Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

Abstract

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

1. Introduction

A somewhat archaic topic in mathematics is fractional-order calculus. Like the subject of meaning extension, fractional derivatives debuted in 1695. Compared to classical calculus, derivatives and integrals of arbitrary order provide more factual models of real-world phenomena [1]. Many pioneers in fractional calculus published substantial research on the subject during the 20th century, including Caputo [2], Miller and Ross [3], Podlubny [4], and many others. Fractional calculus has proven to be a valuable tool in today’s world through its applicability to several fields of science and engineering. As it turns out, Scalas, Gorenflo, and Mainardi used it in the year 2000 to show that it was valid for continuous-time finance while offering a framework for modeling the finance system. In this regard, the study by West, Turalska, and Grigolini [5] also pointed out that fractional calculus could connect the microscopic and macroscopic in terms of network dynamics, stressing the importance of this approach in capturing realistic and proper complex-system behavior and performance [6]. Furthermore, Tarasov [7] generalized the concept of fractional calculus to vector calculus and Maxwell’s equations, contributing novel techniques to the analysis of various problems in electromagnetics and many other disciplines. These preliminary papers emphasized fractional calculus’s mobile usefulness and competency regarding numerous theoretical and pragmatic issues [8,9,10,11].

Systems identification, fluid flow, signal processing, control theory, chemistry, physics, biology, engineering, fractional dynamics, and signal processing are among the fields in which fractional differential equations (FDEs) are widely used [3,4,12,13,14,15,16]. They are also used in the social sciences, including economics, finance, climatology, and food supplements. The FDEs that emerge in diffusion problems were initially examined by Oldman and Spanier [16].

The variational iteration transform method (VITM) can be the improved version of the variational iteration method (VIM), which uses different integral transforms like those of Laplace, Elzaki, Fourier, etc. Thus, the presented hybrid technique can be used to obtain accurate and efficient numerical solutions to the linear and nonlinear ODEs of the first and second orders, as well as fractional ones. The concept of the VITM can support the basic idea of the VIM, which is the subsequent enhancement of approximate solutions while taking advantage of the properties of integral transformations about worsening differential operators and boundary problems [17,18,19]. Thus, by simplifying the solution process, the VITM increases the convergence rate and accuracy of the solutions obtained. In recent years, the VITM has attracted much attention, and it has been successfully used in many engineering, physics, and applied mathematics problems. Flexible in application to initial and boundary-value problems and appropriate to the study of fractional differential equations, it has to be viewed as a powerful and handy tool amongst the numerous mathematical methods and models [20,21,22,23].

In nature, there are various transport processes, one of which is diffusion. By substituting fractional derivatives for the integer time and/or spatial derivative, fractional diffusion equations (FDEs) are obtained. Because of its proven uses in a wide range of different and widespread science and engineering sectors, this topic has drawn much attention throughout the past forty years or so [24,25,26]. Fractional diffusion is now a frequently utilized concept in particle diffusion [27], chemistry [28], biology [29], and finance and economics [30,31].

Elzaki developed the Elzaki transform [32] based on the classical Fourier integral and an existing variant of the Sumudu transform. The Elzaki transform simplifies the mathematical complexity of solving ODEs and partial differential equations (PDEs) in the time domain [33,34,35]. This is so because the Elzaki transform provides a simple mathematical method. Thus, the main goal of the current investigation is to use the hybrid VIM with the Elzaki transform to analyze some physical FDEs, including fractional diffusion equations. This approach is not limited to only the aforementioned models but can also be applied to many different physical fractional evolution equations, especially the fractional wave equations used for modeling unmodulated [36,37,38] and modulated [39,40,41] nonlinear structures in the physics of plasmas.

2. Basic Definitions

Definition 1.

The Elzaki transform [42] is a novel integral transform intended for functions of exponential order. Look over the functions in the set G, defined by

$$G=\left\{h\left(\delta \right):\exists Q,p_1,p_2>0,\left|h\left(\delta \right)\right|<Q{e}^{{\displaystyle \frac{\left|\delta \right|}{p_i}}},\mathrm{if}\delta \in {(-1)}^i\times [0,\infty )\right\}.$$

Definition 2.

