Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Abstract

The recently introduced higher order Haar wavelet method is treated for solving evolution equations. The wave equation, the Burgers’ equations and the Korteweg-de Vries equation are considered as model problems. The detailed analysis of the accuracy of the Haar wavelet method and the higher order Haar wavelet method is performed. The obtained results are validated against the exact solutions.

Higher Order Haar Wavelet Method for Solving Differential Equations

“…The obtained results will be validated by a number of case studies by computing the numerical rates of convergence and comparing the obtained and theoretical results. These results are confirmed in [47][48][49][50][51][52][53] and in other papers.…”

“…The convergence rate of the method was improved from 2 to 2 + 2 s, where s stands for the method parameter. This new method is currently underused, but the first results obtained have shown that a principal growth of the accuracy can be achieved with a minimum growth of complexity [51][52][53]. In [52], the free vibrations analysis of the Euler-Bernoulli nanobeam was performed.…”

Solving Nonlinear Pdes Using the Higher Order Haar Wavelet Method on Nonuniform and Adaptive Grids

“…The time moment t f = max(0.5, t c ) was determined by obtaining the value of critical time moment t c which is the maximal time t for which max x,t<tc |u(x, t) − u e (x, t)| < 10 −3 . These numerical results were compared with the uniform grid results previously published in [76]. Such numerical results are shown in Table 2.…”

“…The KdV equation (4.3) was solved at various resolutions J and the numerical results were compared to the exact solution (4.4). The uniform grid results from [76] were used and new results with the nonuniform grid (2.3) and the adaptive grid (2.4) were calculated. The numerical result itself was used as the basis for the weight function in the adaptive grid for this model equation.…”

“…To the best knowledge of the authors, this is the first time the HOHWM is used on a nonuniform grid. The Burgers' equation [17,18] and the Kortewegde Vries (KdV) equation were chosen as model equations because they have previously been studied using the HOHWM on the uniform grid [76] and have analytical solutions [2,7,49,67] which make comparisons easy to draw. The Burgers' equation finds applications in modelling of turbulence [17] and traffic flow [66] as well as in nonlinear acoustics [37] and non-stationary shock waves in fluids [50].…”