Construction of Interval Shannon Wavelet and Its Application in Solving Nonlinear Black-Scholes Equation

Interval wavelet numerical method for nonlinear PDEs can improve the calculation precision compared with the common wavelet. A new interval Shannon wavelet is constructed with the general variational principle. Compared with the existing interval wavelet, both the gradient and the smoothness near the boundary of the approximated function are taken into account. Using the new interval Shannon wavelet, a multiscale interpolation wavelet operator was constructed in this paper, which can transform the nonlinear partial differential equations into matrix differential equations; this can be solved by the coupling technique of the wavelet precise integration method (WPIM) and the variational iteration method (VIM). At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical results show that this method can decrease the boundary effect greatly and improve the numerical precision in the whole definition domain compared with Yan's method.