Full Text
List of abbreviations| RPSM | residual power series method |
| HPM | homotopy perturbation method |
| ADM | Adomian decomposition method |
| VIM | variational iteration method |
| TFPDEs | time-fractional partial differential equations |
| LRPSM | Laplace residual power series method |
| GCFD | Gerasimov–Caputo fractional derivative |
| DEs | differential equations |
| FODEs | fractional order differential equations |
| Rel-E | relative error |
| Abs-E | absolute error |
| Rec-E | recurrence error |
| LRF | Laplace residual functions |
1. Introduction
Fractional calculus deals with fractional order derivatives and integrations. Fractional calculus was founded by two mathematicians, Leibniz and l'Hopital, and its official birthday is September 30, 1695. The widespread usage of fractional calculus in fields including image processing, physics, engineering, biology, biochemistry, entropy theory, and fluid mechanics has attracted a lot of researchers in recent years [1–5]. There are numerous definitions for fractional order derivatives; however, not all of them are regularly applied. The Grunwald–Letnikov, Hadamard derivative, Riemann–Liouville, conformable, and Gerasimov–Caputo fractional derivative (GCFD) are the most well known fractional order derivatives [6–9].
The GCFD of order $\beta>0$ is given by [10]:
\[ \begin{equation*}
D^{\beta}_{\omega}\aleph(\omega)=
\begin{cases}
\dfrac{1}{\Gamma(r-\beta)}\displaystyle \int_{q}^{\omega}(\omega-q)^{r-\beta-1}\dfrac{d^{r}}{dq^{r}}\aleph(q)dq, & r-1<\beta<r,\\
\dfrac{d^{r}}{d\omega^{r}}\aleph(\omega), & \beta=r\in\mathbb{N}.
\end{cases}
\end{equation*} \]
The following are the important properties of GCFD:
- $D^{\beta}_{\omega}\xi=0$, $\xi\in\mathbb{R}$,
- $D^{\beta}_{\omega}\omega^{\xi}=\dfrac{\Gamma(\xi+1)}{\Gamma(\xi+1-\beta)}\omega^{\xi-\beta}$, $r-1<\beta\le r$, $\xi>r-1$, $r\in\mathbb{N}$, $\xi\in\mathbb{R}$,
- $D^{\beta}_{\omega}(\xi_{1}\aleph_{1}(\omega)+\xi_{2}\aleph_{2}(\omega))=\xi_{1}D_{\omega}^{\beta}\aleph_{1}(\omega)+\xi_{2} D^{\beta}_{\omega}\aleph_{2}(\omega)$.
In some cases, fractional order derivatives are preferable to integer-order derivatives for modeling because they can simulate and examine complex structures with complicated nonlinear processes and higher-order behaviors. There are two main causes of this. First, rather than being limited to an integer order, we can choose any order for the fractional order derivatives. Furthermore, when the mechanism has long-term memory, fractional order derivatives are advantageously based on both past and present situations.
In the disciplines of science and engineering, there are natural and physical phenomena that, when described by mathematical relations, turn into differential equations (DEs). Examples of phenomena that are characterized by DEs include equations of motion, simple harmonic motion, beam deflection, and more. The fractional order differential equations (FODEs) have developed a convenient means of expressing naturally occurring phenomena in artificial intelligence, engineering, physics, earth sciences, bioinformatics, finance systems, and biological systems. Consequently, the study of the approximate or exact solution enables us to recognize the mechanism of these FODEs, and their actual physical intention can be perceived from the graphical depiction of the solution. Applications often come with FODEs that are so complex that exact solutions are frequently impractical. For solving FODEs in the initial circumstances given, methods that give approximate solutions present a potent alternative tool. In recent times, numerous approximate methods for solving FODEs have been presented [11–19].
The residula power series method (RPSM) is a very powerful approach in terms of the construction of power series solutions for FODEs. Many FODEs have been successfully solved by RPSM. Y. Zhang et al. [20] used RPSM to develop series solutions of the time-fractional Schrödinger equation. A. El-Ajou et al. [21] studied the KdV–Burgers equation to find approximate solutions. R. Saadeh et al. [22] discussed the fractional Newell–Whitehead–Segal equation by the RPSM. Approximate analytical solutions of FODEs can be solved effectively using the RPSM. However, to determine the coefficients of the series solution, the $(n-1)\beta$ derivative of the residual function is required. As we all know, it is difficult to compute the fractional order derivatives of a function. This limits the application of the classic RPSM to a certain extent. To overcome this drawback, T. Eriqat et al. [23] introduced a new approach, known as the Laplace residual power series method (LRPSM), which is the coupling of the Laplace transforms and the RPSM for approximate series solutions of linear and nonlinear FODEs. The limit idea is used to establish the series coefficients in the LRPSM. The set of guidelines for this novel approach is based on converting the specified equation into the Laplace transforms space, identifying an expansion solution to the transformed equation, and then obtaining the original equation's solution by using the inverse Laplace transform.
Finding solutions to FODEs with variable coefficients is also an interesting area for researchers. E. Hesameddini and A. Rahimi [24] used the variational iterative approach to solve FODEs with variable coefficients. Y. Keskin et al. [25] used the generalized Taylor collection technique to find solutions for higher-order linear FODEs with variable coefficients.
S. Sarwar et al. [26] found approximate solutions to time-fractional wavelike models with variable coefficients using the definition of the GCFD and with the help of optimal homotopy asymptotic method. With the operational matrix technique, D. Rostamy and K. Karimi [27] established approximate solutions of fractional-order wave and heat equations with variable coefficients. H. Bulut et al. [28] used the Sumudu transform method to obtain approximate solutions for partial differential equations with variable coefficients. M. Nadeem et al. [29] established numerical solutions for the fourth-order parabolic partial differential equation with variable coefficients using the modified Laplace variational iteration method. M. Dehghan and J. Manafian [30] used the homotopy perturbation technique to find a solution for a fourth-order parabolic problem with variable coefficients. T.M. Elzaki and S.M. Ezaki [31] established solutions for ordinary DEs with variable coefficients using the Elzaki transform method. Each of these strategies has particular limitations and drawbacks. These methods require a lot of work and longer running times.
In this research, the LRPSM was used to solve time-fractional partial diffrential equations (TFPDEs) with variable coefficients in the sense of GCFD. This method combines the Laplace transforms with the RPSM, which is based on a revamped version of Taylor's series and yields a convergent series as a solution. By employing the simple limit principle at infinity, the LRPSM excels at calculating the coefficients of terms in a series solution, but other well-known methods such as the variational iteration method (VIM), Adomian decomposition method (ADM), and homotopy perturbation method (HPM) need integration, while the RPSM needs the derivative, both of which are challenging in fractional contexts. LRPSM is also more effective than various series solution methods due to its small processing size and avoidance of Adomian and He polynomials to solve nonlinear problems. Moreover, this method does not require any assumptions about physical parameters, no matter how big or small, for the problem. Therefore, it can be used to handle both mildly and severely nonlinear problems and to circumvent some of the issues that perturbation techniques previously had. To evaluate the efficiency and consistency of the suggested strategy, the relative error (Rel-E), recurrence error (Rec-E), and absolute error (Abs-E) of the three problems are examined. The findings demonstrate that when solving FODEs, the LRPSM can be a viable substitute for the RPSM, VIM, ADM, and HPM.
To emphasize the essential ideas of our suggested method, such as its dependability, capability, and application, we selected the most prevalent forms of TFPDEs with variable coefficients. J. Fourier proposed the heat equation in 1822, which defines how a certain amount of heat diffuses over a region. Consider the TFPDE with variable coefficients shown below [27]:
\[ \begin{equation}
D_{\omega}^{\kappa\beta}\aleph(\theta,\omega)+\Im(\theta)\Psi(\aleph)-\Xi(\theta,\aleph)=0,
\end{equation} \tag{1} \]
subject to the initial condition:
\[ \begin{equation*}
D_{\omega}^{\eta\beta}\aleph(\theta,0)=\Phi_{\eta},
\end{equation*} \]
where
$\eta=0,1,2,3 \ldots,\kappa-1$; $\theta=(\theta_1,\theta_2,\ldots,\theta_e)\in \mathbb{R}^{e}$; $\beta\in \bigl(\frac{\kappa-1}{\kappa}\bigr]$, $\kappa\in \mathbb{N}$, and $\Psi(\aleph) = \Psi(\aleph, D_{\omega}^{\beta}\aleph, D_{\omega}^{2\beta}\aleph, \ldots, D_{\omega}^{(\kappa-1)\beta}\aleph$, $D_{\theta_{1}}^{\ell_{11}}\aleph,$ $D_{\theta_{2}}^{\ell_{12}}\aleph,$ $\ldots, D_{\theta_{e}}^{\ell_{1e}}\aleph,$ $\ldots,$ $D_{\theta_{1}}^{\ell_{p1}}\aleph,$ $ D_{\theta_{2}}^{\ell_{p2}}\aleph, \ldots, D_{\theta_{e}}^{\ell_{pe}}\aleph)$, with $h-1<\ell_{hu}\leqslant h$, $h=1,2,\ldots,p$; $u=1,2,\ldots,e$.
The GCFD of $\omega$ of order $\eta\beta$ and $\theta_{u}$ of order $\ell_{hu}$ are represented by $D_{\omega}^{\eta\beta}$ and $D_{\theta_{u}}^{\ell_{hu}}$, respectively. These kinds of DEs give accurate representations of a wide range of physical events in the fields of fluid dynamics, elastic mechanics, and electrodynamics [26–33].
This study is structured as follows. First, in Section 2, we present significant definitions and findings from FC theory. The LRPSM algorithms for solving TFPDEs with variable coefficients are discussed in Section 3. In Section 4, some linear and nonlinear TFPDE problems are solved to demonstrate the accuracy and reliability of LRPSM. The numerical and graphic results obtained using LRPSM are evaluated in Section 5. Finally, we conclude the paper in Section 6.
