Determination of an unknown coefficient in the Korteweg–de Vries equation

In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L²} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.