Numerical methods for nonlinear fourth-order boundary value problems with applications. (English) Zbl 1131.65064
Summary: We present efficient numerical algorithms for the approximate solution of nonlinear fourth-order boundary value problems. The first algorithm deals with the sinc-Galerkin method (SGM). The sinc basis functions prove to handle well singularities in the problem. The resulting SGM discrete system is carefully developed. The second method, the Adomian decomposition method (ADM), gives the solution in the form of a series solution.
A modified form of the ADM based on the use of the Laplace transform is also presented. We refer to this method as the Laplace Adomian decomposition technique (LADT). The proposed LADT can make the Adomian series solution convergent in the Laplace domain, when the ADM series solution diverges in the space domain. A number of examples are considered to investigate the reliability and efficiency of each method. Numerical results show that the sinc-Galerkin method is more reliable and more accurate.
A modified form of the ADM based on the use of the Laplace transform is also presented. We refer to this method as the Laplace Adomian decomposition technique (LADT). The proposed LADT can make the Adomian series solution convergent in the Laplace domain, when the ADM series solution diverges in the space domain. A number of examples are considered to investigate the reliability and efficiency of each method. Numerical results show that the sinc-Galerkin method is more reliable and more accurate.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
Adomian’s decomposition method; Laplace transform; sinc-Galerkin method; algorithms; nonlinear fourth-order boundary value problems; numerical resultsReferences:
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