Abstract
rom previous Chaps. 2–11, the wavelet-based solution method and its applications are detail introduced to what were conducted by the author and his colleagues. In this chapter, we give a brief introduction of those applications done by else groups using the generalized Coiflets and relevant method proposed by the author’s group.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Poincaré H (1882) Sur les courbes definies par les equations differentielles. Journal De Mathématiques Pures Et Aplliquées 3(8):251–286
Leray J, Schauder J (1934) Topologie et équations fonctionnelles. Annales Scientifiques De l’École Normale Supérieure. 51:45–78
Lahaye E (1934) Une méthode de résolution d'une catégorie d'équations transcendantes. Comptes rendus des séances de l'Académie des sciences. Vie académique 197:1840–1842
Hilton PJ (1953) An introduction to homotopy theory. Cambridge University Press, Cambridge
Sen S (1983) Topology and geometry for physicists. Academic Press, Florida
Alexander JC, Yorke JA (1978) The homotopy continuation method: numerically implementable topological procedures. Trans Ameri Math Soc 242:271–284
Liao SJ (1992) A second-order approximate analytical solution of a simple pendulum by the process analysis method. J Appl Mech 59:970–975
Liao SJ (2004) An analytic approximate approach for free oscillations of self-excited systems. Int J Non-Linear Mech 39(2):271–280
Yang ZC, Liao SJ (2017) A HAM-based wavelet approach for nonlinear ordinary differential equations. Commun Nonlinear Sci Numer Simul 48:439–453
Yang ZC, Liao SJ (2017) A HAM-based wavelet approach for nonlinear partial differential equations: two dimensional Bratu problem as an application. Commun Nonlinear Sci Numer Simul 53:249–262
Wang JZ (2001) Generalized theory and arithmetic of orthogonal wavelets and applications to researches of mechanics including piezoelectric smart structures. Lanzhou University; 2001. PhD Thesis. (advised by Zhou Y H)
Wang JZ, Zhou YH, Gao HJ (2003) Computation of the Laplace inverse transform by application of the wavelet theory. Commun Numer Methods Eng 19(12):959–975
Wang JZ, Wang XM, Zhou YH (2012) A wavelet approach for active-passive vibration control of laminated plates. Acta Mech Sin 28(2):520–531
Liu XJ, Wang JZ, Zhou YH (2013) A wavelet method for solving nonlinear time-dependent partial differential equations. Computer Modeling in Engineering and Science. 94:225–238
Liu XJ, Zhou YH, Zhang L, Wang JZ (2014) Wavelet solutions of Burgers’ equation with high Reynolds numbers. Sci China Technol Sci 57(7):1285–1292
Kosloff D, Kessler D, Filho A, Tessmer E, Behle A, Strahilevitz R (1990) Solution of the equations of dynamic elasticity by a Chebychev spectral method. Geophysics 55(6):734–748
Jaffard S (1992) Wavelet methods for fast resolution of elliptic problems. SIAM J Numerical Anal 29(4):965–986
Alpert B, Beylkin G, Coifman R, Rokhlin V (1993) Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J Scient Comput 14(1):159–184
Xu JC, Shann WC (1992) Galerkin-wavelet methods for two-point boundary value problems. Numer Math 63(1):123–144
Liu XJ (2014) A wavelet method for uniformly solving nonlinear problems and its application to quantitative research on flexible structures with large deformation. Lanzhou University; 2014. PhD Thesis. (Supervised by Zhou Y H)
Zhou YH, Wang XM, Wang JZ, Liu XJ (2011) A wavelet numerical method for solving nonlinear fractional vibration, diffusion and wave equations. Comput Model Eng Sci 7(2):137–160
Zhang L, Liu XJ, Zhou YH, Wang JZ (2013) Influence of vanishing moments on the accuracy of a modified wavelet Galerkin method for nonlinear boundary value problems. AIP Conference Proceedings. Amer Inst Phys 1558(1): 942–945
Ko J, Kurdila AJ, Pilant MS (1995) A class of finite element methods based on orthonormal, compactly supported wavelets. Comput Mech 16(4):235–244
Ma JX, Xue JJ, Yang SJ, He ZJ (2003) A study of the construction and application of a Daubechies wavelet-based beam element. Finite Elem Anal Des 39(10):965–975
Han JG, Ren WX, Huang Y (2006) A spline wavelet finite-element method in structural mechanics. Int J Numer Meth Eng 66(1):166–190
