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Wavelet-Based Solutions for Boundary-Value Problems

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Wavelet Numerical Method and Its Applications in Nonlinear Problems

Part of the book series: Engineering Applications of Computational Methods ((EACM,volume 6))

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Abstract

Nonlinear boundary-value problems (NBVPs) arise from almost every scientific and engineering field.

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References

  1. Cole JD (1968) Perturbation method in applied mathematics. Blaisdell Publishing Company, Waltham

    MATH  Google Scholar 

  2. Nayfeh AH (2000) Perturbation method. Wiley, New York

    Book  MATH  Google Scholar 

  3. Von Dyke M (1975) Perturbation methods in fluid mechanics. The Parabolic Press, Stanford

    MATH  Google Scholar 

  4. Ali AH, Al-Saif ASJ (2008) Adomian decomposition method for solving some models of nonlinear partial differential equations. Basrah J Sci A 26:1-11

    Google Scholar 

  5. Liao S (2012) Homotopy analysis method in nonlinear differential equations. Springer, Berlin

    Book  MATH  Google Scholar 

  6. Von Kármán T (1910) Festigkeits problem in maschinenbau. Encyklopädie Der Mathematischen Wissenschaften. 4:311–385

    Google Scholar 

  7. Way S (1934) Bending of circular plate with large deflection. J Appl Mech 56:627–637

    Google Scholar 

  8. Vincent JJ (1931) The bending of a thin circular plate. Phil Mag 12:185–197

    Article  MATH  Google Scholar 

  9. Chien W (1947) Large deflection of a circular clamped plate under uniform pressure. Chinese J Phys 7:102–113

    MathSciNet  Google Scholar 

  10. Chien W, Yeh K (1954) On the large deflection of circular plates. China Sci 3:405–437

    MathSciNet  Google Scholar 

  11. Qian WC (1987) A selection of scientific papers by Qian Weichang. Fujian Education Press, Fuzhou

    Google Scholar 

  12. Qian WC, Lin HS, Hu HC et al (1954) Large deflection of elastic circular thin plate. Science Press, Beijing

    Google Scholar 

  13. Causon DM, Mingham CG (2010) Introductory finite difference method for PDEs. Ventus Publishing, London

    Google Scholar 

  14. Delfour M, Fortin M, Payr G (1981) Finite-difference solutions of a non-linear Schrödinger equation. J Comput Phys 44:277–288

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen YM (1975) Numerical computation of dynamic stress intensity factors by a Lagrangian finite-difference method. Eng Fract Mech 7:653–660

    Article  Google Scholar 

  16. Brian PLT (1961) A finite-difference method of high-order accuracy for the solution of three-dimensional transient heat conduction problems. AIChE J 7:367–370

    Article  Google Scholar 

  17. Beam RM, Warming RF (1978) An implicit factored scheme for the compressible Navier-Stokes equations. Am Inst Aeron Astron J 16:393–402

    Article  MATH  Google Scholar 

  18. Narasimhan TN, Witherspoon PA (1976) An integrated finite difference method for analyzing fluid flow in porous media. Water Resour Res 12:57–64

    Article  Google Scholar 

  19. Yuste SB, Quintana-Murillo J (2012) A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput Phys Commun 183:2594–2600

    Article  MathSciNet  MATH  Google Scholar 

  20. Buckmire R (2004) Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. Numer Methods Part Diff Equ: Int J 20:327–337

    Article  MathSciNet  MATH  Google Scholar 

  21. Qiu JB (1991) Theory and application of weighted residual value method. Aerospace Press, Beijing

    Google Scholar 

  22. Wang XZ (2007) Computational mechanics. Lanzhou University Press, Lanzhou

    Google Scholar 

  23. Bramble JH, Schatz AH (1971) Least squares methods for 2mth order elliptic boundary-value problems. Math Comput 25:1–32

    MATH  Google Scholar 

  24. Xiu D, Hesthaven JS (2006) High-order collocation methods for differential equations with random inputs. Soc Indus Appl Math J Sci Comput 27:1118–1137

    MathSciNet  MATH  Google Scholar 

  25. Cockburn B, Shu CW (2006) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. Soc Indus Appl Math J Numer Anal 35:2440–2463

    MathSciNet  MATH  Google Scholar 

  26. Wang P, Yao Y, Tulin MP (1995) An efficient numerical tank for non-linear water waves, based on the multi-subdomain approach with BEM. Int J Numer Meth Fluids 20:1315–1337

    Article  MATH  Google Scholar 

  27. Feng X, Neilan M (2009) Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J Sci Comput 38:74–98

    Article  MathSciNet  MATH  Google Scholar 

  28. Atluri SN, Zhu T (1998) A new Meshless Local Petrov-Galerkin approach in computational mechanics. Comput Mech 22:117–127

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhu T, Zhang J, Atluri SN (1998) A meshless local boundary integral equation method for solving nonlinear problems. Comput Mech 22:174–187

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang X, Liu Y (2004) Meshless method. Tsinghua University Press, Beijing

