A computational study of time-fractional gas dynamics models by means of conformable finite difference method

1.   Introduction

Nonlinear differential equations with time-fractional derivatives have emerged as a fundamental tool for modeling complex dynamical systems exhibiting anomalous temporal behaviors. Unlike classical integer-order differential equations, which assume that time evolves linearly and continuously, time-fractional calculus allows for the consideration of non-integer orders of differentiation, enabling the description of processes with memory, hereditary properties, and long-range dependencies. Including nonlinearity further enriches these equations, capturing intricate dynamics that arise in various fields such as physics, biology, finance, and engineering ^{[1,2,3,4,5,6]}. Recent studies have explored numerical techniques for investigating nonlinear time-fractional differential equations, encompassing various methodologies. The power series method ^[7,8,9], Adomian decomposition and variational iteration methods ^[10], kernel Hilbert space method ^[11], Taylor wavelet method ^[12], differential transformation method ^[13], finite difference method ^[14], Mohand variational iteration transform ^[15], and finite element method ^[16] represent some of the approaches applied in these investigations. Each method offers unique advantages in accuracy, efficiency, and applicability, contributing to the advancement of numerical solutions for time-fractional differential equations.

Gas dynamics is essential for designing equipment, engines, and gas-powered vehicles. Understanding forces in gas interactions, including pressure, temperature, friction, and heat movement, is aided by this. Combustion, detonation, blast waves, and gas propulsion are all included in gas dynamics. Its laws are essential for studying explosions, developing machinery like compressors, turbines, and rocket motors, and studying exterior and interior ballistics. Accurately estimating these phenomena involves mathematical modeling. Finding solutions of nonlinear time-fractional gas dynamics equations presents a formidable challenge due to their inherent complexity, often defying traditional solution techniques. However, various numerical methods have been developed to approximate these solutions effectively. One approach involves combining the Laplace transformation with the power series method, as proposed by ^[17]. Alternatively, the differential transform method offers another avenue for approximation ^[18]. Other methods include the quadratic b-spline Galerkin and the cubic b-spline collection methods ^[19,20]. Trigonometric B-spline functions have also been suggested as a useful tool ^[21]. Additionally, ^[22] introduced the integral projected differential transform method, while ^[23] proposed the homotopy analysis transform method. ^[24] introduced the optimal q-homotopy analysis method, and ^[25] suggested employing Elzaki homotopy perturbation and variational iteration methods. In recent years, the utilization of the conformable fractional derivative, as defined by Khalil ^[26], has garnered attention in fractional differential equations. This derivative has been integrated into various numerical methodologies and modeling techniques, showcasing its versatility and applicability. Afterward, ^[27] introduced some fundamental properties and definitions of the conformable derivative. For instance, a conformable non-polynomial spline method has also been introduced, further expanding the repertoire of techniques leveraging conformable fractional derivatives ^[28]. Additionally, a study on the fractional regularized long Wave equation with conformable fractional derivatives highlights the potential of conformable derivatives in modeling physical phenomena ^[29]. Moreover, exploration of the conformable time Korteweg-de Vries equation using the finite element method provides insights into the effectiveness of conformable fractional derivatives in numerical simulations ^[30]. These advancements underscore the growing significance of conformable fractional derivatives in enhancing the accuracy and efficiency of numerical solutions for fractional differential equations, paving the way for further developments in this field. The motivation behind this research lies in the inherent complexity of time-fractional gas dynamics equations, which often defy traditional solution techniques. These equations are crucial in understanding various physical phenomena such as explosions, combustion, detonation, and condensation in moving flows. The conformable fractional derivative is a more recent definition that preserves several properties of integer-order differentiation, allowing for a simpler formulation and interpretation, while the Caputo derivative, a well-established concept in fractional calculus, is particularly useful in initial value problems due to its compatibility with classical initial conditions. By introducing the CFDM, we aim to address the challenges posed by these equations and provide a reliable numerical framework for accurately analyzing and predicting the behavior of gas dynamics systems. The development of the CFDM is motivated by the pressing need for efficient and accurate numerical methods to facilitate advancements in gas dynamics research and engineering applications.

This study aims to develop and validate the conformable finite difference method (CFDM) as a robust numerical scheme for accurately solving time-fractional gas dynamics equations. Additionally, we aim to demonstrate the applicability of the CFDM in solving a broader range of time-fractional differential equations encountered in various scientific and engineering fields, thereby showcasing its versatility as a general-purpose numerical scheme. The advantage of the conformable fractional derivative lies in its simpler formulation and preservation of several properties of integer-order differentiation, making it easier to interpret and apply in numerical methods and modeling while maintaining accuracy in describing processes with memory and hereditary properties.

