Solving ordinary differential equations with higher order Haar wavelet method

2019 ◽  
Author(s):  
Jüri Majak ◽  
Meelis Pohlak ◽  
Martin Eerme ◽  
Boris Shvartsman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Haar Wavelet ◽  
Higher Order ◽  
Wavelet Method ◽  
Haar Wavelet Method

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.

scholarly journals Solving Nonlinear Boundary Value Problems Using the Higher Order Haar Wavelet Method

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2809
Author(s):  
Mart Ratas ◽  
Jüri Majak ◽  
Andrus Salupere
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Boundary ◽  
Haar Wavelet ◽  
Higher Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Boundary Value Problems ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Liénard Equations

The current study is focused on development and adaption of the higher order Haar wavelet method for solving nonlinear ordinary differential equations. The proposed approach is implemented on two sample problems—the Riccati and the Liénard equations. The convergence and accuracy of the proposed higher order Haar wavelet method are compared with the widely used Haar wavelet method. The comparison of numerical results with exact solutions is performed. The complexity issues of the higher order Haar wavelet method are discussed.

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

Reconstruction of the parameter in parabolic partial differential equations using Haar wavelet method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Gopal Priyadarshi ◽  
B.V. Rathish Kumar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Computational Cost ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Wavelet Method ◽  
Content Type ◽  
Haar Wavelet Method

Purpose In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method. Design/methodology/approach The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points. Findings An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time. Originality/value The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

scholarly journals New Numerical Treatment for a Family of Two-Dimensional Fractional Fredholm Integro-Differential Equations

Algorithms ◽  
2020 ◽  
Vol 13 (2) ◽  
pp. 37
Author(s):  
Amer Darweesh ◽  
Marwan Alquran ◽  
Khawla Aghzawi
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Haar Wavelet ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Wavelet Method ◽  
New Approach ◽  
Robust Algorithm ◽  
Haar Wavelet Method ◽  
The Laplace Transform

In this paper, we present a robust algorithm to solve numerically a family of two-dimensional fractional integro differential equations. The Haar wavelet method is upgraded to include in its construction the Laplace transform step. This modification has proven to reduce the accumulative errors that will be obtained in case of using the regular Haar wavelet technique. Different examples are discussed to serve two goals, the methodology and the accuracy of our new approach.

Application of Haar Wavelet Method for Solving the Nonlinear Fuzzy Integro-Differential Equations

2019 ◽  
Vol 16 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Mohamed R. Ali ◽  
Adel R. Hadhoud
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Haar Wavelet ◽  
Test Problems ◽  
Newton Methods ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Homotopy Perturbation ◽  
Proposed Model

Haar wavelet method (HWM) is an essential profitable method for settling the nonlinear Fuzzy Fredholm integro-differential equations (NFIDE). The proposed model converts the NFIDE into to nonlinear equations which tackle by the familiar Newton methods. The authors investigate the convergence of this method. Test problems are solved to show the accuracy of our method where the obtained numerical results are compared with Homotopy perturbation method (HPM) and the exact solutions. Graphical portrayals of the correct and obtained estimated arrangements illuminate the exactness of the methodology.

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