Spectral method for solving nonlinear wave equations |
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Authors: | Chang Qianshun Wu Shengchang |
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Affiliation: | 1. Institute of Applied Mathematics, Academia Sinica, China
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Abstract: | In this paper, we consider the following nonlinear wave equations: $$begin{array}{l} frac{{partial ^2 phi }}{{partial t^2 }} - frac{{partial ^2 phi }}{{partial x^2 }} + mu ^2 phi + v^2 chi ^2 phi + fleft( {|phi |^2 } right)phi = 0, frac{{partial ^2 phi }}{{partial t^2 }} - frac{{partial ^2 chi }}{{partial x^2 }} + alpha ^2 chi + v^2 chi |phi |^2 + g(chi ) = 0 end{array}$$ with the periodic-initial conditions: $$begin{array}{l} phi (x---pi ,t) = phi (x + pi ,t), chi (x---pi ,t) = chi (x + pi ,t), phi (x,0) = hat phi _0 (x), phi _t (x,0) = hat phi _1 (x), chi (x,0) = hat chi _0 (x), chi _t (x,0) = hat chi _1 (x), - infty< x< infty , 0 le t le T. end{array}$$ We put forward the computational method for solving equations (1.1)–(1.4). The method is spectral method for space variable, but is difference method for time variable. We make estimation of error, and prove convergence and stability for this method. |
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