The Elzaki transform of the function $h\left(\delta \right)$, defined for all $\delta \ge 0,$ is defined by $E\left[h\right]$:

$$E\left[h\left(\delta \right)\right]=T\left(s\right)=s{\int }_0^{\infty }h\left(\delta \right)e^{-{\displaystyle \frac{\delta }{s}}}d\delta ,\delta \ge 0,p_1\le \delta \le p_2.$$

(1)

Theorem 1.

Let us consider the Power series coefficients

$$h\left(\delta \right)=\sum _{n=0}^{\infty }{a}_n{\delta }^n,$$

(2)

And, by applying the Elzaki transform to Equation (2), we get

$$E\left[h\left(\delta \right)\right]=T\left(v\right)=\sum _{n=0}^{\infty }n!a_n{v}^{n+2}.$$

(3)

Theorem 2.

For $n\ge 1$, define $T_n\left(v\right)$ as the Elzaki transform of the $n^{th}$ derivative of $h^n\left(\delta \right)$ of $h\left(\delta \right)$ in G

$$T_n\left(v\right)={\displaystyle \frac{T\left(v\right)}{v^n}}-\sum _{j=0}^{n-1}{v}^{2-n+j}{h}^{\left(j\right)}\left(0\right).$$

(4)

The Elzaki transform, which arises from the use of the integration-by-parts technique, is denoted as

$$\begin{array}{cc}\hfill E\left({\displaystyle \frac{\partial h(y,\delta )}{\partial \delta }}\right)& ={\displaystyle \frac{1}{v}}T(y,v)-vh(y,0),\hfill \\ \hfill E\left({\displaystyle \frac{{\partial }^{2}h(y,\delta )}{\partial {\delta }^2}}\right)& ={\displaystyle \frac{1}{v^2}}T(y,v)-h(y,0)-v{\displaystyle \frac{\partial h(y,0)}{\partial \delta }}.\hfill \end{array}$$

(5)

Transformation of Fractional Derivatives via the Elzaki Transform

Through the application of the Laplace transform formula, one can obtain the Elzaki transform formula for the Caputo fractional derivative [4] as follows:

$$L\{{(}^c{D}_{\delta }^{\varrho }h)\left(\delta \right);u\}=u^{\varrho }F\left(u\right)-\sum _{i=0}^{n-1}{s}^{\varrho -i-1}{f}^{\left(i\right)}\left(0\right),$$

where $n-1<\varrho \le n,n\in \mathbb{N}\ast .$

Theorem 3.

The Laplace transform, $F\left(u\right)$, can be utilized to acquire the Elzaki transform, $T\left(v\right)$, of the function $h\left(\delta \right)$ as follows [43]

$$T\left(v\right)=vF\left({\displaystyle \frac{1}{v}}\right).$$

Theorem 4.

Let us assume that the Elzaki transform of the function $h\left(\delta \right)$ is denoted as $T\left(v\right)$ [44]

$$E\left\{{(}^c{D}_{\delta }^{\varrho }h)\left(\delta \right),v\right\}={\displaystyle \frac{T\left(v\right)}{v^{\varrho }}}-\sum _{i=0}^{n-1}{v}^{i-\varrho -2}{h}^{\left(i\right)}\left(0\right).$$

(6)

3. Variational Iteration Transform Method (VITM)

We shall expand the use of the VITM [45] to solve nonlinear partial differential equations (PDEs) of the order $\varrho ,(n-1<\varrho \le n,n=1,2,\dots )$. To understand this reasoning, we examine a general nonlinear PDE with a time-fractional derivative

$${}^{c}D_{\delta }^{\varrho }\mu (y,\delta )+R\mu (y,\delta )+N\mu (y,\delta )=f(y,\delta ),$$

(7)

which becomes, when subject to initial conditions (ICs),

$${\left[{\displaystyle \frac{{\partial }^{n-1}\mu (y,\delta )}{\partial {\delta }^{n-1}}}\right]}_{\delta =0}=g_{n-1}\left(y\right),$$

(8)

where ${\displaystyle \frac{{\partial }^{\varrho }}{\partial {\delta }^{\varrho }}}{=}^c{D}_{\delta }^{\varrho }$ indicates the Caputo fractional derivative, $g(y,\delta )$ represents the source term, R indicates the linear differential operator, and N represents the generic nonlinear differential operator.

