scholarly journals SOLVING NONLINEAR PDES USING THE HIGHER ORDER HAAR WAVELET METHOD ON NONUNIFORM AND ADAPTIVE GRIDS

2021 ◽  
Vol 26 (1) ◽  
pp. 147-169
Author(s):  
Mart Ratas ◽  
Andrus Salupere ◽  
Jüri Majak
Keyword(s):  
Numerical Solutions ◽  
Vries Equation ◽  
Haar Wavelet ◽  
Higher Order ◽  
Nonlinear Pdes ◽  
Nonuniform Grid ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
De Vries ◽  
Model Equations

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

scholarly journals APPLICATION OF HIGHER ORDER HAAR WAVELET METHOD FOR SOLVING NONLINEAR EVOLUTION EQUATIONS

2020 ◽  
Vol 25 (2) ◽  
pp. 271-288 ◽  
Author(s):  
Mart Ratas ◽  
Andrus Salupere
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Haar Wavelet ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Wavelet Method ◽  
Haar Wavelet Method

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.

scholarly journals Higher Order Haar Wavelet Method for Solving Differential Equations

2020 ◽  
Author(s):  
Jüri Majak ◽  
Mart Ratas ◽  
Kristo Karjust ◽  
Boris Shvartsman
Keyword(s):  
Differential Equations ◽  
Computational Cost ◽  
Haar Wavelet ◽  
Higher Order ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Convergence Results ◽  
Key Characteristics ◽  
Parameter Values ◽  
Excellent Accuracy

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the mesh and the method parameter values used.

New higher order Haar wavelet method: Application to FGM structures

2018 ◽  
Vol 201 ◽  
pp. 72-78 ◽  
Author(s):  
J. Majak ◽  
M. Pohlak ◽  
K. Karjust ◽  
M. Eerme ◽  
J. Kurnitski ◽  
...  
Keyword(s):  
Haar Wavelet ◽  
Higher Order ◽  
Wavelet Method ◽  
Haar Wavelet Method

scholarly journals Haar wavelets scheme for solving the unsteady gas-flow in 4-D

2020 ◽  
Vol 24 (2 Part B) ◽  
pp. 1357-1367 ◽  
Author(s):  
Mohamed Ali ◽  
Dumitru Baleanu
Keyword(s):  
Gas Flow ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Collocation Points ◽  
Unsteady Gas Flow ◽  
Theoretical Results

The system of unsteady gas-flow of 4-D is solved successfully by alter the possibility of an algorithm based on collocation points and 4-D Haar wavelet method. Empirical rates of convergence of the Haar wavelet method are calculated which agree with theoretical results. To exhibit the efficiency of the strategy, the numerical solutions which are acquired utilizing the recommended strategy demonstrate that numerical solutions are in a decent fortuitous event with the exact solutions.

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