    Google Scholar 

  31. Sauter SA, Schwab C (2011) Boundary element methods. Springer, Berlin

    Book  MATH  Google Scholar 

  32. Sutradhar A, Paulino GH, Gray LJ (2008) Symmetric Galerkin boundary element method. Springer, Berlin

    MATH  Google Scholar 

  33. Caglar H, Caglar N, Özer M (2008) Fifth-degree B-spline solution for nonlinear fourth-order problems with separated boundary conditions. J Phys: Conf Ser 96:012031

    Google Scholar 

  34. Wang XC (2004) Finite element method. Tsinghua University Press, Beijing

    Google Scholar 

  35. Desai YM, Eldho TI, Shah AH (2011) Finite element method with applications in engineering. Dorling Kindersley Pvt. Ltd., Noida

    Google Scholar 

  36. Wang XC, Shao M (1997) Basic principles and numerical methods of finite element method. Tsinghua University Press, Beijing

    Google Scholar 

  37. Chen Y, Lee J, Eskandarian A (2007) Meshless methods in solid mechanics. Springer Science & Business Media, Inc., New York

    MATH  Google Scholar 

  38. Gu ZQ, Kou GM (1997) Active vibration control. National Defense Industry Press, Beijing

    Google Scholar 

  39. Ja YH (1983) Using the shooting method to solve boundary-value problems involving nonlinear coupled-wave equations. Opt Quant Electron 15:529–538

    Article  Google Scholar 

  40. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28

    Article  Google Scholar 

  41. Tadmor E (2012) A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc 49:507–554

    Article  MathSciNet  MATH  Google Scholar 

  42. Neilan M, Glowinski R, Feng X (2013) Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev 55:205–267

    Article  MathSciNet  MATH  Google Scholar 

  43. Guo Y (2008) Nonlinear partial differential equations. Tsinghua University Press, Beijin

    Google Scholar 

  44. Bebernes J, Eberly D (1987) Mathematical problems from combustion theory. Springer, New York

    MATH  Google Scholar 

  45. Abbott JP (1978) An efficient algorithm for the determination of certain bifurcation points. J Comput Appl Math 4:19–27

    Article  MathSciNet  MATH  Google Scholar 

  46. Frank-Kamenetskii DA (1967) Diffusion and heat transfer in chemical kinetics, 2nd edn. Plenum Press, New York

    Google Scholar 

  47. Jalilian R (2010) Non-polynomial spline method for solving Bratu’s problem. Comput Phys Commun 181:1868–1872

    Article  MathSciNet  MATH  Google Scholar 

  48. Caglar H, Caglar N, Özer M et al (2010) B-spline method for solving Bratu’s problem. Int J Comput Math 87: 885–1891

    Google Scholar 

  49. Abbasbandy S, Hashemi MS, Liu CS (2011) The Lie-group shooting method for solving the Bratu equation. Commun Nonlinear Sci Numer Simul 16:4238–4247

    Article  MathSciNet  MATH  Google Scholar 

  50. Hassan IHAH, Ertürk VS (2007) Applying differential transformation method to the one-dimensional planar Bratu problem. Int J Contemp Math Sci 2:1493–1504

    Article  MathSciNet  MATH  Google Scholar 

  51. Khuri SA (2004) A new approach to Bratu’s problem. Appl Math Comput 147:131–137

    MathSciNet  MATH  Google Scholar 

  52. Deeba E, Khuri SA, Xie S (2000) An algorithm for solving boundary value problems. J Comput Phys 159:125–138

    Article  MathSciNet  MATH  Google Scholar 

  53. 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 18:1939–1948

    Article  MathSciNet  MATH  Google Scholar 

  54. Liu XJ, Wang JZ, Zhou YH (2013) Wavelet solution of a class of two-dimensional nonlinear boundary value problems. CMES-Comput Model Eng Sci 92:493–505

    MathSciNet  MATH  Google Scholar 

  55. Chang SL, Chien CS (2003) A multigrid-lanczos algorithm for the numerical solutions of nonlinear eigenvalue problems. Int J Bifurcation Chaos 13:1217–1228

    Article  MathSciNet  MATH  Google Scholar 

  56. Mohsen A, Sedeek LF, Mohamed SA (2008) New smoother to enhance multigrid-based methods for Bratu problem. Appl Math Comput 204:325–337

    MathSciNet  MATH  Google Scholar 

  57. Doedel EJ, Sharifi H (2000) Collocation methods for continuation problems in nonlinear elliptic PDEs. Notes Numer Fluid Mech 74:105–118

    MATH  Google Scholar 

  58. Fedoseyev AI, Friedman MJ, Kansa EJ (2000) Continuation for nonlinear elliptic partial [differential equations discretized by the multiquadratic method. Int J Bifurcation Chaos 10:481–492

    Article  MATH  Google Scholar 

  59. Odejide SA, Aregbesola YAS (2006) A note on two dimensional Bratu problem. Kragujevac J Math 29:49–57

    MathSciNet  MATH  Google Scholar 

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Zhou, YH. (2021). Wavelet-Based Solutions for Boundary-Value Problems. 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_7

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  • DOI: https://doi.org/10.1007/978-981-33-6643-5_7

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