The time fractional Burgers-Huxley equation plays a significant role in modeling diffusion-reaction processes with memory effects, making it highly relevant in fluid dynamics, chemical processes, and biological systems. Solving the time fractional Burgers-Huxley equation (TFBH) in 2D and 3D is challenging due to its nonlinearity and the complexity of fractional derivatives. Traditional numerical methods often face limitations in accuracy and efficiency. To address these challenges, hybrid analytical techniques combining integral transforms with perturbation methods have gained popularity. The Laplace transform efficiently handles fractional derivatives, while the homotopy perturbation method ensures rapid convergence. This blended work offers a powerful and reliable approach, overcoming computational challenges associated with conventional methods.

In this work, the Laplace transform based homotopy perturbation method is utilized to handle the time-fractional generalized Burgers-Huxley (GBH) equation.

Where$0<\theta \le 1,$ $f\left(\zeta \right)=\mathit{\beta \zeta }\left(1-{\zeta }^{\delta }\right)\left({\zeta }^{\delta }-\gamma \right)$ is reaction term.$\mathit{ \alpha }>0$is the advection coefficient, $\beta >0$ is the reaction coefficient, and $\gamma ,\mathit{ \delta }$ are arbitrary constants which follows the condition $\gamma \in \left(0,1\right),\mathit{ \delta }>0$. If $\beta =0$ and$\mathit{ \delta }=1$then the Eq. (1) transforms into the classical Burgers’ equation.

The homotopy perturbation method (HPM) was first introduced as an efficient approach in . Later, a combined framework integrating the homotopy and perturbation techniques was proposed in to handle nonlinear problems. Later, in , HPM is used to handle fractional order differential equations and nonlinear wave equations, demonstrating its effectiveness. Various studies have focused on solving the Burgers-Huxley equation through analytical and numerical methods. In Hussain et al. and in Inc et al., applied Lie symmetry analysis to explore the Burgers-Huxley equation and time fractional Burgers-Huxley equation respectively. In authors utilized the dynamical approaches whereas in various researchers explored traveling and solitary wave solutions, including generalized and perturbed cases. In Kamboj et al. utilized iterative techniques for analytical solutions of Burgers-Huxley equation, whereas in Hayat et al. employed numerical approaches incorporating the Mittag-Leffler function for the time-fractional form. Additionally, generalized Burgers-Huxley equation is tackled by ADM in , by VIM in , by homotopy perturbation method in to obtain exact and approximate solutions, in authors introduced an algorithm based on the Elzaki transform, and in Az-Zobi utilized the reduced differential transform method, demonstrating diverse solution strategies for the Burgers-Huxley equation. Various researchers have employed the Laplace homotopy perturbation method (LHPM) to tackle nonlinear PDEs. Idowu et al. investigated its effectiveness for multidimensional nonlinear systems, while Johnston et al. extended its application to fractional-order Burgers’ equations. Aminikhah and Hemmatnezhad utilized this technique for heat transfer problems and stiff ODEs. Filobello-Nino et al. demonstrated its suitability for boundary value problems and fluid dynamics models. More recently, Owolabi et al. applied LHPM to fractional ecological models, further illustrating its adaptability across different fields. In Appadu et al. constructed four finite difference techniques to handle 2 dimensional generalized Burgers-Huxley equations.

The structure of this paper is: Section 2 covers essential preliminary concepts, while Section 3 introduces the Laplace transform. Section 4 explains the fundamentals of the homotopy perturbation method (HPM), and Section 5 showcases the hybrid LT-HPM technique. The convergence analysis of HPM is detailed in Section 6, examples are discussed in Section 7, and key conclusions are summarized in Section 8.

Definition 2.1: A real function $g\left(t\right)\in C_{\mu }\mathrm{for}\mathrm{ }t>0 \mathrm{and }\mu \in Ɽ$ if $\exists$ $q\in Ɽ \mathrm{and }q>\mu ,$such that $g\left(t\right)=t^q\mathfrak{m}\left(t\right),$where $\mathfrak{m}\left(t\right)\in C[0,\infty )$ and $g\left(t\right)\in C_{\mu }^n$ if $g^{\left(n\right)}\in C_{\mu },\mathit{ n}\in N$.

