[Paper Review] Solitary wave solutions of FKPP equation using Homogeneous balance method(HB method)

Yirui Yang, Wei Kou+2 more|arXiv (Cornell University)|Sep 24, 2020

Fluid Dynamics Simulations and Interactions4 citations

TL;DR

This paper applies the homogeneous balance (HB) method to derive exact solitary wave solutions for the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation, a nonlinear reaction-diffusion equation. By constructing a functional transformation via the HB principle and solving the resulting ordinary differential equation, the authors obtain explicit solutions in terms of hyperbolic secant and tangent functions, validated through graphical analysis and parameter constraints.

ABSTRACT

In this paper, we use the homogeneous balance(HB) method is used to construct function transformation to solve the nonlinear development equation||Fisher-Kolomogror-Pertrovskii-Piskmov equation (FKPP equation), then the exact solution of FKPP equation is obtained, and the solutionis converted into the form of isolated wave solution. Finally, we have shown the solution of FKPPequation in some picture forms, and the rationality of the solution in this paper can be verified byusing the pictures.

Motivation & Objective

• To develop a systematic approach for solving the nonlinear FKPP equation using the homogeneous balance (HB) method.
• To transform the FKPP equation into a solvable ordinary differential equation (ODE) through functional transformation based on the HB principle.
• To derive exact solitary wave solutions expressed in terms of hyperbolic functions (sech and tanh).
• To validate the analytical solutions through graphical representation and consistency checks with the original equation.
• To determine the parameter constraints (k, c, a, b) that yield physically meaningful and mathematically consistent solutions.

Proposed method

• The HB method is applied to assume a solution form involving higher-order partial derivatives of a function φ(x,t), with the highest derivative order N = 2.
• The ansatz solution is structured as u(x,t) = f''(φ)φ_x² + a f'(φ)φ_x + b, where f(φ) is determined via the balance of nonlinear and highest-order derivative terms.
• The function f(φ) is found to be f(φ) = -6 ln φ, which leads to a solvable ODE system under the HB principle.
• The solution is reduced to a form involving sech² and tanh functions by introducing φ(x,t) = 1 + e^(kx + ct), enabling explicit solitary wave representation.
• The parameters k and c are derived by balancing the highest-order terms and ensuring consistency with the original FKPP equation.
• The final solution is expressed as u(x,t) = -3/2 k² sech²[½(kx + ct)] + 3ak tanh[½(kx + ct)] - 3ak + b, with specific parameter values determined by substitution into the FKPP equation.

Experimental results

Research questions

• RQ1 Can the homogeneous balance method be effectively applied to derive exact solitary wave solutions for the FKPP equation?
• RQ2 What functional form of f(φ) and φ(x,t) satisfies the balance condition between nonlinear and highest-order derivative terms in the FKPP equation?
• RQ3 What are the explicit analytical forms of the solitary wave solutions in terms of hyperbolic functions?
• RQ4 How do the parameters k, c, a, b affect the shape and dynamics of the solitary wave solutions?
• RQ5 Can the derived solutions be verified as exact solutions through substitution and graphical analysis?

Key findings

• The homogeneous balance method successfully reduces the FKPP equation to a solvable ODE system, yielding exact solutions in terms of hyperbolic functions.
• The solution takes the form u(x,t) = -3/2 k² sech²[½(kx + ct)] + 3ak tanh[½(kx + ct)] - 3ak + b, with k and c determined by the parameters a and b.
• For b = 0, c = 5/6, and k = -1/(6a), the solution becomes u(x,t) = -1/(24a²) sech²[½(-x/(6a) + 5t/6)] + (1/2) tanh[½(-x/(6a) + 5t/6)] + 1/2.
• For b = 1, c = -5/6, and k = 1/(6a), the solution is u(x,t) = -1/(24a²) sech²[½(x/(6a) - 5t/6)] - (1/2) tanh[½(x/(6a) - 5t/6)] + 1/2.
• The parameter a is constrained to ±1/√6 to satisfy the original FKPP equation, ensuring consistency of the derived solution.
• Graphical representations (Figure 1) confirm the solitary wave nature of the solutions, showing localized, stable wave profiles for both positive and negative a values.