# A wavelet integral collocation method for nonlinear boundary value problems in Physics

## TL;DR

This paper introduces a high-order wavelet integral collocation method (WICM) for solving nonlinear boundary value problems in physics, demonstrating superior accuracy and efficiency through theoretical analysis and numerical examples.

## Contribution

The paper develops a novel wavelet-based collocation method with proven convergence order N, applicable to general nonlinear physics problems, and shows its advantages over existing methods.

## Key findings

- WICM achieves convergence order N for nonlinear problems.
- Numerical results show accuracy exceeds N and is independent of differential equation order.
- Condition numbers remain stable regardless of collocation points.

## Abstract

A high order wavelet integral collocation method (WICM) is developed for general nonlinear boundary value problems in physics. This method is established based on Coiflet approximation of multiple integrals of interval bounded functions combined with an accurate and adjustable boundary extension technique. The convergence order of this approximation has been proven to be N as long as the Coiflet with N-1 vanishing moment is adopted, which can be any positive even integers. Before the conventional collocation method is applied to the general problems, the original differential equation is changed into its equivalent form by denoting derivatives of the unknown function as new functions and constructing relations between the low and high order derivatives. For the linear cases, error analysis has proven that the proposed WICM is order N, and condition numbers of relevant matrices are almost independent of the number of collocation points. Numerical examples of a wide range of nonlinear differential equations in physics demonstrate that accuracy of the proposed WICM is even greater than N, and most interestingly, such accuracy is independent of the order of the differential equation to be solved. Comparison to existing numerical methods further justifies the accuracy and efficiency of the proposed method.

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Source: https://tomesphere.com/paper/1701.05979