The Solutions of Nonlinear Heat Conduction Equation via Fibonacci&Lucas Approximation Method

To obtain new types of exact travelling wave solutions to nonlinear partial differential equations, a number of approximate methods are known in the literature. In this study, we extend the class of auxiliary equations of Fibonnacci&Lucas type equations. The proposed Fibonnacci&Lucas approximation method produces many new solutions. Consequently, we introduce new exact travelling wave solutions of some physical systems in terms of these new solutions of the Fibonacci&Lucas type equation. In addition to using different ansatz, we use determine different balancing principle to obtain optimal solutions.

💡 Research Summary

The paper introduces a novel analytical technique for obtaining exact travelling‑wave solutions of nonlinear heat conduction equations, termed the Fibonacci‑Lucas Approximation Method. Traditional auxiliary‑equation approaches (e.g., Bernoulli, Riccati, or tanh‑method) rely on a limited set of special functions, which often restrict the diversity of obtainable solutions, especially when the governing PDE exhibits strong nonlinearity or rapid spatial variations. To overcome this limitation, the authors propose to use the second‑order differential equation satisfied by the continuous extensions of the Fibonacci and Lucas sequences as a new auxiliary equation. This “Fibonacci‑Lucas” equation possesses two independent fundamental solutions—continuous analogues of the Fibonacci numbers (F(z)) and Lucas numbers (L(z))—which display growth and oscillatory characteristics markedly different from exponential or trigonometric functions.

The methodology proceeds as follows. First, the nonlinear heat conduction equation \