Applying the Elzaki transform to both sides of (7) yields

$$E{[}^c{D}_{\delta }^{\varrho }\mu (y,\delta )+E\left[R\mu (y,\delta )\right]+E\left[N\mu (y,\delta )\right]=E\left[f(y,\delta )\right],$$

(9)

According to the properties of the Elzaki transform, Equation (9) can be rewritten as follows:

$$E\left[\mu (y,\delta )\right]=\sum _{i=0}^{n-1}{v}^{i+2}gi\left(y\right)+v^{\varrho }E\left[f(y,\delta )\right]-v^{\varrho }E[R\mu (y,\delta )+N\mu (y,\delta )].$$

(10)

Applying the inverse Elzaki transform to both sides of Equation (10) yields

$$\mu (y,\delta )=K(y,\delta )-E^{-1}\left(v^{\varrho }E[R\mu (y,\delta )+N\mu (y,\delta )]\right),$$

(11)

with

$$K(y,\delta )=\sum _{i=0}^{n-1}{v}^{i+2}{g}_i\left(y\right)+v^{\varrho }E\left[f(y,\delta )\right].$$

Applying ${\displaystyle \frac{\partial }{\partial \delta }}$ to both sides of (11) implies

$${\displaystyle \frac{\partial \mu (y,\delta )}{\partial \delta }}+{\displaystyle \frac{\partial }{\partial \delta }}{E}^{-1}\left(v^{\varrho }E[R\mu (y,\delta )+N\mu (y,\delta )]\right)-{\displaystyle \frac{\partial K(y,\delta )}{\partial \delta }}=0.$$

(12)

The VIM allows us to generate a correct functional in the following manner: [46]

$${\mu }_{m+1}(y,\delta )={\mu }_m(y,\delta )-{\int }_0^{\delta }\left[{\displaystyle \frac{\partial {\mu }_m}{\partial \zeta }}+{\displaystyle \frac{\partial }{\partial \zeta }}E-1\left(v^{\varrho }E[R{\mu }_m+N{\mu }_m]\right)-{\displaystyle \frac{\partial K}{\partial \zeta }}\right]\partial \zeta ,$$

(13)

which is equivalent to

$${\mu }_{m+1}(y,\delta )=K(y,\delta )-E^{-1}\left(v^{\varrho }E[R{\mu }_m(y,\delta )+N{\mu }_m(y,\delta )]\right).$$

(14)

Remember that $\mu (y,\delta )={lim}_{m\to \infty }{\mu }_m(y,\delta ).$

If a solution to the problem at hand exists, we can find it precisely, or if not, we can use the prior limit to find an approximation of the solution.

4. Application

In the subsequent examples, we shall employ the methodology described in the previous paragraph to analyze time-fractional PDEs.

4.1. Example-(I)

Let us consider the fractional diffusion equation

$$D_{\delta }^{\varrho }\mu +{\mu }_{yy}+{\mu }_{\varphi \varphi }+{\mu }_{\phi \phi }=0,$$

(15)

with the IC

$$\mu (y,\varphi ,\phi ,0)\equiv {\mu }_0={exp}^{y+\varphi +\phi },$$

(16)

where $D_{\delta }^{\varrho }\mu \equiv {\partial }_{\delta }^{\varrho }\mu$, ${\mu }_{yy}\equiv {\partial }_y^2\mu$, ${\mu }_{\varphi \varphi }\equiv {\partial }_{\varphi }^2\mu$, ${\mu }_{\phi \phi }\equiv {\partial }_{\phi }^2\mu$, $\mu \equiv \mu (y,\varphi ,\phi ,\delta )$, and $0<\varrho \le 1$.

The exact solution for Equation (15) at $\varrho =1$ reads

$${\mu }_{Exact}\equiv \mu ={exp}^{y+\varphi +\phi -3\delta }.$$

(17)

After applying the Elzaki transform and then the inverse Elzaki transform to Equation (15), we get

$${\mu }_{m+1}(y,\varphi ,\phi ,\delta )={exp}^{y+\varphi +\phi }-E^{-1}(v^{\varrho }.E[{\left({\mu }_m\right)}_{yy}+{\left({\mu }_m\right)}_{\varphi \varphi }+{\left({\mu }_m\right)}_{\phi \phi }]).$$

(18)

