Numerical Solution of Fitz Hugh–Nagumo Equations by Wavelet-Based Lifting Schemes

Abstract

One of the many uses for partial differential equations is the Fitzhugh-Nagumo equation. These equations are widely applied to explain or mimic complex real-world phenomena in the fields of science and engineering. This equation is applied in different fields. Some of them include branching Brownian motion, neurophysiology, flame propagation, logistic population development, autocatalytic chemical reaction, and nuclear reactor theory.  With the help of wavelets, which are mathematical operations, data may be divided into several frequency components, each of which can then be examined at the right scale and resolution. In evaluating physical circumstances where there are discontinuities and abrupt spikes in the signal, they have advantages over conventional Fourier approaches. Wavelets were separately developed in applied mathematics, quantum physics, electrical engineering, and seismic geology. Using various wavelet filter coefficients, we presented strategies for lifting this paper to solve the FitzHugh-Nagumo equations numerically. To demonstrate accuracy and convergence at a low computational cost compared to the classical method (i.e., finite difference method) and obtained numerical solutions using lifting schemes are compared with the exact solution. To demonstrate the application and validity of the proposed theories, certain test problems are provided.