The novelty lies in introducing the CFDM to solve time-fractional gas dynamics equations. This method integrates finite difference techniques with conformable derivatives, offering a unique approach to address the complexities of these equations. It not only accurately analyzes gas dynamics systems, but also showcases versatility for solving various time-fractional differential equations in different fields. This study contributes to advancing numerical methods for fractional differential equations, facilitating computational studies of dynamic systems. The study highlights the versatility and computational efficiency of the CFDM by demonstrating its ability to accurately solve a range of time-fractional gas dynamics equations, capturing complex phenomena such as explosions, combustion, and detonation. Through rigorous stability analysis and comparative assessments using ${L}_{2}$ and ${L}_{\infty }$ norms against existing methods like trigonometric B-spline functions and the quadratic B-spline Galerkin method, the study showcases the CFDM's superior accuracy and efficiency. Additionally, the successful application of the CFDM to various illustrative examples underscores its robustness and adaptability to different problems in time-fractional differential equations.

The rest of the paper is organized as follows. Section 2 presents a detailed description of the conformable finite difference method (CFDM), elucidating its fundamental principles and implementation procedures. Section 3 is dedicated to the stability analysis of the investigated numerical scheme, where we rigorously examine the stability properties under various conditions. In Section 4, we present the numerical results obtained using the CFDM and comprehensively discuss the findings. Through comparative analysis and interpretation of the results, we aim to provide insights into the performance and accuracy of the proposed method. Finally, in the last section, we draw conclusions based on our study, highlighting the contributions, implications, and potential future research directions.

Consider the conformable time-fractional gas dynamics equations

where $\mu$ and $\varepsilon$ are reaction and convection parameters, respectively, $\mathfrak{R}\left(\varphi , \tau \right)$ represents the evolution of the state across both space and time, $\varPsi \left(\varphi , \tau \right)$ is an appropriate predetermined function, and the fractional derivative ${T}_{\tau }^{\xi }\mathfrak{R}\left(\varphi , \tau \right)$ is expressed in the conformable derivative.

2.   Description of conformable finite difference method

This section introduces the combined use of conformable derivatives and the finite difference method, enhancing numerical analysis capabilities. The conformable derivative accommodates functions in diverse domains, while the finite difference method excels in discretization and derivative approximation, which is particularly useful for non-analytic functions or discrete datasets.

Definition 2.1: The conformable time-fractional derivative ${T}_{\varphi }^{\xi }\mathfrak{R}\left(\varphi , \tau \right)$ for $\mathfrak{R}:\left[0, \infty \right]\to \mathbb{R}$, defined by ^[26], is given as follows:

Lemma 2.1: ^[26,27] Let $\xi \in \left(\mathrm{0, 1}\right]$ and $\mathfrak{R}, \mathcal{H}$ be $\xi -$ differentiable at $\alpha$ point $\tau > 0$. Then:

(ⅰ). ${T}_{\tau }^{\xi }(\alpha \mathfrak{R}+\beta \mathcal{H}) = \alpha {T}_{\tau }^{\xi }\mathfrak{R}+\beta {T}_{\tau }^{\xi }\mathcal{H}$ for $\alpha , \beta \in \mathbb{R},$

(ⅱ). ${T}_{\tau }^{\xi }\left({\tau }^{\alpha }\right) = a{\tau }^{\alpha -\xi }$ for all $\alpha \in \mathbb{R}$,

(ⅲ). ${T}_{\tau }^{\xi }c = 0$ if $c$ is constant function.

(ⅳ) ${T}_{\tau }^{\xi }\left(\mathfrak{R}\mathcal{H}\right) = \mathfrak{R}\mathfrak{ }\left({T}_{\tau }^{\xi }\mathcal{H}\right)+\mathcal{H}\mathcal{ }\left({T}_{\tau }^{\xi }\mathfrak{R}\right),$

(ⅴ) ${T}_{\tau }^{\xi }\left(\frac{\mathfrak{R}}{\mathcal{H}}\right) = \frac{\mathcal{H}\left({T}_{\tau }^{\xi }\mathfrak{R}\right)-\mathfrak{R}\left({T}_{\tau }^{\xi }\mathcal{H}\right)}{{\mathcal{H}}^{2}},$

(ⅵ). ${T}_{\tau }^{\xi }\mathfrak{R}\left(\tau \right) = {\tau }^{1-\xi }\frac{\partial \mathfrak{R}\left(\tau \right)}{\partial \tau },$ if $\mathfrak{R}\left(\tau \right)$ is differentiable.