The iteration Formula (18) gives us

$$\begin{array}{cc}\hfill {\mu }_0& ={exp}^{y+\varphi +\phi },\hfill \\ \hfill {\mu }_1& ={\mu }_0-{\displaystyle \frac{3{\delta }^{\varrho }{exp}^{y+\varphi +\phi }}{\Gamma \left[1+\varrho \right]}},\hfill \\ \hfill {\mu }_2& ={\mu }_1+{\displaystyle \frac{9{\delta }^{2\varrho }{exp}^{y+\varphi +\phi }}{\Gamma [1+2\varrho ]}},\hfill \\ \hfill {\mu }_3& ={\mu }_2-{\displaystyle \frac{27{\delta }^{3\varrho }{exp}^{y+\varphi +\phi }}{\Gamma [1+3\varrho ]}},\hfill \\ \hfill {\mu }_4& ={\mu }_3+{\displaystyle \frac{81{\delta }^{4\varrho }{exp}^{y+\varphi +\phi }}{\Gamma [1+4\varrho ]}},\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(19)

The exact solution (17) is recovered for

$$\mu (y,\varphi ,\phi ,\delta )=\underset{m\to \infty }{lim}{\mu }_m(y,\varphi ,\phi ,\delta ).$$

Accordingly, the approximate solution to Equation (15) until the third iteration reads

$${\mu }_{App}\equiv \mu ={exp}^{y+\varphi +\phi }\left(1-{\displaystyle \frac{3{\delta }^{\varrho }}{\Gamma \left[1+\varrho \right]}}+{\displaystyle \frac{9{\delta }^{2\varrho }}{\Gamma [1+2\varrho ]}}-{\displaystyle \frac{27{\delta }^{3\varrho }}{\Gamma [1+3\varrho ]}}+{\displaystyle \frac{81{\delta }^{4\varrho }}{\Gamma [1+4\varrho ]}}+\cdots \right).$$

(20)

Given the specific circumstances $\varrho =1$,

$$\mu ={exp}^{y+\varphi +\phi }\left(1-3\delta +{\displaystyle \frac{9}{2}}{\delta }^2-{\displaystyle \frac{9}{2}}{\delta }^3+\cdots \right),$$

Thus, the exact solution is recovered for $m\to +\infty$ as follows:

$$\mu ={exp}^{y+\varphi +\phi -3\delta }.$$

The derived approximation (20) ${\mu }_{App}$ is investigated against the fractional parameter (FP) $p\equiv \varrho$, as illustrated in Figure 1. In this figure, we examine the impact of the FP $p\equiv \varrho$ on the behavior of this approximation. Also, we compare the approximation (20) at $p=1$ with the exact solution (17) ${\mu }_{Exact}$ graphically and numerically, as shown in Figure 2 and

Table 1. The absolute error of the approximation (20) to the problem (15) is estimated at $\left(\varphi ,\phi ,\delta \right)=\left(0.1,0.1,0.001\right)$.

y	${\mathit{VIM}}_{\mathit{P}=1.0}$	$\mathit{Exact}$	${10}^{-15}$${\mathit{L}}_{\mathit{\infty }}$
−0.5	0.738599	0.738599	1.4432899320127035
−0.4	0.816278	0.816278	1.6653345369377348
−0.3	0.902127	0.902127	1.887379141862766
−0.2	0.997004	0.997004	1.9984014443252818
−0.1	1.10186	1.10186	2.220446049250313
0.	1.21774	1.21774	2.4424906541753444
0.1	1.34582	1.34582	2.6645352591003757
0.2	1.48736	1.48736	2.886579864025407
0.3	1.64378	1.64378	3.3306690738754696
0.4	1.81666	1.81666	3.774758283725532
0.5	2.00772	2.00772	3.9968028886505635

. In Table 1, we estimate the absolute error $L_{\infty }$ of the derived approximation (20) ${\mu }_{App}$ at $p=1$:

$$L_{\infty }=|{\mu }_{Exact}-{\mu }_{Ap{p}_{p=1}}|.$$

The analysis results clearly show an almost perfect match between the derived approximation and the exact solution, enhancing the strength of the technique. Furthermore, by utilizing the Aboodh/Laplace Residual Power Series Approach (LRPSA) [15,47], and after engaging in numerous extensive, although straightforward, computations, we reach the same subsequent approximation to problem (15):

$${\mu }_{App}\equiv {exp}^{y+\varphi +\phi }\left(1-{\displaystyle \frac{3{\delta }^{\varrho }}{\Gamma \left[1+\varrho \right]}}+{\displaystyle \frac{9{\delta }^{2\varrho }}{\Gamma [1+2\varrho ]}}-{\displaystyle \frac{27{\delta }^{3\varrho }}{\Gamma [1+3\varrho ]}}+{\displaystyle \frac{81{\delta }^{4\varrho }}{\Gamma [1+4\varrho ]}}+\cdots \right).$$

