computes approximate solutions directly from the kernel functions, minimizing computational cost and avoiding numerical instability.
• Convergence analysis proves that the approximate solution converges uniformly to the exact solution in the reproducing kernel space.

Experimental results

Research questions

• RQ1 Can the reproducing kernel Hilbert space method (RKHSM) effectively solve one-dimensional sine-Gordon equations with initial and boundary conditions without discretization?
• RQ2 How does the accuracy of RKHSM compare to exact solutions and other numerical methods such as collocation with radial basis functions or finite difference schemes?
• RQ3 What is the convergence behavior of the RKHSM approximate solution, and does it uniformly approach the exact solution?
• RQ4 How computationally efficient is RKHSM compared to Newton iteration-based methods that require solving large linear systems?
• RQ5 Can RKHSM be applied to both linear and nonlinear forms of the sine-Gordon equation with high precision?

Key findings

• The RKHSM achieves high accuracy, with absolute errors consistently below 5 × 10^{-7} in tested examples, and relative errors below 3.6 × 10^{-7} in the best cases.
• For Example 5.1, the maximum absolute error was 1.756 × 10^{-7}, and the relative error was 2.987 × 10^{-7}, demonstrating high precision.
• In Example 5.2, the method achieved zero absolute error at symmetric points (x = 0, ±0.4, ±0.8) due to exact symmetry and high numerical stability.
• The CPU time per example ranged from 0.546 to 1.719 seconds, indicating low computational cost despite high accuracy.
• The method showed uniform convergence of the approximate solution to the exact solution, confirming theoretical convergence results.
• Comparison with results from [37] showed that RKHSM produced smaller absolute and relative errors in all tested points, confirming superior accuracy.