Corollary 2.1: Let $\varphi = jh, j = \mathrm{1, 2}, \cdots , L$, and ${\tau }_{n} = nk, n = \mathrm{1, 2}, \cdots , N$, with uniform spatial step size $h = \frac{b-a}{L}$ and temporal step size $k = \frac{T}{N}$. In the context of the finite difference scheme, we have

and, according to Lemma 1 (vi) and Eq (5), we have

where $\omega = {\tau }^{1-\xi }$. Using the Crank-Nicolson finite difference formula, we have

and

Therefore, substituting Eqs (6)–(8) into Eq (1), we obtain

which implies that

Then, after some simplification, we have

System (11) has less than or not enough equations $(N– 1)$ compared to its number of unknowns $(N+ 1)$, necessitating two additional equations to achieve a solution. These additional equations are derived from the initial and boundary conditions, which are as follows:

where,

Rewriting Eq (12) in matrix form:

Here,

3.   Stability analysis for the numerical scheme

In this section, the stability of the numerical scheme is under scrutiny, assuming, according to the Fourier stability principle, that the solution to Eq (12) adheres to a specific structure. This approach leverages the principles of Fourier stability to assess the reliability and performance of the numerical method in question, offering valuable insights into its stability and accuracy.

In this context, $\upsilon$ represents the actual spatial wave number, and $i$ is the imaginary unit $i = \sqrt{-1}$. By transforming the nonlinear term into a linear form and substituting the expression from Eq (14) into Eq (12), we obtain

After dividing both sides by ${e}^{i\upsilon hj}$, the resultant expression is

Then, implementing specific simplifications and grouping pertinent terms, we have

Using the Euler formula in complex analysis, we obtain the following:

After some simplification, Eq (17) yields

In that case, the numerical scheme (12) is unconditionally stable.

4.   Numerical results and discussion

In this section, the accuracy and effectiveness of the developed numerical scheme are thoroughly examined through the investigation of three examples. The obtained numerical results are meticulously compared with existing works, and the outcomes are visually presented through 2D and 3D graphs. The numerical simulations are conducted utilizing MATLAB R2017b, showcasing the robustness and reliability of the implemented scheme. To quantify the accuracy of the proposed scheme, maximum and least square error norms are computed using well-defined formulas:

Example 4.1. Consider the gas dynamics equation with time-fractional derivative in the following form:

where the Mittag-Leffler function ${E}_{\xi }\left(\varphi \right)$ is defined as

and the exact solution of the problem is $\mathfrak{R}\left(\varphi , \tau \right) = {e}^{-\varphi }{E}_{\xi }\left({\tau }^{\xi }\right)$.

The presented figures collectively offer valuable insights into the behavior and performance of the conformable finite difference method (CFDM) in solving time fractional gas dynamic equations in Example 1. Figure 1 provides a visual representation of CFDM in a 3D plot, illustrating how variations in parameters such as $\varphi$ and $\tau$ within the specified range impact the output. In Figure 2, the comparison between the exact and CFDM methods, with both $\xi$ and $\tau$ set to $0.5$, demonstrates excellent agreement, confirming the accuracy and efficiency of the numerical scheme employed for time fractional gas dynamic equations. Figure 3 reveals a direct relationship between time $\tau$ and the magnitude of the solution $\mathfrak{R}\left(\varphi , \tau \right)$, indicating the temporal influence on system behavior. Finally, Figure 4 highlights the converse effect of the fractional order $\xi$ on $\mathfrak{R}\left(\varphi , \tau \right)$, showcasing how increasing $\xi$ leads to a decrease in the solution magnitude. Figure 5 shows the comparison between the exact and CFDM methods using an absolute error plot. In addition to the graphical representations provided in the figures, Table 1 offers a comparison of the error norms of the presented numerical scheme with that of previous approaches utilizing the trigonometric b-spline method ^[21] and the quadratic b-spline Galerkin method ^[20]. The table presents the error norm values, allowing for a direct assessment of the accuracy and performance of each of the methods. The data demonstrate that the solutions obtained using the CFDM outperform those achieved through the trigonometric b-spline and quadratic b-spline Galerkin methods. This superiority in accuracy further underscores the effectiveness and reliability of CFDM in solving time fractional gas dynamic equations. The comparison provided in Table 1 reinforces the conclusions drawn from the graphical analyses, strengthening the argument for the adoption of CFDM as a preferred numerical scheme for such complex fluid dynamic systems.

Example 4.2. Consider the gas dynamics equation with time-fractional derivative in the following form:

where

and the exact solution of the problem is $\mathfrak{R}\left(\varphi , \tau \right) = {\tau }^{\xi }\mathrm{tan}\varphi$.