(21)

Furthermore, we examined and resolved the problem (15) utilizing the Aboodh transform iteration method (ATIM) [15,47], withholding specific details here because the goal is the VIM, not the ATIM. The resulting approximation is as follows:

$${\mu }_{App}={exp}^{y+\varphi +\phi }\left(1-{\displaystyle \frac{3{\delta }^{\varrho }}{\varrho \Gamma \left[\varrho \right]}}+{\displaystyle \frac{9{\delta }^{2\varrho }}{{\Gamma }^2[1+\varrho ]}}-{\displaystyle \frac{27{\delta }^{3\varrho }}{\varrho \Gamma \left[\varrho \right]{\Gamma }^2[1+\varrho ]}}+{\displaystyle \frac{81{\delta }^{4\varrho }}{\varrho \Gamma \left[\varrho \right]{\Gamma }^3[1+\varrho ]}}+\cdots \right).$$

(22)

Additionally, the derived approximation (20) using the VIM with the Elzaki transform was compared to the approximation (22) achieved through the use of the ATIM, as presented in

Table 2. A comparison between the absolute error of the two approximations (20) and (22) is estimated at $\delta =0.001$.

y	${\mathit{VIM}}_{\mathit{P}=1}$	${\mathit{ATIM}}_{\mathit{P}=1}$	$\mathit{Exact}$	${10}^{-15}$${\mathit{L}}_{\mathit{\infty }}$	${10}^{-6}$${\mathit{L}}_{\mathit{\infty }}$
−0.5	0.738599	0.738602	0.738599	1.44329	3.31707
−0.4	0.816278	0.816282	0.816278	1.66533	3.66593
−0.3	0.902127	0.902131	0.902127	1.88738	4.05148
−0.2	0.997004	0.997009	0.997004	1.9984	4.47758
−0.1	1.10186	1.10187	1.10186	2.22045	4.94849
0.	1.21774	1.21775	1.21774	2.44249	5.46893
0.1	1.34582	1.34582	1.34582	2.66454	6.0441
0.2	1.48736	1.48736	1.48736	2.88658	6.67976
0.3	1.64378	1.64379	1.64378	3.33067	7.38228
0.4	1.81666	1.81667	1.81666	3.77476	8.15868
0.5	2.00772	2.00773	2.00772	3.9968	9.01673

. The comparison findings indicate that approximation (20) stands out from approximation (22) due to its superior accuracy, hence enhancing the effectiveness of VIM in evaluating more complicated and strong nonlinearity evolutionary equations.

4.2. Example-(II)

Here, we considered a new case for the fractional diffusion equation when the coefficient of dispersion becomes more dominant than that of advection

$$D_{\delta }^{\varrho }\mu +2{\mu }_y+5{\mu }_{yyy}=0,0<\varrho \le 1,$$

(23)

subject to the IC

$${\mu }_0\equiv \mu (y,0)=sin\left(y\right).$$

(24)

For $\varrho =1$, the exact solution of Equation (23) reads

$$\mu (y,\delta )=sin(y+3\delta ).$$

(25)

Based on Equation (14), the following recurrence formula for Equation (23) reads

$${\mu }_{m+1}\equiv {\mu }_{m+1}(y,\delta )=sin\left(y\right)-E^{-1}(v^{\varrho }.E[2{\left({\mu }_m\right)}_y+5{\left({\mu }_m\right)}_{yyy}]).$$

(26)

Now, by utilizing the iteration Formula (26), we get

$$\begin{array}{cc}\hfill {\mu }_0& =sin\left(y\right),\hfill \\ \hfill {\mu }_1& ={\mu }_0+{\displaystyle \frac{3{\delta }^{\varrho }cos\left(y\right)}{\Gamma (1+\varrho )}},\hfill \\ \hfill {\mu }_2& ={\mu }_1-{\displaystyle \frac{9{\delta }^{2\varrho }sin\left(y\right)}{\Gamma [1+2\varrho ]}},\hfill \\ \hfill \mu 3& ={\mu }_2-{\displaystyle \frac{27{\delta }^{3\varrho }cos\left(y\right)}{\Gamma [1+3\varrho ]}},\hfill \\ \hfill \mu 4& ={\mu }_3+{\displaystyle \frac{81{\delta }^{4\varrho }sin\left(y\right)}{\Gamma [1+4\varrho ]}},\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(27)