Ray SS (2012) On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation. Appl Math Comput 218(9):5239–5248
Ray SS, Gupta AK (2014) A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker-Planck equations in modelling of anomalous diffusion systems. J Math Chem 52(8):2277–2293
Gupta AK, Ray SS (2015) Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method. Appl Math Model 39(17):5121–5130
Ray SS, Gupta AK (2016) Numerical solution of fractional partial differential equation of parabolic type with Dirichlet boundary conditions using two-dimensional Legendre wavelets method. J Comput Nonlinear Dyn 11(1):011012
Amaratunga K, Williams JR, Qian S, Weiss J (1994) Wavelet-Galerkin solutions for one-dimensional partial differential equations. Int J Numer Meth Eng 37(16):2703–2716
Avudainayagam A, Vani C (2000) Wavelet-Galerkin method for integro-differential equations. Appl Numer Math 32(3):247–254
Liu XJ, Zhou YH, Wang XM, Wang JZ (2013) A wavelet method for solving a class of nonlinear boundary value problems. Commun Nonlinear Sci Numer Simul 18(8):1939–1948
Zhang L, Wang JZ, Zhou YH (2015) Wavelet solution for large deflection bending problems of thin rectangular plates. Arch Appl Mech 85(3):355–365
Chen MQ, Hwang C, Shih YP (1996) The computation of wavelet-Galerkin approximation on a bounded interval. Int J Numer Meth Eng 39(17):2921–2944
Yu Q, Xu H, Liao SJ (2018) Analysis of mixed convection flow in an inclined lid-driven enclosure with Buongiorno’s nanofluid model. Int J Heat Mass Transf 126:221–236
Yu Q, Xu H (2018) Novel wavelet-homotopy Galerkin technique for analysis of lid-driven cavity flow and heat transfer with non-uniform boundary conditions. Appl Math Mech 39(12):1691–1718
Yu Q, Xu H, Liao SJ, Yang ZC (2019) A novel homotopy-wavelet approach for solving stream function-vorticity formulation of Navier-Stokes equations. Commun Nonlinear Sci Numer Simul 67:124–151
Yu Q, Xu H, Liao SJ, Yang ZC (2018) Nonlinear analysis for extreme large bending deflection of a rectangular plate on non-uniform elastic foundations. Appl Math Model 61:316–340
Chen QB, Xu H (2020) Coiflet wavelet-Homotopy solution of channel flow due to orthogonally moving porous walls governed by the Navier-Stokes equations. J Math 2020:1–12
Ray SS, Gupta AK (2013) On the solution of Burgers-Huxley and Huxley equation using wavelet collocation method. Comput Model Eng Sci 91:409–424
Koziol P, Hryniewicz Z (2006) Analysis of bending waves in bending on viscoelastic random foundation using wavelet technique. Int J Solids Struct 43:6965–6977
Koziol P, Mares C, Esat I (2006) A wavelet approach for the analysis of bending waves in a beam on viscoelastic random foundation. Appl Mech Mater 5:239–246
Soong TT (1973) Random differential equations in science and engineering. Academic Press, New York
Adomian G (1983) Stochastic systems. Academic Press, New York
Daubechies I (1993) Orthonormal bases of compactly supported wavelets II. Variations on a theme. SIAM J Math Analysis 24(2):499–519
Koziol P, Mares C, Esat I (2008) Wavelet approach to vibratory analysis of surface due to a load moving in the layer. Int J Solids Struct 45:2140–2159
Koziol P, Hryniewicz Z, Mares C (2009) Wavelet analysis of beam-soil structure response for fast moving train. J Phys: Confer Ser. 181(1):012052. IOP Publishing
Koziol P, Mares C (2010) Wavelet approach for vibration analysis of fast moving load on a viscoelastic medium. Shock Vib 17(4, 5):461–472
Hryniewicz Z (2011) Dynamics of Rayleigh beam on nonlinear foundation due to moving load using Adomian decomposition and Coiflet expansion. Soil Dyn Earthq Eng 31(8):1123–1131
Koziol P, Hryniewicz Z (2012) Dynamic response of a beam resting on a nonlinear foundation to a moving load: coiflet-based solution. Shock Vibr 19(5):995–1007
Koziol P, Neves MM (2012) Multilayered infinite medium subject to a moving load: dynamic response and optimization using coiflet expansion. Shock Vibr 19(5):1009–1018
Hryniewicz Z, Kozioł P. Wavelet-based solution for vibrations of a beam on a nonlinear viscoelastic foundation due to moving load. Journal of Theoretical and Applied Mechanics. 2013, 51.
