Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 13 (2018), 183 -- 213

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This work is licensed under a Creative Commons Attribution 4.0 International License.


Man Kwong Mak, Chun Sing Leung and Tiberiu Harko

Abstract. The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schrödinger equation, and its standing wave solutions, have been intensively investigated. In the present paper we consider the applications of the Laplace-Adomian and Adomian Decomposition Methods for obtaining semi-analytical solutions of the generalized biharmonic equations of the type Δ 2y+α Δ y+ω y+b2+g( y) =f, where α , ω and b are constants, and g and f are arbitrary functions of y and the independent variable, respectively. After introducing the general algorithm for the solution of the biharmonic equation, as an application we consider the solutions of the one-dimensional and radially symmetric biharmonic standing wave equation Δ 2R+R-R2σ +1=0, with σ =constant. The one-dimensional case is analyzed by using both the Laplace-Adomian and the Adomian Decomposition Methods, respectively, and the truncated series solutions are compared with the exact numerical solution. The power series solution of the radial biharmonic standing wave equation is also obtained, and compared with the numerical solution.

2010 Mathematics Subject Classification: 34K28; 34L30; 34M25; 34M30; 35C10
Keywords: Biharmonic equation; Laplace-Adomian Decomposition method; One dimensional standing wave equation; Radial standing wave equation

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Man Kwong Mak
Departamento de F\isica, Facultad de Ciencias Naturales,
Universidad de Atacama, Copayapu 485,
Copiapó, Chile.

Chun Sing Leung
Department of Applied Mathematics,
Hong Kong Polytechnic University,
Hong Kong, Hong Kong SAR, P. R. China.

Tiberiu Harko
Department of Physics, Babes-Bolyai University
Kogalniceanu Street, 400084 Cluj-Napoca, Romania.
School of Physics, Sun Yat Sen University,
510275 Guangzhou, P. R. China.
Department of Mathematics, University College London,
Gower Street, London WC1E 6BT, United Kingdom.