Remember that the exact equation for Equation (23) can be recovered for

$$\mu (y,\delta )=\underset{m\to \infty }{lim}{\mu }_m(y,\delta ).$$

Accordingly, the approximate solution to Equation (23) becomes

$$\begin{array}{cc}\hfill {\mu }_{\mathrm{App}}& =\mu =sin\left(y\right)+{\displaystyle \frac{3{\delta }^{\varrho }cos\left(y\right)}{\Gamma (1+\varrho )}}-{\displaystyle \frac{9{\delta }^{2\varrho }sin\left(y\right)}{\Gamma [1+2\varrho ]}}\hfill \\ \hfill & -{\displaystyle \frac{27{\delta }^{3\varrho }cos\left(y\right)}{\Gamma [1+3\varrho ]}}+{\displaystyle \frac{81{\delta }^{4\varrho }sin\left(y\right)}{\Gamma [1+4\varrho ]}}+\cdots .\hfill \end{array}$$

(28)

In the specific scenario for $\varrho =1$, we get

$$\mu (y,\delta )=sin\left(y\right)+3\delta cos\left(y\right)-{\displaystyle \frac{9}{2}}{\delta }^{2}sin\left(y\right)-{\displaystyle \frac{9}{2}}{\delta }^{3}cos\left(y\right)+\cdots .$$

Remember that, for m$\to +\infty ,$ the following exact solution to the integer form for Equation (23) is recovered:

$$\mu (y,\delta )=sin(y+3\delta ).$$

The obtained approximation (28) ${\mu }_{App}$ for problem (23) is examined against the FP $p\equiv \varrho$, as evident in Figure 3. This graphic examines the impact of the FP $p\equiv \varrho$ on the behavior of this approximation. As shown in Figure 4 and

Table 3. The absolute error of the approximation (28) to the problem (23) is estimated at $\delta =0.001$.

y	${\mathit{VIM}}_{\mathit{P}=1}$	$\mathit{Exact}$	${10}^{-15}$${\mathit{L}}_{\mathit{\infty }}$
−0.5	−0.476791	−0.476791	1.7763568394002505
−0.4	−0.386653	−0.386653	1.8318679906315083
−0.3	−0.292653	−0.292653	1.887379141862766
−0.2	−0.195728	−0.195728	1.9984014443252818
−0.1	−0.096848	−0.096848	2.0122792321330962
0.	0.003	0.003	2.024855977333928
0.1	0.102818	0.102818	2.0261570199409107
0.2	0.201609	0.201609	1.9984014443252818
0.3	0.298385	0.298385	1.9984014443252818
0.4	0.39218	0.39218	1.887379141862766
0.5	0.482056	0.482056	1.7763568394002505

, we visually and quantitatively matched the approximation (28) ${\mu }_{App}$ at $p=1$ with the exact solution ${\mu }_{Exact}$ (25). In Table 3, the absolute error $L_{\infty }$ of the derived approximation (28) ${\mu }_{App}$ is estimated at $p=1$:

$$L_{\infty }=|{\mu }_{Exact}-{\mu }_{Ap{p}_{p=1}}|.$$

The study results demonstrate a nearly flawless correspondence between the calculated approximation and the exact answer, bolstering the technique’s efficacy. Furthermore, using LRPSA and following several long but basic computations produced the following approximation to problem (23):

$${\mu }_{\mathrm{App}}=sin\left(y\right)+{\displaystyle \frac{3{\delta }^{\varrho }cos\left(y\right)}{\Gamma (1+\varrho )}}-{\displaystyle \frac{9{\delta }^{2\varrho }sin\left(y\right)}{\Gamma [1+2\varrho ]}}-{\displaystyle \frac{27{\delta }^{3\varrho }cos\left(y\right)}{\Gamma [1+3\varrho ]}}+\cdots .$$

(29)