Koziol P (2014) Wavelet approximation of Adomian’s decomposition applied to the nonlinear problem of a double-beam response subject to a series of moving loads. J Theor Appl Mech 52(3):687–697
Koziol P (2016) Experimental validation of wavelet based solution for dynamic response of railway track subjected to a moving train. Mech Syst Signal Process 79:174–181
Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms I. Commun Pure Appl Math 44(2):141–183
Si LT, Zhao Y, Zhang YH, Kennedy D (2016) A hybrid approach to analyze a beam-soil structure under a moving random load. J Sound Vib 382:179–192
Lu PM, Shi CJ (2012) Analysis solution study of steady response on asphalt road based on the wavelet theory. Chinese J Comput Mech 29(5):806–810 (in Chinses)
Chun-Juan SHI, Peng-min L (2013) Study on the dynamic response of asphalt pavement based on the nonlinear viscoelastic model. Eng Mech 2:47 (in Chinses)
Liu XY, Shi CJ (2012) Random characteristics and reliability analysis of asphalt pavement under vehicle random load. China J Highway Transport 25(6):49–55 (in Chinses)
Meyer Y (1992) Wavelets and operators. Cambridge University Press
Monzón L, Beylkin G, Hereman W (1999) Compactly supported wavelets based on almost interpolating and nearly linear phase filters (coiflets). Appl Comput Harmonic Anal 7(2):184–210
Lieb M, Sudret B (1998) A fast algorithm for soil dynamics calculations by wavelet decomposition. Arch Appl Mech 68(3–4):147–157
Grundmann H, Lieb M, Trommer E (1999) The response of a layered half-space to traffic loads moving along its surface. Arch Appl Mech 69(1):55–67
Zhou YH, Wang JZ, Zheng XJ (1998) Applications of wavelet Galerkin FEM to bending of beam and plate structures. Appl Math Mecha 19(8):745–755
Zhou YH, Wang JZ (1998) A dynamic control model of piezoelectric cantilevered beam-plate based on wavelet theory. Acta Mech Sin 30(6):719–727 (in Chinese)
Wang JZ, Zhou YH (1998) Error estimation of the generalized wavelet Gaussian integral method. J Lanzhou Univ 34:26–30. (in Chinese)
Zhou YH, Wang JZ (1999) Generalized Gaussian integral methods for calculations of scaling function transform of wavalets and its applications. Acta Math Scientia 19(2):293–300 (in Chinese)
Zhou YH, Wang JZ, Zheng XJ (2001) A wavelet-based approach for dynamic control of intelligent piezoelectric plate structures with linear and non-linear deformation. IUTAM Symposium on Smart Structures and Structronic Systems. Springer, Dordrecht, pp 179–187
Zhou YH, Wang JZ, Zheng XJ, Jiang Q (2000) Vibration control of variable thickness plates with piezoelectric sensors and actuators based on wavelet theory. J Sound Vib 237(3):395–410
Zhou YH, Wang JZ, Zheng XJ (2001) A numerical inversion of the Laplace transform by use of the scaling function transform of wavelet theory. Acta Math Scientia 21A(1):86–93 (in Chinese)
Yang SJ, Ma JX, Xue JJ, He ZJ (2002) Simulation of temperature field of copier paper based on wavelet finite element method. Acta Simulate Systematica Sinica 14(9):1243–1245 (in Chinese)
Si HW, Li DX (2003) Researches on wavelet-based vibration control of large space smart structures. J National Univ Defense Technol 25(3):14–18 (in Chinese)
Ma JX, Xue JJ, Yang SJ, HE ZJ (2004) Study of the construction and application of Daubechies wavelet-based beam element. Minimicro Syst-ShenYang 25:663–666
Fenik S, Starek L (2010) Optimal PI controller with position feedback for vibration suppression. J Vib Control 16(13):2023–2034
Hegewald T, Inman DJ (2001) Vibration suppression via smart structures across a temperature range. J Intell Mater Syst Struct 12(3):191–203
Rew KH, Han JH, Lee I (2002) Multi-modal vibration control using adaptive positive position feedback. J Intell Mater Syst Struct 13(1):13–22
Ma XL, Wu B, Zhang JH, Shi X (2019) A new numerical scheme with wavelet-Galerkin followed by spectral deferred correction for solving string vibration problems. Mech Mach Theory 142:103623
Wang JZ, Zhang L, Zhou YH (2018) A simultaneous space-time wavelet method for nonlinear initial boundary value problems. Appl Math Mech 39(11):1547–1566
Zhang L, Wang JZ, Liu XJ, Zhou YH (2017) A wavelet integral collocation method for nonlinear boundary value problems in physics. Comput Phys Commun 215:91–102
Wang XM (2014) A Coiflets-based wavelet Laplace method for solving the Riccati differential equations. J Appl Math 2014:257049
Wang XM (2014) A new wavelet method for solving a class of nonlinear Volterra-Fredholm integral equations. Abstract Appl Analy 2014:975985
Wang XM (2014) A wavelet method for solving Bagley-Torvik equation. Comput Model Eng Sci 102(2):169–182
Momani S, Shawagfeh N (2006) Decomposition method for solving fractional Riccati differential equations. Appl Math Comput 182(2):1083–1092
Li Y, Sun N, Zheng B, Wang Q, Zhang Y (2014) Wavelet operational matrix method for solving the Riccati differential equation. Commun Nonlinear Sci Numer Simul 19(3):483–493
Cang J, Tan Y, Xu H, Liao SJ (2009) Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals 40(1):1–9
HosseinNia SH, Ranjbar A, Momani S (2008) Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Comput Math Appl 56(12):3138–3149
Odibat Z, Momani S (2008) Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fractals 36(1):167–174
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Zhou, YH. (2021). Brief Introduction in Applications of Other Groups. In: Wavelet Numerical Method and Its Applications in Nonlinear Problems. Engineering Applications of Computational Methods, vol 6. Springer, Singapore. https://doi.org/10.1007/978-981-33-6643-5_12
Download citation
DOI: https://doi.org/10.1007/978-981-33-6643-5_12
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-33-6642-8
Online ISBN: 978-981-33-6643-5
eBook Packages: EngineeringEngineering (R0)