Title: A wavelet integral collocation method for nonlinear boundary value problems in Physics,
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.