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Coiflet solution of strongly nonlinear p-Laplacian equations

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Abstract

A new boundary extension technique based on the Lagrange interpolating polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coiflet-based solution procedure is established for general two-dimensional p-Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.

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References

  1. Andreianov, B., Boyer, F., and Hubert, F. Finite volume schemes for the p-Laplacian on Cartesian meshes. Esaim Mathematical Modelling and Numerical Analysis, 38, 27–28 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Frisch, M., Matarrese, S., Mohayaee, R., and Sobolevskii, A. A reconstruction of the initial conditions of the universe by optimal mass transportation. nature, 417, 260–262 (2002)

    Article  Google Scholar 

  3. Hoskins, B. J. The geostrophic momentum approximation and the semigeostrophic equations. Journal of Atmospheric Sciences, 32, 233–242 (1975)

    Article  Google Scholar 

  4. Villani, C. Optimal transport, old and new. Grundlehren der Mathematischen Wissenschaften, Springer, Berlin (2009)

    Google Scholar 

  5. Feng, X. and Neilan, M. A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations. Journal on Numerical Analysis, 47, 2952–2981 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, J. H. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B, 13, 1141–1199 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Urbano, J. M. A free boundary problem with convection for the p-Laplacian. Rendiconti Lincei-Matematica e Applicazioni, 17, 1–19 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Acker, A. and Meyer, R. A free boundary problem for the p-Laplacian: uniqueness, convexity, and successive approximation of solutions. Electronic Journal of Differential Equations, 1995, 1–19 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Cuccu, F., Emamizadeh, B., and Porru, G. Nonlinear elastic membranes involving the p-Laplacian operator. Electronic Journal of Differential Equations, 2006, 285–296 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Huang, Y. Q., Li, R., and Liu, W. B. Preconditioned descent algorithms for p-Laplacian. Journal of Scientific Computing, 32, 343–371 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chow, S. S. Finite element error estimates for non-linear elliptic equations of monotone type. Numerische Mathematik, 54, 373–393 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barrett, J. W. and Liu, W. B. Finite element approximation of the p-Laplacian. Mathematics of Computation, 61, 523–537 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Ainsworth, M. and Kay, D. Approximation theory for the hp-finite element method and application to the nonlinear Laplacian. Applied Numerical Mathematics, 34, 329–344 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhou, G. M., Huang, Y. Q., and Feng, C. S. Preconditioned hybrid conjugate gradient algorithm for p-Laplacian. International Journal of Numerical Analysis and Modeling, 2, 123–130 (2005)

    MathSciNet  Google Scholar 

  15. Oberman, A. M. Finite difference methods for the infinity Laplace and p-Laplace equations. Journal of Computational and Applied Mathematics, 254, 65–80 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lefton, L. and Wei, D. Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method. Numerical Functional Analysis and Optimization, 18, 389–399 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pezzo, L. M., Lombardi, A. L., and Martínez, S. Interior penalty discontinuous Galerkin FEM for the p(x)-Laplacian. Journal on Numerical Analysis, 50, 2497–2521 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, Y. and Han, B. A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media. Applied Mathematics and Mechanics (English Edition), 29(11), 1495–1504 (2008) DOI 10.1007/s10483-008-1110-y

    Article  MathSciNet  MATH  Google Scholar 

  19. Ding, L., Han, B., and Liu, J. Q. A wavelet multiscale method for inversion of Maxwell equations. Applied Mathematics and Mechanics (English Edition), 30(8), 1035–1044 (2009) DOI 10.1007/s10483-009-0810-1

    Article  MATH  Google Scholar 

  20. Xiang, J.W., Chen, X. F., and Li, X. K. Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval. Applied Mathematics and Mechanics (English Edition), 30(10), 1325–1334 (2009) DOI 10.1007/s10483-009-1012-x

    Article  MathSciNet  MATH  Google Scholar 

  21. Bertoluzza, S. and Naldi, G. A wavelet collocation method for the numerical solution of partial differential equations. Applied and Computational Harmonic Analysis, 3, 1–9 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lazaar, S., Ponenti, P. J., Liandrat, J., and Tchamitchian, P. Wavelet algorithms for numerical resolution of partial differential equations. Computer Methods in Applied Mechanics and Engineering, 116, 309–314 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ren, X. and Xanthis, L. S. A dynamically adaptive wavelet method of arbitrary lines for nonlinear evolutionary problems capturing steep moving fronts. Computer Methods in Applied Mechanics and Engineering, 195, 4962–4970 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Vasilyev, O. V. and Kevlahan, N. R. An adaptive multilevel wavelet collocation method for elliptic problems. Journal of Computational Physics, 206, 412–431 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, J. Z. Generalized Theory and Arithmetic of Orthogonal Wavelets and Applications to Researches of Mechanics Including Piezoelectric Smart Structures, Ph.D. dissertation, Lanzhou University, Lanzhou (2001)

    Google Scholar 

  26. Wang, X. M. A new wavelet method for solving a class of nonlinear Volterra-Fredholm integral equations. Abstract and Applied Analysis, 2014, 1–6 (2014)

    MathSciNet  Google Scholar 

  27. Liu, X. J., Zhou, Y. H., Wang, X. M., and Wang, J. Z. A wavelet method for solving a class of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation, 18, 1939–1948 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, X. J., Wang, J. Z., Wang, X. M., and Zhou, Y. H. Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions. Applied Mathematics and Mechanics (English Edition), 35(1), 49–62 (2014) DOI 10.1007/s10483-014-1771-6

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, X. J., Wang, J. Z., and Zhou, Y. H. Wavelet solution of a class of two-dimensional nonlinear boundary value problems. Computer Modeling in Engineering and Sciences, 92, 493–505 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Wang, J. Z., Wang, X. M., and Zhou, Y. H. A wavelet approach for active-passive vibration control of laminated plates. Acta Mechanica Sinica, 28, 520–531 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, X. M., Liu, X. J., Wang, J. Z., and Zhou, Y. H. A wavelet method for bending of circular plate with large deflection. Acta Mechanica Solida Sinica, 28, 83–90 (2015)

    Article  Google Scholar 

  32. Zhang, L., Wang, J. Z., and Zhou, Y. H. Wavelet solution for large deflection bending problems of thin rectangular plates. Archive of Applied Mechanics, 85, 355–365 (2015)

    Article  MATH  Google Scholar 

  33. Bermejo, R. and Infante, J. A multigrid algorithm for the p-Laplacian. SIAM Journal on Scientific Computing, 21, 1774–1789 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fiedler, M. A note on the Hadamard product of matrices. Linear Algebra and Its Applications, 49, 233–235 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. 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

    Cong Xu, Jizeng Wang, Xiaojing Liu, Lei Zhang & Youhe Zhou

Authors
  1. Cong Xu
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  2. Jizeng Wang
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  3. Xiaojing Liu
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  4. Lei Zhang
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  5. Youhe Zhou
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Corresponding author

Correspondence to Jizeng Wang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11472119 and 11421062)

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Xu, C., Wang, J., Liu, X. et al. Coiflet solution of strongly nonlinear p-Laplacian equations. Appl. Math. Mech.-Engl. Ed. 38, 1031–1042 (2017). https://doi.org/10.1007/s10483-017-2212-6

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  • DOI: https://doi.org/10.1007/s10483-017-2212-6

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Chinese Library Classification

2010 Mathematics Subject Classification

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