Haar wavelet method for solving Fisher's equation

In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher's equation , a prototypical reaction–diffusion equation. The solutions of Fisher's equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computation-ally attractive. The problems of the propagation of nonlinear waves have fascinated scientists for over 200 years. The modern theory of nonlinear waves, like many areas of mathematics, had its beginnings in attempts to solve specific problems, the hardest among them being the propagation of waves in water. There was significant activity on this problem in the 19th century and the beginning of the 20th century, including the classic work of Stokes, Lord Rayleigh, Korteweg and de Vries, Boussin-esque, Benard and Fisher to name some of the better remembered examples [15,30]. We would like to solve the well-known and classical Kolmogorov–Petrovski–Piscounov reaction–diffusion equation known as KPP equation. The solitons appear as a result of a balance between weak nonlinearity and dispersion. Soliton is defined as a nonlinear wave characterized by the following properties: (i) A localized wave propagates without change of its properties (shape, velocity, etc.). (ii) Localized wave are stable against mutual collisions and retain their identities. On the other hand, the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with compact support that are called campactons [22,25,26]. Unlike soliton that narrows as the amplitude increases, the compacton's width is independent of the amplitude. However, when diffusion takes part instead of dispersion, energy release by nonlinearity balances energy consumption by diffusion, which results in traveling waves or fronts [14]. Traveling wave fronts are an important and much studied solution form for reaction–diffusion equations, with important applications to chemistry, biology and medicine [23]. Such solutions were first studied in the 1930s by Fisher for the scalar equation