2. Preliminaries
In this section, basic definitions, Laplace transform characteristics, and LRPSM-related theorems that help establish approximate series solutions are included.
Definition 1 [34]. The Laplace transforms of $\aleph(\theta,\omega)$ is defined as follows:
\[ \begin{equation*}
\mathcal{L}[\aleph(\theta,\omega)]=\Omega(\theta,\nu)=
\int_{0}^{\infty}\aleph(\theta,\omega)e^{-\omega\nu}d\omega,
\end{equation*} \]
and the inverse Laplace transforms is defined by
\[ \begin{equation*}
\mathcal{L}^{-1}[\Omega(\theta,\nu)]=\aleph(\theta,\omega)=
\int_{b- i \infty}^{b+ i\infty}e^{\nu\omega}\Omega(\theta,\nu)d\nu,\quad b=\operatorname{Re}(\nu)>b_{0},
\end{equation*} \]
where $\theta=(\theta_{1},\theta_{2},\ldots,\theta_{e})\in \mathbb{R}^{e}$ and $e\in \mathbb{N}$ and $b_{0}$ lies in the right half plane of the absolute convergence of the Laplace integral.
Lemma 1 [23]. Consider that $\aleph_{1}(\theta,\omega)$ and $\aleph_{2}(\theta,\omega)$ satisfy the axioms of the existence of Laplace transforms. Suppose that $\mathcal{L}[\aleph_{1}(\theta,\omega)]=\Omega_{1}(\theta,\nu),$, $\mathcal{L}[\aleph_{2}(\theta,\omega)]=\Omega_{2}(\theta,\nu)$ and the constants $\zeta_{1}$, $\zeta_{2}$. When this occurs, the following criteria are met [23, 24]:
- $\mathcal{L}[\zeta_{1}\aleph_{1}(\theta,\omega)+\zeta_{2}\aleph_{2}(\theta,\omega)]=\zeta_{1}\Omega_{1}(\theta,\nu)+ \zeta_{2}\Omega_{2}(\theta,\nu)$;
- $\mathcal{L}^{-1}[\zeta_{1}\Omega_{1}(\theta,\nu)+\zeta_{2}\Omega_{2}(\theta,\nu)]=\zeta_{1}\aleph_{1}(\theta,\omega)+ \zeta_{2}\aleph_{2}(\theta,\omega)$;
- $\xi_{0}(\theta)=\lim\limits_{\nu\to\infty}\nu\Omega(\theta,\nu)=\aleph(\theta,0)$;
- $\mathcal{L}[D_{\omega}^{\beta}\aleph(\theta,\omega)]=\nu^{\beta}\Omega(\theta,\nu)- \displaystyle\sum_{j=0}^{r-1}\dfrac{\aleph^{(j)}(\theta,0)}{\nu^{j-\beta+1}}$,
$r-1<\beta\leqslant r$, $r\in \mathbb{N}$; - $\mathcal{L}[D_{\omega}^{r\beta}\aleph(\theta,\omega)]=\nu^{r\beta}\Omega(\theta,\nu)-\displaystyle \sum_{j=0}^{r-1}\nu^{\beta(r-j)-1} D_{\omega}^{j\beta}\aleph(\theta,0)$, $0<\beta\leq1$.
Definition 2 [35]. The multiple fractional power series is defined as follows:
\[ \begin{equation*}
\sum_{r=0}^{\infty}\xi_{r}(\theta)(\omega-\omega_{0})^{r\beta}=
\xi_{0}(\omega-\omega_{0})^{0}+\xi_{1}(\omega-\omega_{0})^{\beta}+
\xi_{2}(\omega-\omega_{0})^{2\beta}+\cdots,
\end{equation*} \]
where $\theta=(\theta_{1},\theta_{2},\ldots,\theta_{e})\in \mathbb{R}^{e}$ and $e\in \mathbb{N}$, $\omega$ represent a variable and $\xi_{r}(\theta)$ represent the coefficients of the multiple fractional power series.
We present a new type of multiple fractional power series in the sense of Laplace transform space in the following lemma, which is the main pillar for the LRPSM.
Lemma 2 [23]. Assume that the multiple fractional power series demonstration is in Laplace transforms space for the function $\mathcal{L} [\aleph(\theta,\omega)]=\Omega(\theta,\nu)$ as shown follows:
\[ \begin{equation*}
\Omega(\theta,\nu)=\sum_{r=0}^{\infty}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}},\quad \nu>0,
\end{equation*} \]
where, $\theta=(\theta_{1},\theta_{2},\ldots, \theta_{e})\in \mathbb{R}^{e}$, $e\in \mathbb{N}$.
Theorem 1 [23]. The coefficients of the multiple fractional power series can be determined as follows:
\[ \begin{equation*}
\xi_{r}(\theta)=D_{\omega}^{r\beta}\aleph(\theta,0),
\end{equation*} \]
where $D_{\omega}^{r\beta}=D_{\omega}^{\beta}\cdot D_{\omega}^{\beta}\cdots D_{\omega}^{\beta}$ $(r$ times$)$.
The following theorem establishes the prerequisite for the convergence of the new form of multiple fractional power series.
Theorem 2 [23]. Let $\mathcal{L}[\aleph(\theta,\omega)]=\Omega(\theta,\nu)$ can be denoted as the new form of multiple fractional power series given in Lemma 2. If $\vert\nu\mathcal{L}[D_{\omega}^{(j+1)\beta}\aleph(\theta,\omega)]\vert \leqslant \mathcal{P}$ on $0<\nu\leqslant y$ with $0<\beta\leqslant 1$, then the remainder $R_{j}(\theta,\nu)$ of the multiple fractional power series satisfies the following inequality:
\[ \begin{equation*}
\vert R_{j}(\theta,\nu)\vert \leqslant \frac{\mathcal{P}}{\nu^{(j+1)\beta+1}},
\quad 0< \nu\leqslant y.
\end{equation*} \]
3. LRPSM for Solving TFPDEs with Variable Coefficients
In this section, the set of steps for employing the suggested method to acquire approximative analytical solutions to TFPDEs is addressed. First of all, we apply the Laplace transforms to the TFPDEs with variable coefficients and get an algebraic expression as a result. Then, we take into account the multiple fractional power series as the new space solution for the resulting expression obtained in the first step, which is the fundamental idea behind the LRPSM. The way in which the coefficients of this series are obtained by employing the limit idea is the primary difference between the LRPSM and the RPSM. The resulting implications are then translated into actual space using the inverse Laplace transforms. The rules for using the LRPSM to locate solutions are listed below.
The LRPSM Algorithm for Linear and Nonlinear TFPDEs. We describe the LRPSM's set of guidelines for solving Eq. (1).
Step 1. Reformatted Eq. (1):
\[ \begin{equation}
D_{\omega}^{\kappa\beta}\aleph(\theta,\omega)+\Im(\theta)\Psi(\aleph)-\Xi(\theta,\aleph)=0.
\end{equation} \tag{2} \]
Step 2. Utilize $\mathcal{L}$ on both sides of Eq. (2):
\[ \begin{equation}
\mathcal{L}\big[D_{\omega}^{\kappa\beta}\aleph(\theta,\omega)+\Im(\theta)\Psi(\aleph)-\Xi(\theta,\aleph)\big]=0.
\end{equation} \tag{3} \]
Using the Lemma 1(v) and the initial condition, we get the following from Eq. (3):
\[ \begin{equation}
\Omega(\theta,\nu)=\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)}{\nu^{j\beta+1}}-\frac{\Im(\theta)B(\nu)}{\nu^{j\beta}}+\frac{A(\theta,\nu)}
{\nu^{j\beta}},
\end{equation} \tag{4} \]
where $\mathcal{L}[\Xi(\theta,\aleph)]=\mathcal{A}(\theta,\nu)$, $\mathcal{L}[\Psi(\aleph)]=\mathcal{B}(\nu)$.
Step 3. Assume that the multiple fractional power series solution of Eq. (4) in Laplace transforms space is below:
\[ \begin{equation*}
\Omega(\theta,\nu)=\sum_{r=0}^{\infty}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}},\quad \nu>0.
\end{equation*} \]
Step 4. As a result of applying the Lemma 1(iii) and Theorem 1 we obtained the following results:
\[ \begin{equation*}
\xi_{0}(\theta)=\lim_{\nu\to \infty}\nu\Omega(\theta,\nu)=\aleph(\theta,0),
\end{equation*} \]
\[ \begin{equation*}
\xi_{1}(\theta)=D_{\omega}^{\beta}\aleph(\theta,0),
\end{equation*} \]
\[ \begin{equation*}
\xi_{2}(\theta)=D_{\omega}^{2\beta}\aleph(\theta,0),
\end{equation*} \]
\[ \begin{equation*}
\xi_{\eta}(\theta)=D_{\omega}^{\eta\beta}\aleph(\theta,0).
\end{equation*} \]
Step 5. Assume that the $j$th truncated multiple fractional power series solution of Eq. (4) is below:
\[ \begin{equation*}
\Omega_{j}(\theta,\nu)=\sum_{r=0}^{j}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}}, \quad \nu>0,
\end{equation*} \]
\[ \begin{equation*}
\Omega_{j}(\theta,\nu)=\frac{\xi_{0}(\theta)}{\nu}+\frac{\xi_{1}(\theta)}{\xi^{\beta+1}}+\cdots+\frac{\xi_{\eta}(\theta)}{\nu^{\eta\beta+1}}+ \sum_{r=\eta+1}^{j}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}}.
