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. 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 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.

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

“…Following our previous works [24,25,[31][32][33][34], a continuous function f (x) defined on the interval [0, T] can be approximated by the wavelet expansion…”

“…Recently, a wavelet integral collocation method (WICM) was developed to solve nonlinear boundary value problems, and it shows high-order convergence for problems with various nonlinearities [31]. For this method, various derivatives of unknown function in the equation are denoted as new functions, and the equation is directly discretized by being satisfied at certain given points.…”

“…Although nonlinearity, space-time coupling, and high-order derivatives pose major challenges to the accurate and efficient solution of nonlinear wave problems, most recent studies have demonstrated that these difficulties can be resolved very effectively when using the function expansion scheme based on Coiflet-type basis functions [31]. Therefore, for the solution of the nonlinear wave problem shown in Equation (1), we adopted the WICM method [31] in spatial discretization and a multi-step method [32] based on wavelet approximation in time discretization.…”

“…Although nonlinearity, space-time coupling, and high-order derivatives pose major challenges to the accurate and efficient solution of nonlinear wave problems, most recent studies have demonstrated that these difficulties can be resolved very effectively when using the function expansion scheme based on Coiflet-type basis functions [31]. Therefore, for the solution of the nonlinear wave problem shown in Equation (1), we adopted the WICM method [31] in spatial discretization and a multi-step method [32] based on wavelet approximation in time discretization. This processing strategy was expected to allow the resulting algorithm possessing the following advantages: (1) spatial discretization and temporal discretization are completely decoupled; (2) both the spatial discretization and temporal discretization maintain high accuracy consistent with the direct approximation of a function, and almost independent of the order of the equation; (3) nonlinearity can be dealt with in a simple and unified way.…”

“…In other words, a fully decoupling between spatial and temporal discretization was achieved. Next, a wavelet linear multi-step method [31] was employed to further solve these resulting nonlinear ODEs. Eventually, the solution method on nonlinear wave problems is established accordingly.…”