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.
Keyword : Haar wavelets, evolution equations, higher order wavelet expansion
How to Cite
Ratas, M., & Salupere, A. (2020). Application of higher order Haar wavelet method for solving nonlinear evolution equations. Mathematical Modelling and Analysis, 25(2), 271-288. https://doi.org/10.3846/mma.2020.11112
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.F. Hamilton and D. T. Blackstock (Eds.). Nonlinear acoustics. Academic Press, San Diego, 1998.
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H. Hein and L. Feklistova. Computationally efficient delamination detection in composite beams using Haar wavelets. Mech. Syst. Signal Process., 25(6):2257– 2270, 2011.
C.H. Hsiao. State analysis of linear time delayed systems via Haar wavelets. Math. Comput. Simulation, 44(5):457–470, 1997. https://doi.org/10.1016/S0378-4754(97)00075-X
G. Jin, X. Xie and Z. Liu. The Haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory. Compos. Struct., 108:435–448, 2014. https://doi.org/10.1016/j.compstruct.2013.09.044
R. Jiwari. A Haar wavelet quasilinearization approach for numerical simulation of Burgers equation. Comput. Phys. Comm., 183(11):2413–2423, 2012. https://doi.org/10.1016/j.cpc.2012.06.009
M. Kirs, K. Karjust, I. Aziz, E. Ounapuu and E. Tungel. Free vibration analysis˜ of a functionally graded material beam: evaluation of the Haar wavelet method. Proc. Estonian Acad. Sci., 67(1):1–9, 2018.
P.A. Lagerstrom, J.D. Cole and L. Trilling. Problems in the theory of viscous compressible fluids. Guggenheim Aeronautical Laboratory, unnumbered report, California Institute of Technology, 1949.
G. L. Lamb Jr. Elements of Soliton Theory. John Wiley & Sons, 1980.
Ü. Lepik. Numerical solution of differential equations using Haar wavelets. Math. Comput. Simulation, 68(2):127–143, 2005. https://doi.org/10.1016/j.matcom.2004.10.005
Ü. Lepik. Haar wavelet method for nonlinear integro-differential equations. Appl. Math. Comput., 176(1):324–333, 2006. https://doi.org/10.1016/j.amc.2005.09.021
Ü. Lepik. Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci. Phys. Math., 56(1):28–46, 2007. https://doi.org/10.1117/12.736416
Ü. Lepik. Numerical solution of evolution equations by the Haar wavelet method. Appl. Math. Comput., 185(1):695–704, 2007. https://doi.org/10.1016/j.amc.2006.07.077
Ü. Lepik. Solving fractional integral equations by the Haar wavelet method. Appl. Math. Comput., 214(2):468–478, 2009. https://doi.org/10.1016/j.amc.2009.04.015
Ü. Lepik. Solving PDEs with the aid of two-dimensional Haar wavelets. Comput. Math. Appl., 61(7):1873–1879, 2011. https://doi.org/10.1016/j.camwa.2011.02.016
Ü. Lepik and H. Hein. Haar wavelets: with applications. Springer, 2014. https://doi.org/10.1007/978-3-319-04295-4
J. Majak, M. Pohlak and M. Eerme. Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells. Mech. Compos. Mater., 45(6):631–642, 2009. https://doi.org/10.1007/s11029-010-9119-0
J. Majak, M. Pohlak, M. Eerme and T. Lepikult. Weak formulation based Haar wavelet method for solving differential equations. Appl. Math. Comput., 211(2):488–494, 2009. https://doi.org/10.1016/j.amc.2009.01.089
J. Majak, M. Pohlak, K. Karjust, M. Eerme, J. Kurnitski and B.S. Shvartsman. New higher order Haar wavelet method: Application to FGM structures. Compos. Struct., 201:72–78, 2018. https://doi.org/10.1016/j.compstruct.2018.06.013
J. Majak, B. Shvartsman, K. Karjust, M. Mikola, A. Haavaj oe and M. Pohlak. On the accuracy of the Haar wavelet discretization method. Compos. Part B, 80:321–327, 2015. https://doi.org/10.1016/j.compositesb.2015.06.008
J. Majak, B. Shvartsman, M. Pohlak, K. Karjust, M. Eerme and E. Tungel. Solution of fractional order differential equation by the Haar Wavelet method. numerical convergence analysis for most commonly used approach. AIP Conference Proceedings, 1738(1):480110, 2016. https://doi.org/10.1063/1.4952346
J. Majak, B.S. Shvartsman, M. Kirs, M. Pohlak and H. Herranen. Convergence theorem for the Haar wavelet based discretization method. Compos. Struct., 126:227–232, 2015. https://doi.org/10.1016/j.compstruct.2015.02.050
T. Musha and H. Higuchi. Traffic current fluctuation and the Burgers equation. Jpn. J. Appl. Phys., 17(5):811–816, 1978. https://doi.org/10.1143/JJAP.17.811
S. Nazir, S. Shahzad, R. Wirza, R. Amin, M. Ahsan, N. Mukhtar, Iván. García-Magariño and J. Lloret. Birthmark based identification of software piracy using Haar wavelet. Math. Comput. Simulation, 2019. https://doi.org/10.1016/j.matcom.2019.04.010
Ö. Oruç. A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations. Comput. Math. Appl., 77(7):1799–1820, 2019. https://doi.org/10.1016/j.camwa.2018.11.018
Ö. Oruç¸ F. Bulut and A. Esen. A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers equation. J. Math. Chem., 53(7):1592–1607, 2015. https://doi.org/10.1007/s10910-015-0507-5
Ö. Oruç, F. Bulut and A. Esen. Numerical solution of the KdV equation by Haar wavelet method. Pramana - J. Phys., 87(6):94, Nov 2016. https://doi.org/10.1007/s12043-016-1286-7
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