\end{equation*} \]
Step 6. The Laplace residual functions (LRF) and $j$th truncated LRF for the Eq. (4) are defined as follows:
\[ \begin{equation*}
\mathcal{L}\operatorname{Res}(\theta,\nu)=
\Omega(\theta,\theta)-\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)}{\nu^{j\beta+1}}+
\frac{\Im(\theta)\mathcal{B}(\nu)}{\nu^{j\beta}}-
\frac{\mathcal{A}(\theta,\nu)}{\nu^{j\beta}},
\end{equation*} \]
and
\[ \begin{equation}
\mathcal{L}\operatorname{Res}_{j}(\theta,\nu)=
\Omega_{j}(\theta,\nu)-\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)}{\nu^{j\beta+1}}+\frac{\Im(\theta)\mathcal{B}(\nu)} {\nu^{j\beta}}-\frac{\mathcal{A}(\theta,\nu)}{\nu^{j\beta}}.
\end{equation} \tag{5} \]
Step 7. By inserting the $j$th truncated multiple fractional power series into the Eq. (5) we get the following:
\[ \begin{equation}
\mathcal{L}\operatorname{Res}_{j}(\theta,\nu)= \Bigl(\frac{\xi_{0}(\theta)}{\nu}+\frac{\xi_{1}(\theta)}{\xi^{\beta+1}}+\cdots+\frac{\xi_{\eta}(\theta)}{\nu^{\eta\beta+1}}+
\sum_{r=\eta+1}^{j}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}}\Bigr)-\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)}{\nu^{j\beta+1}}+\frac{\Im(\theta)
\mathcal{B}(\nu)}{\nu^{j\beta}}-\frac{\mathcal{A}(\theta,\nu)}{\nu^{j\beta}}.
\end{equation} \tag{6} \]
Step 8. Multiplying the resulting expression $\mathcal{L}\operatorname{Res}_{j}(\theta,\nu)$ by $\nu^{j\beta+1}$ on both sides of Eq. (6):
\[ \begin{equation}
\nu^{j\beta+1}\mathcal{L}\operatorname{Res}_{j}(\theta,\nu)=
\nu^{j\beta+1}\Bigl(\frac{\xi_{0}(\theta)}{\nu}+\frac{\xi_{1}(x)}{\xi^{\beta+1}}+\cdots+\frac{\xi_{\eta}(\theta)}{\nu^{\eta\beta+1}}+
\sum_{r=\eta+1}^{j}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}}-\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)}{\nu^{j\beta+1}}+
\frac{\Im(\theta)\mathcal{B}(\nu)} {\nu^{j\beta}}-\frac{\mathcal{A}(\theta,\nu)}{\nu^{j\beta}}\Bigr).
\end{equation} \tag{7} \]
Step 9. Applying $\lim\limits_{\nu\to \infty}$ on both sides of Eq. (7):
\[ \begin{equation*}
\lim_{\nu\to\infty}\nu^{j\beta+1}\mathcal{L}\operatorname{Res}_{j}(\theta,\nu)=
\lim_{\nu\to\infty}\nu^{j\beta+1} \Bigl(
\frac{\xi_{0}(\theta)}{\nu}+\frac{\xi_{1}(\theta)}{\xi^{\beta+1}}+
\cdots+\frac{\xi_{\eta}(\theta)}{\nu^{\eta\beta+1}} +\sum_{r=\eta+1}^{j}\frac{\xi_{r}(\theta)}{\nu^{r\beta+1}}-\sum_{j=0}^{\kappa-1}\frac{D_{\omega}^{j}\aleph(\theta,0)} {\nu^{j\beta+1}}+\frac{\Im(\theta)\mathcal{B}(\nu)} {\nu^{j\beta}}-\frac{\mathcal{A}(\theta,\nu)}{\nu^{j\beta}}\Bigr).
\end{equation*} \]
Step 10. Solve the following expression for $\xi_{j}(\theta)$:
\[ \begin{equation*}
\lim_{\nu\to\infty}\big(\nu^{j\beta+1}\mathcal{L} \operatorname{Res}_{j}(\theta,\nu)\big)=0,
\end{equation*} \]
where $j=\eta+1, \eta+2, \ldots$.
Step 11. To obtain the $j$th step approximate solution of Eq. (4), substitute the obtained values of $\xi_{j}(\theta)$ into a $j$th truncated series of $\Omega(\theta,\nu)$.
Step 12. Apply the $\mathcal{L}^{-1}$ on $\Omega_{j}(\theta,\nu)$ to attain the $j$th approximate solution $\aleph_{j}(\theta,\omega)$.
4. Applications to Linear and Nonlinear TFPDEs with Variable Coefficients
In this section, to show the applicability and reliability of the novel algorithm, three linear and nonlinear TFPDEs are solved.
Problem 1. In our first illustration, we take the nonlinear TFPDE provided below [33]:
\[ \begin{equation}
D_{\omega}^{2\beta}\aleph(\theta_{1},\omega)=
\aleph^{2}(\theta_{1},\omega)\frac{\partial^{2}}{\partial \theta_{1}^{2}}
\bigl(\aleph_{\theta_{1}}(\theta_{1},\omega)\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\omega)\aleph_{\theta_{1}\theta_{1}\theta_{1}} (\theta_{1},\omega)\bigr) + \bigl(\aleph_{\theta_{1}}(\theta_{1},\omega)\bigr)^{2}\frac{\partial^{2}}{\partial \theta_{1}^{2}}\bigl(\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\omega)\bigr)^{3}-18\aleph^{5}(\theta_{1},\omega)+\aleph(\theta_{1},\omega),
\end{equation} \]
where $0<\beta\leqslant 1$, $\omega\geqslant 0$, $\theta_{1}\in \mathbb{R}$, and with the following initial conditions:
\[ \begin{equation*}
\aleph(\theta_{1},0)={\rm e}^{\theta_{1}}, \quad D_{\omega}^{\beta}\aleph(\theta,0)={\rm e}^{\theta_{1}}.
\end{equation*} \]
By applying $\mathcal{L}$ to Eq. (8),
\[ \begin{equation}
\mathcal{L}\bigl[D_{\omega}^{2\beta}\aleph(\theta_{1},\omega)\bigr]=
\mathcal{L}\Bigl[\aleph^{2}(\theta_{1},\omega)\frac{\partial^{2}}{\partial \theta_{1}^{2}}\bigl(\aleph_{\theta_{1}}(\theta_{1},\omega)\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\omega)\aleph_{\theta_{1}\theta_{1}\theta_{1}} (\theta_{1},\omega)\bigr) +
\bigl(\aleph_{\theta_{1}}(\theta_{1},\omega)\bigr)^{2}\frac{\partial^{2}}{\partial\theta^{2}}\bigl(\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\omega)\bigr)^{3}- 18\aleph^{5}(\theta_{1},\omega)+\aleph(\theta_{1},\omega)\Bigr].
\end{equation} \tag{9} \]
By following the steps that are established in Section 3, we get the following result from Eq. (9):
\[ \begin{multline}
\Omega(\theta_{1},\nu)=
\frac{{\rm e}^{\theta_{1}}}{\nu}+\frac{{\rm e}^{\theta_{1}}}{\nu^{\beta+1}}+
\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[\bigl(\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]\bigr)^{2}
\frac{\partial^{2}} {\partial{\theta_{1}}^{2}}
\Bigl(\frac{\partial}{\partial{\theta_{1}}}\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)] \cdot
\frac{\partial^{2}}{\partial{\theta_{1}}^{2}}
\mathcal{L}^{-1} [\Omega(\theta,\nu)] \frac{\partial^{3}}{\partial{\theta_{1}}^{3}}\mathcal{L}^{-1}
[\Omega(\theta_{1},\nu)]\bigr)\Bigr] + {}
\\
+\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[
\Bigl(\frac{\partial}{\partial\theta_{1}}\mathcal{L}^{-1} [\Omega(\theta_{1},\omega)]\Bigr)^{2}
\frac{\partial^{2}}{\partial{\theta^{2}}}
\Bigl(\frac{\partial^{2}}{\partial{\theta_{1}^{2}}} \mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]\Bigr)^{3}-18\bigl(\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]\bigr)^{5}\Bigr]+
\frac{1}{\nu^{2\beta}}\Omega(\theta_{1},\nu).
\end{multline} \tag{10} \]
Consider the series solutions of Eq. (10), which have the following form:
\[ \begin{equation*}
\Omega(\theta_{1},\nu)=\sum_{r=0}^{\infty}\frac{\xi_{r}(\theta_{1})}{\nu^{r\beta+1}}, \quad \nu>0.
\end{equation*} \]
The $j$th truncated expansion is as
\[ \begin{equation*}
\Omega_{j}(\theta_{1},\nu)=\sum_{r=0}^{j}\frac{\xi_{r}(\theta_{1})}{\nu^{r\beta+1}},\quad \nu>0.
\end{equation*} \]
As a result of applying the Lemma 1(iii) and Theorem 1, we obtained the following results:
\[ \begin{equation*}
\lim_{\nu\to\infty}\big(\nu\Omega(\theta_{1},\nu)\big)=\aleph(\theta_{1},0)=\xi_{0}(\theta_{1})={\rm e}^{\theta_{1}},
\quad
\xi_{1}(\theta_{1})=D_{\omega}^{\beta}\aleph(\theta_{1},0)={\rm e}^{\theta_{1}}.
\end{equation*} \]
So, Eq. (10) becomes as follows:
\[ \begin{equation}
\Psi_{j}(\theta_{1},\nu)=\frac{{\rm e}^{\theta_{1}}}{\nu}+\frac{{\rm e}^{\theta_{1}}}{\nu^{\beta+1}}+\sum_{r=2}^{j}\frac{\xi_{r}(\theta_{1})}{\nu^{r\beta+1}}.
