New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

Abstract

The generalized time fractional Kolmogorov-Petrovsky-Piskunov equation (FKPP),, which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet-Neumann, and Neumann-Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

Effect of the nonlinear dispersive coefficient on time-dependent variable coefficient soliton solutions of the Kolmogorov-Petrovsky-Piskunov model arising in biological and chemical science

“…On contrast with the solutions of the KPP model in Refs. [ [33] , [34] , [35] , [36] , [37] , [38] , [39] ], we observed that all of the authors used to find exact traveling wave solutions by the relation . But our solution covers all of their solutions when we consider both are linear in the transformation relation .…”

“…12 ). Besides this, we can use fractional power of the time variable in the function taking constant which cover the fractional solutions obtained by the solutions of [ 33 , 38 ]. Actually, arbitrariness of time varying function suggested appreciating machinery for chaotic behaviors in ocean engineering phenomena in identifiable complications.…”

“…(1) converts to the FitzHugh–Nagumo model, and for , it converts to the case form of Fisher model. Recently diverse methods are applied to solve the KPP model such as Thanon Korkiatsakul applied the Chebyshev Wavelet technique [ 33 ] to analyze the new analytical solutions for the t-FKPP model, Yu-Ming Chu used the effectual approach, the first integral technique [ 34 ] for this equation, Ben applied analytical and numerical method [ 35 ] with initial-boundary value problems of KPP equation, Ma found explicit and exact solutions by applying Cole-Hopf transformation and Backlund transformations [ 36 ] of the KPP model, Zhang found the approximate analytical solution of the GKPP equation by using a hypothesis undetermined technique [ 37 ], Song used a new form of GT-DDT technique [ 38 ] for nonlinear fractional KPP equations, Feng applied expansion method [ 39 ] to seek the traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation, and so on.…”