Definition 2.2: Caputo fractional derivative of $\mathcal{h}\mathcal{\left(}\tau \right)$ is defined as

where$\mathcal{ h}\mathcal{\in }{C}_{-1}^n,\mathit{ n}-1<\alpha \le n,\mathit{ n}\mathbb{\in }\mathbb{N}\mathbb{,}\mathbb{ }\tau >0.$

The Laplace transform of $f\left(t\right)$, represented as $\mathcal{L}\left\{f\left(t\right)\right\}$is mathematically expressed by the integral:

The inverse Laplace transform is determined along a specific contour $\mathrm{\Gamma },$ referred to as the Bromwich contour, and is expressed as:

The contour $\mathrm{\Gamma }$is selected to enclose all the singularities of $\zeta \left(s\right)$.

This paper utilizes several well-known properties of the Laplace transform, including:

1. $\mathcal{L}\left[\alpha {\zeta }_1\left(t\right)+\mathrm{ }\beta {\zeta }_2\left(t\right)\right]=\alpha \mathcal{L}\left[{\zeta }_1\left(t\right)\right]+\beta \mathcal{L}\left[{\zeta }_2\left(t\right)\right]$,

Where $\alpha$ and $\beta$ are constants.

2. Let the Laplace transform of $\zeta \left(\mathfrak{t}\right)=\zeta \left(s\right),$ then

1) $\mathcal{L}\left[{\zeta }^{\text{'}}\left(t\right)\right]=\mathit{s\zeta }\left(s\right)-F\left(0\right)$

2) $\mathcal{L}\left[{\zeta }^{"}\left(t\right)\right]=s^2\zeta \left(s\right)-\mathit{s\zeta }\left(0\right)-{\zeta }^{\text{'}}\left(0\right)$

3) $\mathcal{L}\left[{\zeta }^{\left(n\right)}\left(t\right)\right]=s^n\zeta \left(s\right)-s^{n-1}\zeta \left(0\right)-{\mathtt{s}s}^{n-2}{\zeta }^{\text{'}}\left(0\right)-\dots -{\zeta }^{\left(n-1\right)}\left(0\right)$

Where ${\zeta }^{\left(n\right)}\left(\mathtt{t}\right)$ denotes the nth derivatives of $\zeta \left(\mathtt{t}\right)$.

3. $\mathcal{L}\left(1\right)=\frac{1}{s}, s>0$

4. $\mathcal{L}\left(s^n\right)=\frac{n!}{s^{n+1}}, s>0$

5. $L\left(e^{at}\right)=\frac{1}{s-a},$

J. H. He is recognized for introducing the homotopy perturbation method (HPM) , which has been widely applied in solving nonlinear partial differential equations. To illustrate the fundamental idea of this technique, let us examine a nonlinear partial differential equation.

Subject to the following boundary conditions as:

Here$\mathit{ \psi }$ as a differential operator and $g$ as a function dependent on $r$. The operator $\psi$ can be decomposed into two parts: $\rho$ which represents the linear component, and $\sigma$, which accounts for the nonlinear component. Thus, Eq. (2) can be reformulated as:

In topology, two continuous mappings between different topological spaces are said to be homotopic if it is possible to continuous transform one function into the other. This transform is referred to as a homotopy between two functions. According to the homotopy technique as discussed in let us define a homotopy:

which satisfies

Or

It takes the form

The constant $b\in [0, 1]$ is an embedding parameter. Consider the initial guess as $u_{\mathsf{0}}$, which satisfies the boundary conditions. Now from (4),

and

In topology, as the parameter $b$ varies from 0 to 1, the function $w(r,\mathit{ b})$ transitions from ${\zeta }_0\left(r\right)$to$\mathit{ \zeta }\left(r\right)$. This transformation referred to as a deformation. Consequently the expressions $\rho \left(w\right)-\rho \left({\zeta }_0\right)$ and $\psi \left(w\right)-g\left(r\right)$ are identified as homotopies. Assuming the solution of Eq. (2) can be expressed as a power series in terms of $b$:

If $b=1$, then the solution of Eq. (2) is:

Let us explore the generalized time fractional Burgers-Huxley equation in this form

With initial condition

By using Laplace transform to equation (5),

By applying differential property of the Laplace transform along with the given Initial condition

Now apply the inverse Laplace transform

Where $G\left({\mathrm{ӽ}}_1,{\mathrm{ӽ}}_2,\dots {\mathrm{ӽ}}_k,\mathtt{t}\right)$ denotes the term that originates from the source term and the given initial conditions.

