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

Abstract

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.

Coiflet wavelet-homotopy solution of free convection ina closed cavity subjected to an inclined externalmagnetic field

“…As suggested by Liu [17], the modified Coiflet wavelet function for the boundary value problems is expressed by…”

“…Owing to the combined effects of the heating and magnetic field and the particular configuration of magnetic field, the non-linearity of the cavity problem is very strong, especially when physical parameters such as the Rayleigh number and the inclined angle are large. Very recently, inspired by Yang and Liao [11,12], Yu and Xu and their collaborators [13,14] developed a very efficient approach based on the homotopy analysis method (HAM) [15] and the Coiflet wavelet [16][17][18], as well as the wavelet Galerkin method [19][20][21] for cavity flow and heat transfer problems. So far, the Coiflet wavelet-homotopy technique has been successfully applied to several nonlinear problems with both homogenous and nonhomogenous boundary conditions.…”

Shannon Wavelet Precision Integration Method for Pathologic Onion Image Segmentation Based on Homotopy Perturbation Technology

“…Introducing the extension operators 1, 2, and 3 and substituting (10) into (8), the wavelet coefficients can be rewritten as…”

“…But the object boundary and target pixels obtained by these methods are often unclosed, which is makes it difficult to analyze geometric characteristics of the target for connection and econometric analysis of segmentation results. In the recent years, the wavelet precise integration method [2][3][4][5][6][7][8][9][10] has been developed to solve the nonlinear PDEs for image processing, which can improve the efficiency and precision of the image processing effectively. WPIM is an image segmentation variational method based on the C-V model, with which the segmentation results with closed object contour can be obtained.…”

Haar wavelet transform and variational iteration method for fractional option pricing models

“…Yan proposed the wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for the nonlinear model 13 . In order to eliminate the boundary effect in WPIM, an interval wavelet was constructed by Liu 14 based on the restricted variational principle, which can greatly improve the solution precision of the PDEs, especially for PDEs with steep wave solutions. The shortcoming of the method in solving the nonlinear evolution PDEs is that the wavelet transform should be calculated in each time step as the solution varies with the time parameter.…”