\end{equation} \tag{11} \]
The following is the LRF for Eq. (10):
\[ \begin{multline*}
\mathcal{L}\operatorname{Res}(\theta_{1},\nu)=
\Omega(\theta_{1},\nu)-\frac{{\rm e}^{\theta_{1}}}{\nu}-\frac{{\rm e}^{\theta_{1}}}{\nu^{\beta+1}}-
\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[ \bigl(\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]\bigr)^{2}
\frac{\partial^{2}}{\partial{\theta_{1}}^{2}}
\Bigl(\frac{\partial}{\partial{\theta_{1}}}\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)] \cdot
\frac{\partial^{2}}{\partial{\theta_{1}}^{2}} \mathcal{L}^{-1}
[\Omega(\theta_{1},\nu)]
\frac{\partial^{3}}{\partial{\theta_{1}}^{3}}\mathcal{L}^{-1}
[\Omega(\theta_{1},\nu)]\Bigr)\Bigr] - {}
\\
{}-\frac{1}{\nu^{2\beta}}\mathcal{L}
\Bigl[\Bigl(\frac{\partial}{\partial\theta_{1}}\mathcal{L}^{-1} [\Omega(\theta_{1},\omega)]\Bigr)^{2}\frac{\partial^{2}}{\partial{\theta_{1}^{2}}}
\Bigl(\frac{\partial^{2}}{\partial{\theta_{1}^{2}}} \mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]\Bigr)^{3} -18\Bigl(\mathcal{L}^{-1}[\Omega(\theta_{1},\nu)]
\Bigr)^{5}\Bigr] - \frac{1}{\nu^{2\beta}}\Omega(\theta_{1},\nu).
\end{multline*} \]
The following is the $j$th truncated LRF for Eq. (10):
\[ \begin{multline}
\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\nu)=
\Omega_{j}(\theta_{1},\nu)-\frac{{\rm e}^{\theta_{1}}}{\nu}-\frac{{\rm e}^{\theta_{1}}}{\nu^{\beta+1}} -\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[\bigl(\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\nu)]\bigr)^{2}\frac{\partial^{2}} {\partial{\theta_{1}}^{2}}
\Bigl(\frac{\partial}{\partial{\theta_{1}}}\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\nu)] \cdot
\frac{\partial^{2}}{\partial{\theta_{1}}^{2}} \mathcal{L}^{-1} [\Omega_{j}(\theta_{1},\nu)]
\frac{\partial^{3}} {\partial{\theta_{1}}^{3}}\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\nu)]\Bigr)\Bigr]- {}
\\
{} -
\frac{1}{\nu^{2\beta}}\mathcal{L}
\Bigl[\Bigl(\frac{\partial}{\partial\theta_{1}}\mathcal{L}^{-1} [\Omega_{j}(\theta_{1},\omega)]\Bigr)^{2}\frac{\partial^{2}} {\partial{\theta_{1}^{2}}} \Bigl(\frac{\partial^{2}}{\partial{\theta_{1}^{2}}} \mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\nu)]\Bigr)^{3} - 18\Bigl(\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\nu)]\Bigr)^{5}\Bigr]-
\frac{1}{\nu^{2\beta}}\Omega_{j}(\theta_{1},\nu).
\end{multline} \tag{12} \]
In Eq. (11) and Eq. (12), use $j=\overline{1, 7}$ to find the undetermined coefficients $\xi_{j}(\theta_{1},\theta_{2})$, and then solve the following:
\[ \begin{equation*}
\lim_{\nu\to\infty}\bigl(\nu^{j\beta+1}\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\nu)\bigr)=0.
\end{equation*} \]
The following outcomes obtained:
\[ \begin{equation*}
\xi_{2}(\theta_{1})={\rm e}^{\theta_{1}},\quad
\xi_{3}(\theta_{1})={\rm e}^{\theta_{1}},\quad
\xi_{4}(\theta_{1})={\rm e}^{\theta_{1}},
\end{equation*} \]
\[ \begin{equation*}
\xi_{5}(\theta_{1})={\rm e}^{\theta_{1}},\quad
\xi_{6}(\theta_{1})={\rm e}^{\theta_{1}},\quad
\xi_{7}(\theta_{1})={\rm e}^{\theta_{1}}.
\end{equation*} \]
In this way, we obtained the following 7th step approximate solution of Eq. (10) in Laplace transforms space:
\[ \begin{equation}
\Omega_{7}(\theta_{1},\nu)={\rm e}^{\theta_{1}}\Bigl(\frac{1}{\nu}+\frac{1}{\nu^{\beta+1}}+\frac{1}{\nu^{2\beta+1}}+\frac{1}{\nu^{3\beta+1}} +\frac{1}{\nu^{4\beta+1}}+\frac{1}{\nu^{5\beta+1}}+\frac{1}{\nu^{6\beta+1}}+\frac{1}{\nu^{7\beta+1}}\Bigr).
\end{equation} \tag{13} \]
By applying $\mathcal{L}^{-1}$ to Eq. (13), we obtain the approximate 7th step solution in the original space, which has the following form:
\[ \begin{equation*}
\aleph_{7}(\theta_{1},\omega) ={\rm e}^{\theta_{1}}
\Bigl(1+\frac{\omega^{\beta}}{\Gamma[\beta+1]}+\frac{\omega^{2\beta}}{\Gamma[2\beta+1]}+\frac{\omega^{3\beta}}{\Gamma[3\beta+1]} +
\frac{\omega^{4\beta}}{\Gamma[4\beta+1]}+\frac{\omega^{5\beta}}{\Gamma[5\beta+1]} +\frac{\omega^{6\beta}}{\Gamma[6\beta+1]}+ \frac{\omega^{7\beta}}{\Gamma[7\beta+1]}\Bigr).
\end{equation*} \]
When $\beta=1.0$, the 7th step approximate solution is
\[ \begin{equation}
\aleph_{7}(\theta_{1},\omega)={\rm e}^{\theta_{1}}\Bigl(1+\frac{\omega}{\Gamma[{2}]}+\frac{\omega^{2}}{\Gamma[{3}]}+\frac{\omega^{3}}{\Gamma[{4}]}+\frac{\omega^{4}}{\Gamma[{5}]}+\frac{\omega^{5}}{\Gamma[{6}]}+
\frac{\omega^{6}}{\Gamma[{7}]}+\frac{\omega^{7}}{\Gamma[{8}]}\Bigr).
\end{equation} \tag{14} \]
The first eight terms of the series of the exact solution to $\aleph(\theta,\omega)={\rm e}^{\theta_{1}+\omega}$ are represented by Eq. (14).
Problem 2. In our second illustration, we take the linear TFPDE provided below [32]:
\[ \begin{equation}
D_{\omega}^{2\beta}\aleph(\theta_{1},\theta_{2},\theta_{3},\omega)=
\frac{1}{2}\theta_{1}^{2}\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\theta_{2},\theta_{3},
\omega)+\frac{1}{2}\theta_{2}^{2}\aleph_{\theta_{2}\theta_{2}}(\theta_{1},\theta_{2},\theta_{3},\omega) +
\frac{1}{2}\theta_{3}^{2}\aleph_{\theta_{3}\theta_{3}} (\theta_{1},\theta_{2},\theta_{3},\omega)+\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2},
\end{equation} \tag{15} \]
where $0<\beta\leq1, (\theta_{1},\theta_{2},\theta_{3},\omega)\in\big(\mathbb{R}^{+}\big)^{4}$, and with the following initial conditions:
\[ \begin{equation*}
\aleph(\theta_{1},\theta_{2},\theta_{3},0)=0,
\quad
D_{\omega}^{\beta}\aleph(\theta_{1},\theta_{2},\theta_{3},0)= \theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}.
\end{equation*} \]
By applying $\mathcal{L}$ to Eq. (15),
\[ \begin{equation}
\mathcal{L}[D_{\omega}^{2\beta}\aleph(\theta_{1},\theta_{2},\theta_{3},\omega)] =
\mathcal{L} \Bigl[\frac{1}{2}\theta_{1}^{2}\aleph_{\theta_{1}\theta_{1}}(\theta_{1},\theta_{2},\theta_{3},\omega)+
\frac{1}{2}\theta_{2}^{2}\aleph_{\theta_{2}\theta_{2}}(\theta_{1},\theta_{2},\theta_{3},\omega)+
\frac{1}{2}\theta_{3}^{2}\aleph_{\theta_{3}\theta_{3}} (\theta_{1},\theta_{2},\theta_{3},\omega)+\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}\Bigr],
\end{equation} \tag{16} \]
By following the steps that are established in Section 3, we get the following result from Eq. (16):
\[ \begin{equation}
\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)=\frac{1}{\nu^{\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})+\frac{\theta_{1}^{2}}{2\nu^{2\beta}}
D_{\theta_{1}\theta_{1}}\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)+\frac{\theta_{2}^{2}}{2\nu^{2\beta}}
D_{\theta_{2}\theta_{2}}\Omega(\theta_{1},\theta_{2},\theta_{3},\omega)+\frac{\theta_{3}^{2}}{2\nu^{2\beta}}
D_{\theta_{3}\theta_{3}}\Omega(\theta_{1},\theta_{2},\theta_{3},\omega)
+ \frac{1}{\nu^{2\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}).
\end{equation} \tag{17} \]
Consider the series solutions of Eq. (17), which have the following form:
\[ \begin{equation*}
\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)=\sum_{r=0}^{\infty}\frac{\xi_{r}(\theta_{1},\theta_{2},\theta_{3})}{\nu^{r\beta+1}}, \quad \nu>0.
\end{equation*} \]
The $j$th truncated series is as
\[ \begin{equation}
\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)=\sum_{r=0}^{j}\frac{\xi_{r}(\theta_{1},\theta_{2},\theta_{3})}{\nu^{r\beta+1}}, \nu>0.
