Articles

A simultaneous space-time wavelet method for nonlinear initial boundary value problems

Expand
  • Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China

Received date: 2018-01-22

  Revised date: 2018-06-15

  Online published: 2018-11-01

Supported by

Project supported by the National Natural Science Foundation of China (No. 11472119), the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-ot11), and the 111 Project (No. B14044)

Abstract

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the Nth-order accuracy, as long as the Coiflet with[0, 3N-1] compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.

Cite this article

Jizeng WANG, Lei ZHANG, Youhe ZHOU . A simultaneous space-time wavelet method for nonlinear initial boundary value problems[J]. Applied Mathematics and Mechanics, 2018 , 39(11) : 1547 -1566 . DOI: 10.1007/s10483-018-2384-6

References

[1] WILLIAMS, J. R. and AMARATUNGA, K. Introduction to wavelets in engineering. Internal Journal for Numerical Methods in Engineering, 37, 2365-2388(1994)
[2] SCHNEIDER, K. and VASILYEV, O. V. Wavelet methods in computational fluid dynamics. Annual Review of Fluid Mechanics, 42, 473-503(2010)
[3] LIU, X., ZHOU, Y., WANG, X., and WANG, J. A wavelet method for solving a class of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation, 18, 1939-1948(2012)
[4] WANG, X., LIU, X., WANG, J., and ZHOU, Y. A wavelet method for bending of circular plate with large deflection. Acta Mechanica Solida Sinica, 28, 83-90(2015)
[5] ZHANG, L., WANG, J., and ZHOU, Y. Wavelet solution for large deflection bending problems of thin rectangular plates. Archive of Applied Mechanics, 85, 355-365(2015)
[6] LIU, X., ZHOU, Y., ZHANG, L., and WANG, J. Wavelet solutions of Burgers' equation with high Reynolds numbers. Science China Technological Science, 57, 1285-1292(2014)
[7] ZHOU, Y. and ZHOU, J. A modified wavelet approximation for multi-resolution AWCM in simulating nonlinear vibration of MDOF systems. Computer Methods in Applied Mechanics and Engineering, 197, 1466-1478(2008)
[8] BABOLIAN, E. and FATTAHZADEH, F. Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Applied Mathematics and Computation, 188, 417-426(2007)
[9] MAHDAVI, S. and RAZAK, H. A. A wavelet-based approach for vibration analysis of framed structures. Applied Mathematics and Computation, 220, 414-428(2013)
[10] KAUR, H., MITTAL, R., and MISHRA, V. Haar wavelet solutions of nonlinear oscillator equations. Applied Mathematical Modelling, 38, 4958-4971(2014)
[11] ALAM, J., KEVLAHAN, N. K. R., and VASILYEV, O. V. Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations. Journal of Computational Physics, 214, 829-857(2006)
[12] HARIHARAN, G. and KANNAN, K. Haar wavelet method for solving some nonlinear parabolic equations. Journal of Mathematical Chemistry, 48, 1044-1061(2010)
[13] ZHOU, D., CAI, W., and ZHANG, W. An adaptive wavelet method for nonlinear circuit simulation. IEEE Transactions on Circcuits and Systems, 46, 931-938(1999)
[14] HSIAO, C. and WANG, W. Haar wavelet approach to nonlinear stiff systems. Mathematics and Computers in Simulation, 57, 347-353(2001)
[15] BUJURKE, N., SALIMATH, C., and SHIRALASHETTI, S. Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series. Nonlinear Dynamics, 51, 595-605(2008)
[16] PERNOT, S. and LAMARQUE, C. A wavelet-Galerkin procedure to investigate time-periodic systems:transient vibration and stability analysis. Journal of Sound and Vibration, 245, 845-875(2001)
[17] ZHOU, J. and ZHOU, Y. A new simple method of implicit time integration for dynamic problems of engineering structures. Acta Mechanica Sinica, 23, 91-99(2007)
[18] ELGOHARY, T., DONG, L., JUNKINS, J., and ATLURI, S. A simple, fast, and accurate timeintegrator for strongly nonlinear dynamical systems. CMES-Computer Modeling in Engineering and Science, 100, 249-275(2014)
[19] JIWARI, R., MITTAL, R., and SHARMA, K. K. A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers' equation. Applied Mathematics and Computation, 219, 6680-6691(2013)
[20] CHEN, R. and WU, Z. Applying multiquadric quasi-interpolation to solve Burgers' equation. Applied Mathematics and Computation, 172, 472-484(2006)
[21] ZHU, C. G. and WANG, R. H. Numerical solution of Burgers' equation by cubic B-spline quasiinterpolation. Applied Mathematics and Computation, 208, 260-272(2009)
[22] KUTLUAY, S., ESEN, A., and DAG, I. Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method. Journal of Computational and Applied Mathematics, 167, 21-33(2004)
[23] MITTAL, R. and JAIN, R. Numerical solutions of nonlinear Burgers' equation with modified cubic B-splines collocation method. Applied Mathematics and Computation, 218, 7839-7855(2012)
[24] LAKESTANI, M. and DEHGHAN, M. Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation. Computer Physics Communications, 181, 1392-1401(2010)
[25] KHURI, S. and SAYFY, A. A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation. Applied Mathematics and Computation, 216, 1047-1056(2010)
[26] HUSSAIN, A., HAQ, S., and UDDIN, M. Numerical solution of Klein-Gordon and sine-Gordon equations by meshless method of lines. Engineering Analysis with Boundary Element, 37, 1351-1366(2013)
[27] DEHGHAN, M. and SHOKRI, A. Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. Journal of Computational and Applied Mathematics, 230, 400-410(2009)
[28] BURRUS, C. S. and ODEGARD, J. E. Coiflet systems and zero moments. IEEE Transactions on Signal Processing, 46, 761-766(1998)
[29] WANG, J. Generalized Theory and Arithmetic of Orthogonal Wavelets and Applications to Researches of Mechanics Including Piezoelectric Smart Structures (in Chinese), Ph. D. dissertation, Lanzhou Univeristy, Lanzhou (2001)
[30] ZHOU, Y., WANG, X., WANG, J., and LIU, X. A wavelet numerical method for solving nonlinear fractional vibration, diffusion, and wave equations. CMES-Computer Modeling in Engineering and Science, 1981, 1-24(2011)
[31] HAIRER, E., NORSETT, S. P., and WANNER, G. Solving Ordinary Differential Equations I:Nonstiff Problems, Springer-Verlag, Berlin (2009)
[32] CVETIĆANIN, L. Ninety years of Duffng's equation. Theoretical and Applied Mechanics, 40, 49-63(2013)
[33] ZHANG, L., WANG, J., LIU, X., and ZHOU, Y. A wavelet integral collocation method for nonlinear boundary value problems in physics. Computer Physics Communications, 215, 91-102(2017)
Outlines

/

APS Journals | CSTAM Journals | AMS Journals | EMS Journals | ASME Journals