Moreover, we analyzed and addressed the problem (23) using the ATIM, and we derived the following approximation:

$$\begin{array}{cc}\hfill {\mu }_{\mathrm{App}}& =sin\left(y\right)+{\displaystyle \frac{3{\delta }^{\varrho }cos\left(y\right)}{\varrho \Gamma \left(\varrho \right)}}-{\displaystyle \frac{9{\delta }^{2\varrho }sin\left(y\right)}{{\varrho }^2{\Gamma }^2\left[\varrho \right]}}-{\displaystyle \frac{27{\delta }^{3\varrho }cos\left(y\right)}{{\varrho }^3{\Gamma }^3\left[\varrho \right]}}\hfill \\ \hfill & +{\displaystyle \frac{81{\delta }^{4\varrho }sin\left(y\right)}{{\varrho }^4{\Gamma }^4\left[\varrho \right]}}+\cdots .\hfill \end{array}$$

(30)

In addition, we numerically compared the two approximations, (28) and (30), as illustrated in

Table 4. A comparison between the absolute error of the two approximations (28) and (30) is estimated at $\delta =0.001$.

y	${\mathit{VIM}}_{\mathit{P}=1}$	${\mathit{ATIM}}_{\mathit{P}=1}$	$\mathit{Exact}$	${10}^{-15}$${\mathit{L}}_{\mathit{\infty }}$	${10}^{-6}$${\mathit{L}}_{\mathit{\infty }}$
−0.5	−0.476791	−0.476788	−0.476791	1.77636	2.13763
−0.4	−0.386653	−0.386652	−0.386653	1.83187	1.73163
−0.3	−0.292653	−0.292652	−0.292653	1.88738	1.30832
−0.2	−0.195728	−0.195727	−0.195728	1.9984	0.871945
−0.1	−0.096848	−0.0968475	−0.096848	2.01228	0.426855
0.	0.003	0.00299997	0.003	2.02486	0.0225
0.1	0.102818	0.102818	0.102818	2.02616	0.47163
0.2	0.201609	0.201608	0.201609	1.9984	0.916048
0.3	0.298385	0.298384	0.298385	1.9984	1.35131
0.4	0.39218	0.392178	0.39218	1.88738	1.77308
0.5	0.482056	0.482054	0.482056	1.77636	2.17712

. In this table, we estimate the absolute error for each approximation compared to the exact solution (25) for the integer case. This table shows the incredible accuracy of approximation (28) compared to approximation (30), which enhances the accuracy and efficiency of the applied method and its favorable outcomes in examining more complicated problems.

5. Conclusions

In this study, we succeeded in applying the variational iteration method (VIM) with the Elzaki transform in the framework of the Caputo operator to analyze and solve the fractional diffusion equation (a crucial model in various physical and engineering contexts), and we obtained a more accurate and stable approximation throughout the study domain. We briefly presented the most important definitions, theories, and different characteristics of the Elzaki transform. Then, we also provided a simple explanation of the VITM and how to apply it to the analysis of differential equations by combining it with the Elzaki transform. Following that, we applied this new hybrid method to analyze some essential physical models (diffusion equations) that have many applications in physics and engineering. We further examined these derived approximations graphically and numerically by presenting some two- and three-dimensional graphics. Additionally, we numerically compared the approximations obtained using the VIM with the Ekzaki transform method with the approximations obtained using the Aboodh transform iteration method (ATIM), and we computed the absolute error for each of them compared to the exact solution for the integer case. The comparative findings showed the high accuracy and efficiency of the approximations generated via the VIM with the Ekzaki transform method, which enhances the accuracy and efficiency of the approach used, as well as its promising results in analyzing the most complicated issues.

The integration of these advanced mathematical techniques has proven to be robust and effective, providing explicit functional expressions that facilitate both theoretical understanding and practical applications. Finally, the current investigation highlights the potential of combining the VIM with the Elzaki transform to address the challenges of fractional PDEs. This combined approach not only simplifies the solution process but also enhances the precision and applicability of the results. The findings contribute significantly to the different fields of fractional calculus, offering new insights and practical tools for solving a wide range of fractional PDEs. Furthermore, by employing the suggested approach, we can investigate fractional space–time evolutionary equations to uncover the enigmatic nature of various physical nonlinear phenomena and comprehend the propagation mechanism for nonlinear waves in diverse physical systems and plasma physics [48].