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A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation. (English) Zbl 1510.65264

Summary: In this paper, a new mixed method based on Lucas and Fibonacci polynomials is developed for numerical solutions of 1D and 2D sinh-Gordon equations. Firstly time variable discretized by central finite difference and then unknown function and its derivatives are expanded to Lucas series. With the help of these series expansion and Fibonacci polynomials, matrices for differentiation are derived. With this approach, finding the solution of sinh-Gordon equation transformed to finding the solution of an algebraic system of equations. Lucas series coefficients are acquired by solving this system of algebraic equations. Then by plugginging these coefficients into Lucas series expansion numerical solutions can be obtained consecutively. The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems. By calculating \(L_2\) and \(L_\infty\) error norms of some 1D and 2D test problems efficiency and performance of the proposed method is monitored. Acquired accurate results confirm the applicability of the method.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations

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References:

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