Wavelet-Based Homotopy Analysis Method for Nonlinear Matrix System and Its Application in Burgers Equation

Abstract

To generalize the homotopy analysis method (HAM) to multidegree-of-freedom nonlinear system, the adaptive precise integration method (APIM) is introduced into the HAM, with which the almost exact value of the exponential matrix can be obtained. Combining the interval interpolation wavelet collocation method, HAM-based APIM can be employed to solve the nonlinear PDEs. As an example, Burgers equation is spatially discretized by the interval quasi-Shannon wavelet collocation method and solved by the proposed method to illustrate the effectiveness and great potential of the homotopy analysis method in nonlinear problems.

Haar wavelet transform and variational iteration method for fractional option pricing models

“…[31][32][33] Unlike analytical perturbation methods, HPM does not depend on small parameter which is difficult to find, and so it has been widely used to solve various nonlinear problems. 34,35 Majak et al proposed a new high-order Haar wavelet method (HOHWM) to solve differential equations and integro-differential equations. 36 HOHWM allows to achieve the same accuracy with HWM with several orders lower mesh.…”

“…The HPM and VIM are some of the most effective tools to solve nonlinear problems, which has been developed and applied to solve various nonlinear problems by He 29,30 and by others 31–33 . Unlike analytical perturbation methods, HPM does not depend on small parameter which is difficult to find, and so it has been widely used to solve various nonlinear problems 34,35 …”

Interval Wavelet Numerical Method on Fokker-Planck Equations for Nonlinear Random System

“…Mei proposed a general construction method of the interval wavelet based on the general variational principle [18] and took the Shannon-Gabor wavelet as example to illustrate its correctness, which has been employed to eliminate the boundary effect in solving PDEs such as the image processing [19], stochastic analysis [20,21], option pricing [22], hydrodynamics [23][24][25][26], and image segmentation [27]. The purpose of this paper is to construct an interval Shannon-Gabor wavelet collocation method on solving Fokker-Planck equation for nonlinear oscillators and a time fractional Fokker-Planck equation describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath [28].…”