Now we proceed by applying the homotopy

According to HPM, the solution takes the form of an infinite series

$\zeta \left({\mathrm{ӽ}}_1,{\mathrm{ӽ}}_2,\dots {\mathrm{ӽ}}_k,\mathtt{t}\right)=\sum _{j=0}^{\infty }{p}^j{\zeta }_j\left({\mathrm{ӽ}}_1,{\mathrm{ӽ}}_2,\dots {\mathrm{ӽ}}_k,\mathtt{t}\right)$,

The decomposition of non-linear term may be as follow

Here $H_n\left(u\right)$ is the He’s polynomial and is given as:

Here $p\in \left[0,1\right]$ is an embedding parameter and${\zeta }_j=0,1,2,\dots$ are unknown functions that need to be determined.

This is an amalgamation of Laplace transform and homotopy perturbation method. Compare the coefficients associated with corresponding indices of p,

And so on. Continue the process, the solution is:

This implies

Here, in the section we have explored the theorems that illustrate the convergence of HPM.

Theorem: Let $Ҥ$and $Ҡ$ be Banach spaces, consider a mapping $\Phi : Ҥ\to Ҡ$ that is contractive and non-linear, and $\forall ⱴ, ῡ\in Ҥ,$

As the Banach fixed point theorem, the mapping $\Phi$ possesses a unique fixed point $\zeta$, that is

In the context of homotopy perturbation method.

assume, $V_0={ⱴ}_0={\zeta }_0\in B_r\left(ⱴ\right)$

where${\mathit{ B}}_r\left(ⱴ\right)=\left\{{\zeta }^{*}\in x: ‖{\zeta }^{*}-\zeta ‖<r\right\}$ then

Proof:

1) By utilizing an inductive approach with the base case when $n=1$

Assume that

So as

Using (ⅰ)

2) Because of $‖V_n-\zeta ‖\le {\gamma }^n‖{ⱴ}_0-\zeta ‖$ and

$\underset{n\to \infty }{\lim }{\gamma }^n=0, \underset{n\to \infty }{\mathrm{ lim}}‖V_n-\zeta ‖=0$,

that is $\underset{n\to \infty }{\lim }{V}_n=\zeta$.$∎$

We now examine numerical experiments that illustrate the accuracy and ease of implementation of the suggested method.

Example 1. Consider the (2+1)-D time fractional Burgers-Huxley equation ($a=0,\mathit{ \beta }=1,\mathit{ \delta }=1,\mathit{ \gamma }=1$) of the form:

Initial condition: $\zeta \left(\mathrm{ӽ},y,0\right)=\frac{e^{\frac{\sqrt{2}}{4}\left(\mathrm{ӽ}+y\right)}}{e^{\frac{\sqrt{2}}{4}\left(\mathrm{ӽ}+y\right)}+e^{\frac{-\sqrt{2}}{4}\left(\mathrm{ӽ}+y\right)}}$

Solution: Applying the Laplace transform homotopy perturbation method on Eq. (9), we get

Compare the coefficients associated with corresponding indices of p,

and so on.

and so on. After simplification we obtain,

The solution is

The solution converges to

Figures 1, 3, 5 show the physical behavior of the solutions at different range of $x$ and $y$ for $\mathtt{t}=0.1, 10$ and 30 respectively. Figures 2, 4, 6 show the contour diagram of the solutions at different range of $x$ and $y$ for $\mathtt{t}=0.1, 10$ and 30 respectively.

Table 1 shows the absolute errors obtained by using exact solutions and the solutions obtained by Laplace transform based homotopy perturbation method (Taking first four terms of an infinite series) at $\mathrm{t}$ = 0.5. Table 2 shows the absolute errors obtained by using exact solutions and the solutions obtained by Laplace transform based homotopy perturbation method (Taking first four terms of an infinite series) at $t$ = 3s.

Example 2. Consider the (3+1)-D time fractional Burgers-Huxley equation ($a=-1,\mathit{ \beta }=1,\mathit{ \delta }=1,\mathit{ \gamma }=1$) of the form:

Initial condition: $u\left(\mathrm{ӽ},y,0\right)=\frac{e^{\frac{-\left(\mathrm{ӽ}+y+z\right)}{4}}}{e^{\frac{\left(\mathrm{ӽ}+y+z\right)}{4}}+e^{\frac{-\left(\mathrm{ӽ}+y+z\right)}{4}}}$

Solution: Applying the Laplace transform homotopy perturbation method on Eq. (10), we get Eq. number (8).

After simplification, we get

The solution is