Many problems exist in different fields of study such as wave propagation, fluid mechanics, mechanical engineering, dynamical systems, chemistry, image processing, plasma physics, hydrodynamics, finance, biology, optics and other fields of engineering and science. Some scientists have proposed and researched nonlinear fractional partial differential equations (NFPDEs) (Zubair et al.
2018; Raza et al.
2019; Hosseini et al.
2020). Suggested numerous different definitions have been presented in the study (Yang et al.
2019; Park et al.
2020). Lately, investigators have begun to view a deficit at most of the fractional derivative definitions (Samko et al.
1993; Kilbas et al.
2006). Since fractional differential equations are an important field in plasma physics, mathematical physics, mathematical biology, nonlinear optics, applied mathematics, and quantum field theory, the solutions of these equations, along with their soliton-type solutions, have become very important. Fractional derivatives have an important place in the study of real-world problems so there is a different types of fractional derivatives for example Caputo fractional derivative, conformable derivative, modified Riemann-Liouville derivative, Riemann-Liouville derivative, beta derivative and many others. The universe is filled with events nonlinear and inner behaviors of this nonlinear are modeled for samples next to nonlinear alongside differential equations of fractional order as well as whole number order (Miller and Ross
1993; Hassani et al.
2020). It is thought that fractional-order equations analyze the intricate properties of complicated physical phenomena that occur in many fields of study on a small scale (Gomez-Aguilar et al.
2018; Bonyah et al.
2018). Later, scientists searched deeply to find approximate and suitable solutions nonlinear evolution equations, and as a result of their research, many methods have been developed recently to solve nonlinear evolution equations due to the differing opinions of scientists. On the instant, Seadawy handled the extended auxiliary equation procedure (Seadawy
2017), Duran et al. were interested in the modified
\(\left( 1/G^{\prime }\right)\)-expansion practice (Duran et al.
2021a), Jianming et al. investigated the Backlund transformation procedure (Jianming et al.
2011), Yokus analyzed the extended finite difference procedure (Yokus
2018), Ablowitz and Clarkson handled the inverse scattering transformation practice (Ablowitz and Clarkson
1991), Duran et al. investigated the Bernoulli sub-equation function method (Duran et al.
2021b), Helal and Mehana investigated the Adomian decomposition practice (Helal and Mehana
2006), Al-Mdallal and Syam were interested in the sine–cosine procedure (Al-Mdallal and Syam
2007), Das and Ghosh paid attention to the
\(\left( G^{\prime }/G\right)\) -expansion procedure (Das and Ghosh
2019), Islam and Akter handled the rational fractional
\(\left( D_{\xi }^{\alpha }G/G\right)\)-expansion practice (Islam and Akter
2020), Mohyud-Din et al. handled the variational iteration practice (Mohyud-Din et al.
2009), Hashemi and Mirzazadeh were interested in the Lie symmetry practice (Hashemi and Mirzazadeh
2023), Wazwaz analyzed the sine-cosine procedure (Wazwaz
2004), Bekir investigated the
\((G^{\prime }/G)\)-expansion procedure (Bekir
2008), Arshed et al. were interested in the first integral practice (Arshed et al.
2020), Biswas et al. handled the modified simple equation method (Biswas et al.
2018), Celik analyzed the F expansion practice (Çelik
2021), Kudryashov acquired the exact solutions of the Fisher model by the Kudryashov method (Kudryashov
2012), Kudryashov handled the first integral method (Kudryashov
2020), Kudryashov explored the general projective Riccati equations and the enhanced Kudryashov’s methods (Kudryashov
2023), Wang et al. applied the semi-inverse method to fractal (2 + 1)-Dimensional Zakharov–Kuznetsov model (Wang
2023a; Wang and Xu
2023; Wang et al.
2023a), Alquran applied the Maclaurin series to nonlinear equations (Alquran
2023a), Alquran applied the rational sine-cosine approach to second fourth-order Wazwaz equation (Alquran
2023b), Jaradat and Alquran applied the Kudryashov expansion method to (2 + 1)-dimensional two-mode Zakharov–Kuznetsov equation (Jaradat and Alquran
2020), Ghanbari applied the algorithm of the new method to the Oskolkov and the Oskolkov–Benjamin–Bona–Mahony–Burgers equations (Ghanbari
2021a), Ghanbari and Gómez–Aguilar applied the generalized exponential rational function procedure to the nonlinear Radhakrishnan–Kundu–Lakshmanan model (Ghanbari and Gómez-Aguilar
2019a), Sadaf et al. applied the
\(\left( \frac{G^{\prime }}{G}, \frac{1}{G}\right)\)-expansion method to CI equation (Sadaf et al.
2023a), Mahmood et al. applied the modified Khater method to the (2 + 1)-dimensional Chaffee–Infante equation (Mahmood et al.
2023), Akram et al. applied the extended
\(\left( \frac{G^{\prime }}{G^{2}}\right)\)-expansion method to the higher order nonlinear Schrödinger equation (Akram et al.
2023a,
2023b,
c) and so on (Wang
2023b,
c,
d,
e,
f; Wang and Shi
2022; Wang et al.
2023b,
c; Ali et al.
2019; Jaradat et al.
2018; Jaradat and Alquran
2022; Ghanbari
2022; Ghanbari and Gómez-Aguilar
2019b; Ghanbari and Baleanu
2019,
2020,
2023a,
2023b; Khater and Ghanbari
2021; Ghanbari
2019,
2021b; Ghanbari et al.
2018; Ghanbari and Akgül
2020; Ghanbari and Kuo
2019; Tian et al.
2022; Sadaf et al.
2023b.
The Chaffee–Infante (CI) model, which is useful for studying the diffusion formation of a gas in a uniform medium, is very important. Therefore, it has an important role in the field of mathematics and physics (Raza et al.
2021). The CI equation was first studied by Nathaniel Chafee and Ettore Infante. The most interesting aspect is a bifurcation in the system parameter that indicates the steepness of the potential. The CI model is very important in many areas, for example; such as ion-acoustic waves in plasma, fluid dynamics, plasma physics, sound waves, and electromagnetic waves Sriskandarajah and Smiley (
1996). This model is the standard representation of endless-dimensional gradient systems in which the structure of the spherical attractor can be exactly characterized (Caraballo et al.
2007). The appropriate derivative time-fractional CI equation is as follows:
$$\begin{aligned} \left( \frac{\partial ^{\beta }u}{\partial t^{\beta }}\right) _{x}-\left( \frac{\partial ^{2}u}{\partial x^{2}}-\alpha u^{3}+\alpha u\right) _{x}+\theta \frac{\partial ^{2}u}{\partial y^{2}}=0 \end{aligned}$$
(1)
where
\(\alpha\) represents the coefficient of diffusion and
\(\theta\) represent degradation coefficient. The diffusion of a gas in a homogeneous medium is an important phenomenon in a physical context and the CI model provides a useful model to study such phenomena.