\end{equation} \tag{18} \]
As a result of applying the Lemma 1(iii) and Theorem 1, we obtained the following results:
\[ \begin{equation*}
\lim_{\nu\to\infty}\bigl(\nu\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)\bigr)=\xi_{0}(\theta_{1},\theta_{2},\theta_{3})= \aleph(\theta_{1},\theta_{2},\theta_{3},0)=0,
\end{equation*} \]
\[ \begin{equation*}
\xi_{1}(\theta_{1},\theta_{2},\theta_{3})=D_{\omega}^{\beta}\aleph(\theta_{1},\theta_{2},\theta_{3},0)=\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}.
\end{equation*} \]
So, Eq. (18) becomes as follows:
\[ \begin{equation}
\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)=\frac{(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})}{\nu^{\beta+1}}+
\sum_{r=2}^{\kappa}\frac{\xi_{r}(\theta_{1},\theta_{2},\theta_{3})}{\nu^{r\beta+1}},\quad \nu>0.
\end{equation} \tag{19} \]
The following is the LRF for Eq. (17):
\[ \begin{multline*}
L\operatorname{Res}(\theta_{1},\theta_{2},\theta_{3},\nu)=
\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)-\frac{1}{\nu^{\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})-
\frac{\theta_{1}^{2}}{2\nu^{2\beta}}D_{\theta_{1}\theta_{1}}\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)- {} \\
{} - \frac{\theta_{2}^{2}}{2\nu^{2\beta}}
D_{\theta_{2}\theta_{2}}\Omega(\theta_{1},\theta_{2},\theta_{3},\nu) - \frac{\theta_{3}^{2}}{2\nu^{2\beta}}D_{\theta_{3}\theta_{3}}
\Omega(\theta_{1},\theta_{2},\theta_{3},\nu)-\frac{1}{\nu^{2\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}).
\end{multline*} \]
The following is the $j$th truncated LRF for Eq. (17):
\[ \begin{multline}
\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)=
\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)-\frac{1}{\nu^{\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})-
\frac{\theta_{1}^{2}}{2\nu^{2\beta}} D_{\theta_{1}\theta_{2}}\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu) - {} \\
{} - \frac{\theta_{2}^{2}}{2\nu^{2\beta}}
D_{\theta_{2}\theta_{2}}\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu) -
\frac{\theta_{3}^{2}}{2\nu^{2\beta}}
D_{\theta_{3}\theta_{3}}\Omega_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)-\frac{1}{\nu^{2\beta+1}}(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}).
\end{multline} \tag{20} \]
In Eq. (19) and Eq. (20), use $j=\overline{1, 7}$ to find the undetermined coefficients $\xi_{j}(\theta_{1},\theta_{2})$, and then solve the following:
\[ \begin{equation*}
\lim_{\nu\to\infty}\big(\nu^{j\beta+1}\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\theta_{2},\theta_{3},\nu)\big)=0.
\end{equation*} \]
The following outcomes obtained:
\[ \begin{equation*}
\xi_{2}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2},\qquad
\xi_{3}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2},
\end{equation*} \]
\[ \begin{equation*}
\xi_{4}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2},\qquad
\xi_{5}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2},
\end{equation*} \]
\[ \begin{equation*}
\xi_{6}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2},\qquad
\xi_{7}(\theta_{1},\theta_{2},\theta_{3})=\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}.
\end{equation*} \]
In this way, we obtained the following 7th step approximate solution of Eq. (17) in Laplace transforms space:
\[ \begin{equation}
\Omega_{7}(\theta_{1},\theta_{2},\theta_{3},\nu)=\frac{\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}}{\nu^{\beta+1}}+
\frac{\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}}{\nu^{2\beta+1}}+
\frac{\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}}{\nu^{3\beta+1}}+
\frac{\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}} {\nu^{4\beta+1}} +
\frac{\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}}{\nu^{5\beta+1}}+
\frac{\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}}{\nu^{6\beta+1}}+\frac{\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2}}{\nu^{7\beta+1}}.
\end{equation} \tag{21} \]
By applying $\mathcal{L}^{-1}$ to Eq. (21), we obtain the approximate 7th step solution in the original space, which has the following form:
\[ \begin{equation*}
\aleph_{7}(\theta_{1},\theta_{2},\theta_{3},\omega)=
(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})\Bigl(\frac{\omega^{\beta}}{\Gamma[\beta+1]}+
\frac{\omega^{3\beta}}{\Gamma[3\beta+1]}+
\frac{\omega^{5\beta}}{\Gamma[5\beta+1]}+\frac{\omega^{7\beta}}{\Gamma[7\beta+1]}\Bigr) +
(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2})\Bigl(\frac{\omega^{2\beta}}{\Gamma[2\beta+1]}+\frac{\omega^{4\beta}}{\Gamma[4\beta+1]}+
\frac{\omega^{6\beta}}{\Gamma[6\beta+1]}\Bigr).
\end{equation*} \]
When $\beta=1.0$, the 7th step approximate solution is
\[ \begin{equation}
\aleph_{7}(\theta_{1},\theta_{2},\theta_{3},\omega)=(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})\Bigl(\frac{\omega}{\Gamma[{2}]}+\frac{\omega^{3}}
{\Gamma[{4}]}+\frac{\omega^{5}}{\Gamma[{6}]}+\frac{\omega^{7}}{\Gamma[{8}]}\Bigr) +
(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2})\Bigl(\frac{\omega^{2}}{\Gamma[{3}]}+\frac{\omega^{4}}{\Gamma[{5}]}+\frac{\omega^{6}}{\Gamma[{7}]}\Bigr).
\end{equation} \tag{22} \]
The first seven terms of the series of the exact solution to
\[ \begin{equation*}
\aleph(\theta_{1},\theta_{2},\theta_{3},\omega)=(\theta_{1}^{2}+\theta_{2}^{2}-\theta_{3}^{2})\sinh\omega+(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2})
(\cosh\omega-1)
\end{equation*} \]
are represented by Eq. (22).
Problem 3. In our third illustration, we take the nonlinear TFPDE provided below [33]:
\[ \begin{equation}
D_{\omega}^{2\beta}\aleph(\theta_{1},\theta_{2},\omega)=\frac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}\bigl(\aleph_{\theta_{1}\theta_{1}}
(\theta_{1},\theta_{2},\omega)\aleph_{\theta_{2}\theta_{2}}(\theta_{1},\theta_{2},\omega)\bigr) -
\frac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}} \bigl(\theta_{1}\theta_{2}\aleph_{\theta_{1}}(\theta_{1},\theta_{2},\omega)\aleph_{\theta_{2}}(\theta_{1},\theta_{2},\omega)\bigr)-
\aleph(\theta_{1},\theta_{2},\omega),
\end{equation} \tag{23} \]
where $0<\beta\leqslant 1$, $(\theta_{1},\theta_{2},\omega)\in(\mathbb{R^{+}})^3$, and with the following initial conditions:
\[ \begin{equation*}
\aleph(\theta_{1},\theta_{2},0)={\rm e}^{\theta_{1}\theta_{2}}, \quad
D_{\omega}^{\beta}\aleph(\theta_{1},\theta_{2},0)={\rm e}^{\theta_{1}\theta_{2}}.
\end{equation*} \]
By applying $\mathcal{L}$ to Eq. (23),
\[ \begin{equation}
\mathcal{L}[D_{\omega}^{2\beta}\aleph(\theta_{1},\theta_{2},\omega)=
\mathcal{L}\Bigl[\frac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}
\big(\aleph_{\theta_{1}\theta_{1}}
(\theta_{1},\theta_{2},\omega)\aleph_{\theta_{2}\theta_{2}}(\theta_{1},\theta_{2},\omega)\big) -
\frac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}
\big(\theta_{1}\theta_{2}\aleph_{\theta_{1}}(\theta_{1},\theta_{2},\omega)\aleph_{\theta_{2}}(\theta_{1},\theta_{2},\omega)\big)-
\aleph(\theta_{1},\theta_{2},\omega)\Bigr].
\end{equation} \tag{24} \]
By following the steps that are established in Section3, we get the following result from Eq. (24):
\[ \begin{multline}
\Omega(\theta_{1},\theta_{2},\nu)=
\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu}+\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu^{\beta+1}}+
\frac{1}{\nu^{2\beta}}\mathcal{L}
\Bigl[\frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}
\bigl(D_{\theta_{1}\theta_{1}}\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2},\nu)]D_{\theta_{2}\theta_{2}}\mathcal{L}^{-1}
[\Omega(\theta_{1},\theta_{2},\nu)]\bigr)\Bigr] - {}
\\
{}
-\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[\frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}\bigl(\theta_{1}\theta_{2}D_{\theta_{1}}\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2},\nu)]D_{\theta_{2}}\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2}
,\nu)]\bigr)\Bigr] - \frac{1}{\nu^{2\beta}}\Omega(\theta_{1},\theta_{2},\nu).
\end{multline} \tag{25} \]
Consider the series solutions of Eq. (25), which have the following form:
\[ \begin{equation*}
\Omega(\theta_{1},\theta_{2},\nu)=\sum_{r=0}^{\infty}\frac{\xi_{r}(\theta_{1},\theta_{2})}{\nu^{r\beta+1}},
\quad
\nu>0.
\end{equation*} \]
The $j$th truncated expansion is as
\[ \begin{equation*}
\Omega_{j}(\theta_{1},\theta_{2},\nu)=\sum_{r=0}^{j}\frac{\xi_{r}(\theta_{1},\theta_{2})}{\nu^{r\beta+1}},
\quad
\nu>0.
\end{equation*} \]
As a result of applying the Lemma 1(iii) and Theorem 1, we obtained the following results:
\[ \begin{equation*}
\lim_{\nu\to\infty}\big(\nu\Omega(\theta_{1},\theta_{2},\nu)\big)=
\aleph(\theta_{1},\theta_{2},0)=\xi_{0}(\theta_{1})={\rm e}^{\theta_{1}\theta_{2}},\\
\xi_{1}(\theta_{1})=D_{\omega}^{\beta}\aleph(\theta_{1},\theta_{2},0)= {\rm e}^{\theta_{1}\theta_{2}}.
