Solving PDEs with the Aid of Two-Dimensional Haar Wavelets

A numerical treatment based on Haar wavelets for coupled KdV equation

“…After this achievement researchers have been using Haar wavelets to obtain numerical solutions of differential equations because of their simplicity and computational features. Recently, many authors have used Haar wavelet method for solving ordinary and partial differential equations [20][21][22][23][24][25][26][27][28][29][30][31]. Especially high order pdes like KdV and fractional coupled KdV equations are considered in [32,33].…”

A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

“…For convenience, we transform the fourth-order matrix equation to a secondorder matrix equation. After the a ii are calculated, we substitute them back into (15) to obtain the solution 7 .…”

“…Lepik adapted the method of Chen and Hsiao 1 to solve various types of differential equations such as nonlinear ODEs 3 , evolution equations 4 , integral equations 5 , higher-order ODEs 6 , and PDEs 3,7 . Lepik 7 proposed a procedure to solve PDEs by using the two-dimensional Haar wavelet and claimed that the proposed method was mathematically simple and computationally efficient for solving the diffusion and Poisson equations. The main feature is to expand the highest derivative into the 2-dimensional Haar wavelet series.…”

“…To the best of our knowledge, the convergence analysis of a wavelet-based method for solving PDEs has not been presented before. Since the method of Lepik 7 is simple and efficient, we perform a convergence analysis on this method when applied to solve 2D PDE boundary value problems.…”

Wavelet methods for solving three-dimensional partial differential equations

“…Numerical solutions of two-dimensional PDEs using Haar wavelet have been presented in Lepik [16]. The fundamental idea behind the Haar wavelet method is to convert the given problem into a system of equations which involves finite number of variables.…”