\end{equation*} \]
So, Eq. (25) becomes
\[ \begin{equation}
\Omega_{j}(\theta_{1},\theta_{2},\nu)=\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu}+\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu^{\beta+1}}+\sum_{r=2}^{j}\frac{\xi_{r}(\theta_{1},\theta_{2})}{\nu^{r\beta+1}},
\quad \nu>0.
\end{equation} \tag{26} \]
The following is the LRF for Eq. (25):
\[ \begin{multline*}
\mathcal{L}\operatorname{Res}(\theta_{1},\theta_{2},\nu)=
\Omega(\theta_{1},\theta_{2},\nu)-\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu}-\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu^{\beta+1}}-
\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[ \frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}\bigl(
D_{\theta_{1}\theta_{1}}\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2},\nu)]
D_{\theta_{2}\theta_{2}}\mathcal{L}^{-1} [\Omega(\theta_{1},\theta_{2},\nu)]\bigr)\Bigr] + {} \\
{} + \frac{1}{\nu^{2\beta}}\mathcal{L}
\Bigl[\frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}\bigl(\theta_{1}\theta_{2}D_{\theta_{1}}\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2},\nu)] D_{\theta_{2}}
\mathcal{L}^{-1}[\Omega(\theta_{1},\theta_{2},\nu)]\bigr)\Bigr]+
\frac{1}{\nu^{2\beta}}\Omega(\theta_{1},\theta_{2},\nu).
\end{multline*} \]
The following is the $j$th truncated LRF for Eq. (25):
\[ \begin{multline}
\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\theta_{2},\nu)=
\Omega_{j}(\theta_{1},\theta_{2},\nu)-\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu}-\frac{{\rm e}^{\theta_{1}\theta_{2}}}{\nu^{\beta+1}} -
\frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[ \frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}\bigl(
D_{\theta_{1}\theta_{1}}\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\theta_{2},\nu)]
D_{\theta_{2}\theta_{2}}\mathcal{L}^{-1}
[\Omega_{j}(\theta_{1},\theta_{2},\nu)]\bigr)\Bigr] + {} \\
{} + \frac{1}{\nu^{2\beta}}\mathcal{L}\Bigl[
\frac{\partial^{2}}{\partial\theta_{1}\partial \theta_{2}}\bigl(\theta_{1}\theta_{2}D_{\theta_{1}}\mathcal{L}^{-1}[\Omega_{j}(\theta_{1},\theta_{2},\nu)] D_{\theta_{2}}\mathcal{L}^{-1}
[\Omega_{j}(\theta_{1},\theta_{2},\nu)]\bigr)\Bigr] +
\frac{1}{\nu^{2\beta}}\Omega_{j}(\theta_{1},\theta_{2},\nu).
\end{multline} \tag{27} \]
In Eq. (16) and Eq. (17), use $j=\overline{1, 7}$ to find the undetermined coefficients $\xi_{j}(\theta_{1},\theta_{2})$, and then solve the following:
\[ \begin{equation*}
\lim_{\nu\to\infty}\big(\nu^{j\beta+1}\mathcal{L}\operatorname{Res}_{j}(\theta_{1},\theta_{2},\nu)\big)=0.
\end{equation*} \]
The following outcomes obtained:
\[ \begin{equation*}
\begin{array}{lll}
\xi_{2}(\theta_{1},\theta_{2})=-{\rm e}^{\theta_{1}\theta_{2}}, &
\xi_{3}(\theta_{1},\theta_{2})=-{\rm e}^{\theta_{1}\theta_{2}}, &
\xi_{4}(\theta_{1},\theta_{2})={\rm e}^{\theta_{1}\theta_{2}},\\
\xi_{5}(\theta_{1},\theta_{2})={\rm e}^{\theta_{1}\theta_{2}},&
\xi_{6}(\theta_{1},\theta_{2})=-{\rm e}^{\theta_{1}\theta_{2}},&
\xi_{7}(\theta_{1},\theta_{2})=-{\rm e}^{\theta_{1}\theta_{2}}.
\end{array}
\end{equation*} \]
In this way, we obtained the following 7th step approximate solution of Eq. (25) in Laplace transforms space:
\[ \begin{equation}
\Omega_{7}(\theta_{1},\theta_{2},\nu)={\rm e}^{\theta_{1}\theta_{2}}
\Bigl(\frac{1}{\nu}+\frac{1}{\nu^{\beta+1}}-\frac{1}{\nu^{2\beta+1}}-\frac{1}{\nu^{3\beta+1}} +
\frac{1}{\nu^{4\beta+1}}+\frac{1}{\nu^{5\beta+1}}-\frac{1}{\nu^{6\beta+1}}-\frac{1}{\nu^{7\beta+1}}\Bigr).
\end{equation} \tag{28} \]
By applying $\mathcal{L}^{-1}$ to Eq. (28), we obtain the approximate 7th step solution in the original space, which has the following form:
\[ \begin{equation*}
\aleph_{7}(\theta_{1},\theta_{2},\omega)=
{\rm e}^{\theta_{1}\theta_{2}}\Bigl(
1+\frac{\omega^{\beta}}{\Gamma[\beta+1]}-\frac{\omega^{2\beta}}{\Gamma[2\beta+1]}-\frac{\omega^{3\beta}}{\Gamma[3\beta+1]}+
\frac{\omega^{4\beta}}{\Gamma[4\beta+1]} +
\frac{\omega^{5\beta}}{\Gamma[5\beta+1]}-\frac{\omega^{6\beta}}{\Gamma[6\beta+1]}-\frac{\omega^{7\beta}}
{\Gamma[7\beta+1]}\Bigr).
\end{equation*} \]
When $\beta=1.0$, the 7th step approximate solution is
\[ \begin{equation}
\aleph_{7}(\theta_{1},\theta_{2},\omega)={\rm e}^{\theta_{1}\theta_{2}}
\Bigl(1+\frac{\omega}{\Gamma[{2}]}-\frac{\omega^{2}}{\Gamma[{3}]}-\frac{\omega^{3}}{\Gamma[{4}]}+\frac{\omega^{4}}{\Gamma[{5}]}+
\frac{\omega^{5}}{\Gamma[{6}]}-\frac{\omega^{6}}
{\Gamma[{7}]}-\frac{\omega^{7}}{\Gamma[{8}]}
\Bigr).
\end{equation} \tag{29} \]
The first eight terms of the series of the exact solution to ${\rm e}^{\theta_{1}\theta_{2}}(\cos\omega+\sin\omega)$ are represented by Eq. (29).
5. Numerical Simulation and Discussion
In this section, the results of the approximate and exact solutions to the problems are examined graphically and numerically. Error functions can be used to evaluate the accuracy of the approximate analytical approach, so it is necessary to specify the errors in the approximate solutions provided by the LRPSM. We used the Abs-E, Rel-E, and Rec-E functions to demonstrate the accuracy and efficiency of LRPSM.
Figure 1. The approximate and exact results of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for various amounts of \(\beta\) in the range \(\omega \in [0{,}1.0]\), when \(\theta_1=0.5\) (a), \(\theta_1=\theta_2=\theta_3=0.5\) (b), and \(\theta_1=\theta_2=0.5\) (c)
Fig. 1 depicts the 2D graphs of the comparative study of the exact and approximate solutions obtained by the proposed method in Problems 1–3. These figures show the 2D plots of the exact and 7th step approximate solutions attained by LRPSM for Problems 1–3, when $\beta=0.6$, 0.7, 0.8, 0.9, and 1.0 in the range $\omega \in [0{,} 1.0]$. These graphs indicate that when $\beta\to 1.0$ is applied, the approximative solution converges to the exact solution. The exact and approximation solutions overlap at $\beta=1.0$, demonstrating the accuracy and reliability of the proposed method.
Fig. 2 shows the Abs-E of the proposed method's 7th step approximate and exact solutions to Problems 1–3 in the range $\omega \in [0{,} 0.5]$ for $\beta = 1.0$.
Figure 2. The 2D curves of the Abs-E graph of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and the exact result in the range \(\omega \in [0{,}0.5]\), when \(\theta_1=0.25\) (a), \(\theta_1=\theta_2=\theta_3=0.25\) (b), and \(\theta_1=\theta_2=0.25\) (c)
Fig. 3 depicts a comparison of the Rel-E of the exact and 7th step approximate solutions to problems $1-3$ with $\beta = 1.0$ in the range $\omega \in [0{,} 1.0]$. The graphical analysis of the approximate and exact findings in the form of Abs-E and Rel-E demonstrates the reliability and precision of LRPSM.
Fig. 4 depicts the comparison study using the 3D plots in terms of the Abs-E of the approximate finding from the seven iterations and the exact result found using the suggested method to Problems 1–3, respectively, at $\beta = 1.0$ in the ranges $\omega \in [0{,} 0.5]$ and $\theta \in [0{,} 0.5]$.
Fig. 5 depicts the comparison study using the 3D curve in terms of the Rel-E of the approximate finding from the seven iterations and the exact result found using the suggested method to Problems 1–3, respectively, at $\beta = 1.0$ in the range $\omega \in [0{,} 0.5]$ and $\theta \in [0{,} 0.5]$.
The study has revealed that the 7th step approximate solutions of the proposed method are very similar to the exact solution. The reliability and accuracy of LRPSM is demonstrated by graphical analysis of approximate and exact results in the form of Abs-E and Rel-E.
Figure 3. The 2D curves of the Rel-E graph of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and the exact result in the range \(\omega \in [0{,}0.5]\), when \(\theta_1=0.25\) (a), \(\theta_1=\theta_2=\theta_3=0.25\) (b), and \(\theta_1=\theta_2=0.25\) (c)
Figure 4. The 3D plots of the Abs-E of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and exact result in the ranges \(\omega \in [0{,}0.5]\) and \(\theta_1 \in [0{,}0.5]\), when \(\theta_2=\theta_3=0.25\) (b), and \(\theta_2=0.25\) (c)
Figure 5. The 3D plots of the Rel-E of \(\aleph(\theta_1,\omega)\) (a), \(\aleph(\theta_1,\theta_2,\theta_3,\omega)\) (b), and \(\aleph(\theta_1,\theta_2,\omega)\) (c) for the approximate finding obtained through seven iterations and exact result in the ranges \(\omega \in [0{,}0.5]\) and \(\theta_1 \in [0{,}0.5]\), when \(\theta_2=\theta_3=0.25\) (b), and \(\theta_2=0.25\) (c)
Tables 1–3 show Abs-E and Rel-E in the range $\omega\in[0, 1.0]$ between the approximate solution obtained from the seven iterations and the exact solution obtained by LRPSM at $\beta = 1.0$ for appropriately chosen values. The amplitudes of Abs-E and Rel-E are shown in Table 1 which range from $2.29781 \cdot 10^{-11}$ to $ {3.04568 \cdot 10^{-3}}$ and from $ 2.07914 \cdot 10^{-11}$ to $4.55539 \cdot 10^{-4}$, respectively, for Problem 1. The intervals of Abs-E and Rel-E are shown in Table 2 which are from $ 1.63156 \cdot 10^{-13}$ to $ 2.93588 \cdot 10^{-3}$ and from $1.21370 \cdot 10^{-9}$ for $1.27164 \cdot 10^{-3}$, respectively, to Problem 2. Table 3 shows the intervals of Abs-E and Rel-E which are from $2.19100 \cdot 10^{-11}$ to $ 2.81468 \cdot 10^{-3}$ and $2.08398 \cdot 10^{-11}$ for $8.18227 \cdot 10^{-4}$, respectively, to Problem 3. Tables 1–3 demonstrate that the 7th step approximate solutions to all numerical equations have remarkably minimal Abs-E and Rel-E. So, the suggested method is very useful for the analysis of different FODEs with a physical interest in the areas of applied mathematics and engineering. As shown in Tables 4–6, Rec-E has been used to numerically demonstrate the process by which the approximate solutions of the Problems 1–3 converge to the exact solution for selected values in the range $\omega \in [0{,} 1]$. As the order of the fractional derivative is increased, the approximate solution of the seven iterations produced using the suggested method quickly converges to the exact solution. The graphical and numerical findings show the accuracy and reliability of the LRPSM.
Table 1 displays the Abs-E and Rel-E at appropriate grid locations in the ranges $\omega\in[0, 1.0]$ and $\theta_{1} \in [0{,} 1.0]$ of the approximate solution attained from the seven iterations and the exact solution to Problem 1 at $\beta=1.0$ using the LRPSM.
Table 1. The Abs-E and Rel-E for Problem 1| $(\theta_{1},\omega)$ | Abs. Errors | Rel. Errors |
| $(0.05,0.05)$ | $2.29781 \cdot 10^{-11}$ | $2.07914 \cdot 10^{-11}$ |
| $(0.15,0.15)$ | $1.87820 \cdot 10^{-8}$ | $1.39140 \cdot 10^{-8}$ |
| $(0.25,0.25)$ | $4.51442 \cdot 10^{-7}$ | $2.73814 \cdot 10^{-7}$ |
| $(0.35,0.35)$ | $3.81249 \cdot 10^{-6} $ | $1.89322 \cdot 10^{-6}$ |
| $(0.45,0.45)$ | $1.93189 \cdot 10^{-5}$ | $7.85450 \cdot 10^{-6}$ |
| $(0.55,0.55)$ | $7.22544 \cdot 10^{-5}$ | $2.40514 \cdot 10^{-5}$ |
| $(0.65,0.65)$ | $2.20912 \cdot 10^{-4}$ | $6.02057 \cdot 10^{-5}$ |
| $(0.75,0.75)$ | $5.85105 \cdot 10^{-4}$ | $1.30554 \cdot 10^{-4}$ |
| $(0.85,0.85)$ | $1.39181 \cdot 10^{-3}$ | $2.54261 \cdot 10^{-4}$ |
| $(0.95,0.95)$ | $3.04568 \cdot 10^{-3}$ | $4.55539 \cdot 10^{-4}$ |
Table 2. The Abs-E and Rel-E for Problem 2| $(\theta_{1},\theta_{2},\theta_{3},\omega)$ | Abs. Errors | Rel. Errors |
| $(0.05,0.05,0.05,0.05)$ | $1.63156 \cdot 10^{-13}$ | $1.21370 \cdot 10^{-9}$ |
| $(0.15,0.15,0.15,0.15)$ | $1.07593 \cdot 10^{-9}$ | $2.59356 \cdot 10^{-7}$ |
| $(0.25,0.25,0.25,0.25)$ | $6.44068 \cdot 10^{-8}$ | $2.97104 \cdot 10^{-6}$ |
| $(0.35,0.35,0.35,0.35)$ | $9.56001 \cdot 10^{-7}$ | $1.43769 \cdot 10^{-5}$ |
| $(0.45,0.45,0.45,0.45)$ | $7.18225 \cdot 10^{-6}$ | $4.58092 \cdot 10^{-5}$ |
| $(0.55,0.55,0.55,0.55)$ | $3.59959 \cdot 10^{-5}$ | $1.14039 \cdot 10^{-4}$ |
| $(0.65,0.65,0.65,0.65)$ | $1.37909 \cdot 10^{-4}$ | $2.41227 \cdot 10^{-4}$ |
| $(0.75,0.75,0.75,0.75)$ | $4.36368 \cdot 10^{-4}$ | $4.54630 \cdot 10^{-4}$ |
| $(0.85,0.85,0.85,0.85)$ | $1.19656 \cdot 10^{-3}$ | $7.86128 \cdot 10^{-4}$ |
| $(0.95,0.95,0.95,0.95)$ | $2.93588 \cdot 10^{-3}$ | $1.27164 \cdot 10^{-3}$ |
Table 3. The Abs-E and Rel-E for Problem 3| $(\theta_{1},\theta_{2},\omega)$ | Abs. Errors | Rel. Errors |
| $(0.05,0.05,0.05)$ | $2.19100 \cdot 10^{-11}$ | $2.08398 \cdot 10^{-11}$ |
| $(0.15,0.05,0.15)$ | $1.65204 \cdot 10^{-8}$ | $1.41915 \cdot 10^{-8}$ |
| $(0.25,0.05,0.25)$ | $3.73431 \cdot 10^{-7}$ | $2.88416 \cdot 10^{-7}$ |
| $(0.35,0.05,0.35)$ | $3.02361 \cdot 10^{-6}$ | $2.08615 \cdot 10^{-6}$ |
| $(0.45,0.05,0.45)$ | $1.49760 \cdot 10^{-5}$ | $9.15876 \cdot 10^{-6}$ |
| $(0.55,0.05,0.55)$ | $5.58162 \cdot 10^{-5}$ | $2.99928 \cdot 10^{-5}$ |
| $(0.65,0.05,0.65)$ | $1.73376 \cdot 10^{-4}$ | $8.10919 \cdot 10^{-5}$ |
| $(0.75,0.05,0.75)$ | $4.75627 \cdot 10^{-4}$ | $1.91749 \cdot 10^{-4}$ |
| $(0.85,0.05,0.85)$ | $1.19473 \cdot 10^{-3}$ | $4.11041 \cdot 10^{-4}$ |
| $(0.95,0.05,0.95)$ | $2.81468 \cdot 10^{-3}$ | $8.18227 \cdot 10^{-4}$ |
Table 2 displays the Abs-E and Rel-E at appropriate grid locations in the ranges $\omega \in [0{,} 1.0]$, $\theta_{1} \in [0{,} 1.0]$, $\theta_{2} \in [0{,} 1.0]$ as well as $\theta_{3} \in [0{,} 1.0]$ of the approximate solution obtained from the seven iterations and the exact solution to Problem 2 at $\beta=1.0$ using the LRPSM.
Table 3 displays the Abs-E and Rel-E at appropriate grid locations in the ranges $\omega \in [0{,} 1.0]$, $\theta_{1} \in [0{,} 1.0]$ as well as $\theta_{2} \in [0{,} 1.0]$ of the approximate solution obtained from the seven iterations and the exact solution to Problem 3 at $\beta=1.0$ using the LRPSM.
Table 4 shows the Rec-E between the approximate solution attained from the seven iterations and exact solutions of Problem 1 acquired by LRPSM at appropriate grid locations in the ranges $\omega \in [0{,} 1.0]$, and $\theta_{1} \in [0{,} 1.0]$.
Table 4. The Rec-E for $\aleph(\theta_{1},\omega)$ at different values of $\beta$ for Problem 1| $(\theta_{1},\omega)$ | $\beta=0.7$ | $\beta=0.8$ | $\beta=0.9$ | $\beta=1.0$ |
| $(0.02,0.02)$ | $9.92308 \cdot 10^{-8}$ | $6.80134 \cdot 10^{-9}$ | $4.41026 \cdot 10^{-10}$ | $2.72054 \cdot 10^{-11}$ |
| $(0.12,0.12)$ | $5.80237 \cdot 10^{-5}$ | $9.74157 \cdot 10^{-6}$ | $1.54730 \cdot 10^{-6}$ | $2.33798 \cdot 10^{-7}$ |
| $(0.22,0.22)$ | $5.35032 \cdot 10^{-4}$ | $1.21625 \cdot 10^{-4}$ | $2.61571 \cdot 10^{-5}$ | $5.35152 \cdot 10^{-6}$ |
| $(0.32,0.32)$ | $2.19460 \cdot 10^{-3}$ | $6.01676 \cdot 10^{-4}$ | $1.56060 \cdot 10^{-4}$ | $3.85073 \cdot 10^{-5}$ |
| $(0.42,0.42)$ | $6.28249 \cdot 10^{-3}$ | $1.97328 \cdot 10^{-3}$ | $5.86366 \cdot 10^{-4}$ | $1.65756 \cdot 10^{-4}$ |
| $(0.52,0.52)$ | $1.46622 \cdot 10^{-2}$ | $5.12431 \cdot 10^{-3}$ | $1.69430 \cdot 10^{-3}$ | $7.39038 \cdot 10^{-4}$ |
| $(0.62,0.62)$ | $2.99909 \cdot 10^{-2}$ | $1.14451 \cdot 10^{-2}$ | $4.13208 \cdot 10^{-3}$ | $1.80237 \cdot 10^{-3}$ |
| $(0.72,0.72)$ | $5.59386 \cdot 10^{-2}$ | $2.30044 \cdot 10^{-2}$ | $8.95017 \cdot 10^{-3}$ | $3.90397 \cdot 10^{-3}$ |
| $(0.82,0.82)$ | $9.74598 \cdot 10^{-2}$ | $4.27726 \cdot 10^{-2}$ | $1.77593 \cdot 10^{-2}$ | $7.74645 \cdot 10^{-3}$ |
| $(0.92,0.92)$ | $1.61125 \cdot 10^{-1}$ | $7.49016 \cdot 10^{-2}$ | $3.29412 \cdot 10^{-2}$ | $1.43686 \cdot 10^{-2}$ |
Table 5 shows the Rec-E between the approximate solution obtained from the seven iterations and exact solutions of Problem 2 acquired by LRPSM at appropriate grid locations in the ranges $\omega \in [0{,} 1.0]$, $\theta_{1} \in [0{,} 1.0]$, $\theta_{2} \in [0{,} 1.0]$, and $\theta_{3} \in [0{,} 1.0]$.
Table 6 shows the Rec-E between the approximate solution from the 5th iteration and exact solutions of Problem 3 acquired by LRPSM at appropriate grid locations in the ranges $\omega \in [0{,} 1.0]$, $\theta_{1} \in [0{,} 1.0]$, and $\theta_{2} \in [0{,} 1.0]$.
Table 5. The Rec-E for $\aleph(\theta_{1},\theta_{2},\omega)$ at different values of $\beta$ for Problem 2| $(\theta_{1},\theta_{2},\theta_{3},\omega)$ | $\beta=0.7$ | $\beta=0.8$ | $\beta=0.9$ | $\beta=1.0$ |
| $(0.02,0.02,o.02,0.02)$ | $3.89064 \cdot 10^{-11}$ | $2.66667 \cdot 10^{-12}$ | $ 1.72917 \cdot 10^{-13}$ | $1.06667 \cdot 10^{-14}$ |
| $(0.12,0.12,0.12,0.12)$ | $7.41059 \cdot 10^{-7}$ | $1.24416 \cdot 10^{-7}$ | $1.97616 \cdot 10^{-8}$ | $2.98598 \cdot 10^{-9}$ |
| $(0.22,0.22,0.22,0.22)$ | $2.07817 \cdot 10^{-5}$ | $4.72416 \cdot 10^{-6}$ | $1.01599 \cdot 10^{-6}$ | $2.07863 \cdot 10^{-7}$ |
| $(0.32,0.32,0.32,0.32)$ | $1.63185 \cdot 10^{-4}$ | $4.47392 \cdot 10^{-5}$ | $1.16043 \cdot 10^{-5}$ | $2.86331 \cdot 10^{-6}$ |
| $(0.42,0.42,0.42,0.42)$ | $7.28160 \cdot 10^{-4}$ | $2.28710 \cdot 10^{-4}$ | $6.79616 \cdot 10^{-5}$ | $1.92116 \cdot 10^{-5}$ |
| $(0.52,0.52,0.52,0.52)$ | $2.35708 \cdot 10^{-3}$ | $8.23775 \cdot 10^{-3}$ | $2.72374 \cdot 10^{-4}$ | $8.56726 \cdot 10^{-5}$ |
| $(0.62,0.62,062,0.62)$ | $6.20169 \cdot 10^{-3}$ | $2.36668 \cdot 10^{-3}$ | $8.54455 \cdot 10^{-4}$ | $2.93468 \cdot 10^{-4}$ |
| $(0.72,0.72,0.72,0.72)$ | $1.41151 \cdot 10^{-2}$ | $5.80475 \cdot 10^{-3}$ | $2.25842 \cdot 10^{-3}$ | $8.35884 \cdot 10^{-4}$ |
| $(0.82,0.82,0.82,0.82)$ | $2.88624 \cdot 10^{-2}$ | $1.26669 \cdot 10^{-2}$ | $5.25936 \cdot 10^{-3}$ | $2.07738 \cdot 10^{-3}$ |
| $(0.92,0.92,0.92,0.92)$ | $5.43487 \cdot 10^{-2}$ | $2.52648 \cdot 10^{-2}$ | $1.11113 \cdot 10^{-2}$ | $4.64872 \cdot 10^{-3}$ |
Table 6. The Rec-E for $\aleph(\theta_{1},\theta_{2},\omega)$ at different values of $\beta$ for Problem 3| $(\theta_{1},\theta_{2},\omega)$ | $\beta=0.7$ | $\beta=0.8$ | $\beta=0.9$ | $\beta=1.0$ |
| $(0.02,0.02,0.02)$ | $9.73048 \cdot 10^{-8}$ | $6.66933 \cdot 10^{-9}$ | $4.32466 \cdot 10^{-10}$ | $2.66773 \cdot 10^{-11}$ |
| $(0.12,0.12,0.12)$ | $5.22088 \cdot 10^{-5}$ | $8.76532 \cdot 10^{-6}$ | $1.39224 \cdot 10^{-6}$ | $2.10368 \cdot 10^{-7}$ |
| $(0.22,0.22,0.22)$ | $5.06660 \cdot 10^{-4}$ | $1.02447 \cdot 10^{-4}$ | $2.20326 \cdot 10^{-5}$ | $4.50767 \cdot 10^{-6}$ |
| $(0.32,0.32,0.32)$ | $1.76544 \cdot 10^{-3}$ | $4.84017 \cdot 10^{-4}$ | $1.25542 \cdot 10^{-4}$ | $3.09771 \cdot 10^{-5}$ |
| $(0.42,0.42,0.42)$ | $4.92422 \cdot 10^{-3}$ | $1.54666 \cdot 10^{-3}$ | $4.59594 \cdot 10^{-4}$ | $1.29920 \cdot 10^{-4}$ |
| $(0.52,0.52,0.52)$ | $1.14235 \cdot 10^{-2}$ | $3.99241 \cdot 10^{-3}$ | $1.32005 \cdot 10^{-3}$ | $4.15211 \cdot 10^{-4}$ |
| $(0.62,0.62,0.62)$ | $2.36957 \cdot 10^{-2}$ | $9.04270 \cdot 10^{-3}$ | $3.26474 \cdot 10^{-3}$ | $1.12130 \cdot 10^{-3}$ |
| $(0.72,0.72,0.72)$ | $4.57254 \cdot 10^{-2}$ | $1.88043 \cdot 10^{-2}$ | $7.31607 \cdot 10^{-3}$ | $2.70782 \cdot 10^{-3}$ |
| $(0.82,0.82,0.82)$ | $8.40860 \cdot 10^{-2}$ | $3.69032 \cdot 10^{-2}$ | $1.53223 \cdot 10^{-2}$ | $6.05212 \cdot 10^{-3}$ |
| $(0.92,0.92,0.92)$ | $1.49692 \cdot 10^{-1}$ | $6.95868 \cdot 10^{-2}$ | $3.06038 \cdot 10^{-2}$ | $1.28040 \cdot 10^{-2}$ |
6. Conclusion
We used the Laplace transform with the residual power series method to solve time-fractional partial differential equations with variable coefficients in the sense of the Gerasimov–Caputo fractional derivative. Graphics and tables show that the 7th step approximate and exact solutions are in perfect agreement, which demonstrated the efficiency and reliability of the Laplace residual power series method.
Finally, in conclusion, the main features of our recommended method are the following. The residual power series method is useful for obtaining approximate analytical solutions to fractional order problems, but it requires the residual function's $(n-1)\beta$ derivative to determine the coefficients of the series solution, whereas more widely used methods such as the homotopy perturbation, variational iterational, and Adomian decomposition methods require integration. We are all aware of how challenging it is to find the fractional derivatives and the integration, but our method only requires the concept of an infinite limit, which is relatively simple. Therefore, the problem can be solved using our recommended method, which provides a quick and simple way to figure out the coefficients of the series solution. It is not essential to make any significant physical or parametric assumptions related to the problem. Therefore, our method overcomes some of the inherent limitations of conventional perturbation approaches and works with both weak and strongly nonlinear systems. The He and Adomian polynomials do not need our method, so nonlinear problems can be solved with a relatively small number of calculations. As a result, it performs noticeably better than various series solution methods based on Adomian decomposition and homotopy perturbation methods. In contrast to various analytic approximate methods, the Laplace residual power series method may give analytical expansion solutions for both linear and nonlinear problems without the need for perturbation, linearization, or discretization.
Therefore, our method is simple to apply, accurate and reliable in its results. In the future, as new fractional-order problems emerge in various situations, we plan to solve them using the Laplace residual power series method.
Competing interests. The authors declare that they have no competing interests.
Authors’ contributions and responsibilities. The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Availability of Data and Materials. No data were generated or analyzed during the current study.
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