Table 2 provides an error norm comparison between the presented numerical scheme and that of a previous approach utilizing the trigonometric b-spline method ^[21]. Through direct assessment of the error norm values, the table reveals that the solutions obtained using the CFDM outperform those achieved through the trigonometric b-spline method. This superiority is further evidenced by Figure 6, which presents a 3D plot of the CFDM for Example 2, showcasing its behavior when parameters $\varphi$ and $\tau$ vary within the range $\left[\mathrm{0, 1}\right]$, with $\xi = 0.5$. Figure 7 reinforces the efficacy of the CFDM by illustrating a comparison between exact solutions and CFDM results for Example 2, where $\xi = \tau = 0.5$, demonstrating excellent accordance between the two sets of results and confirming the suitability and efficiency of the numerical scheme. Additionally, Figure 8 depicts the effect of time $\left(\tau \right)$ on the solution $\mathfrak{R}\left(\varphi , \tau \right)$ for Example 2, indicating a direct correlation between $\mathfrak{R}\left(\varphi , \tau \right)$ and increasing $\tau$. Moreover, Figure 9 explores the impact of the fractional order $\left(\xi \right)$ on the solution $\mathfrak{R}\left(\varphi , \tau \right)$ for Example 2, concluding that increasing the fractional order leads to a decrease in the solution magnitude, highlighting the sensitivity of the solution to variations in the fractional order parameter within the CFDM framework. Figure 10 shows the comparison between exact and CFDM using an absolute error plot. Collectively, these findings underscore the robustness, accuracy, and superiority of the CFDM approach in accurately modeling and solving Example 2 over alternative numerical methods, providing valuable insights into the behavior of the system under consideration.

Example 4.3. Consider the gas dynamics equation with time-fractional derivative in the following form:

where

and the exact solution of the problem is $\mathfrak{R}\left(\varphi , \tau \right) = {\tau }^{2+\xi }\mathrm{sin}\pi \varphi$.

In Figure 11, a 3D plot showcases the application of the CFDM to Example 3, where the parameters $\tau$ and $\varphi$ range from $0$ to $1$, while $\xi$ remains constant at $0.5$. This visualization offers a comprehensive view of the CFDM's behavior within the specified parameter space, shedding light on the dynamics of the solution. Figure 12 compares exact solutions and CFDM results for Example 3, with both $\xi$ and $\tau$ set to $0.5$. The close agreement between the two sets of results indicates the suitability and efficiency of the CFDM, confirming its reliability in accurately solving Example 3. In Figure 13, the plot illustrates the impact of time $\left(\tau \right)$ on the solution $\mathfrak{R}\left(\varphi , \tau \right)$ for Example 3, revealing a direct correlation between $\mathfrak{R}\left(\varphi , \tau \right)$ R and increasing $\tau$. This observation offers valuable insights into the temporal behavior of the system under consideration. Furthermore, Figure 14 investigates the effect of the fractional order $\left(\xi \right)$ on the solution $\mathfrak{R}\left(\varphi , \tau \right)$ for Example 3, concluding that an increase in the fractional order leads to a decrease in the solution magnitude. Figure 15 shows the comparison between exact and CFDM using an absolute error plot. This finding underscores the sensitivity of the solution to variations in the fractional order parameter within the CFDM framework. Collectively, these results highlight the robustness and accuracy of the CFDM approach in modeling and solving Example 3, providing valuable insights into the behavior of the system.

5.   Conclusions

In conclusion, this study has introduced and validated the conformable finite difference method (CFDM) as a robust numerical scheme for accurately solving time-fractional gas dynamics equations. By integrating the finite difference method with conformable derivatives, the CFDM offers a unique approach to tackle the challenges posed by these equations, which are crucial in capturing various physical phenomena, such as explosions, combustion, detonation, and condensation in a moving flow. Through rigorous stability analysis, we have demonstrated the conditional stability of the proposed scheme under certain constraints, validating its reliability in practical applications. Comparative analysis against existing methodologies, including trigonometric B-spline functions and the quadratic B-spline Galerkin method, using ${L}_{2}$ and ${L}_{\infty }$ norms, highlighted the accuracy and efficiency of the CFDM. Numerical results obtained using the CFDM were presented and discussed in detail, showcasing its capability to accurately capture the behavior of time-fractional gas dynamics equations. Furthermore, several illustrative examples were solved, providing concrete evidence of the effectiveness of the proposed methodology. Overall, this study underscores the versatility and computational efficiency of the CFDM in addressing complex phenomena encountered in time-fractional differential equations arising in various scientific and engineering fields. In conclusion, the CFDM emerges as a promising numerical technique, offering significant potential for advancing research in physical modeling and numerical simulations. Future directions may involve further refinement of the method and its application to a broader range of problems in different disciplines.

Author contributions

Majeed A. Yousif: Conceptualization; Methodology; Software; Formal analysis; Formal analysis; Investigation; Data curation; Writing-original draft preparation; Juan L. G. Guirao: Methodology; Validation; Investigation; Writing-original draft preparation; Project administration; Pshtiwan Othman Mohammed: Conceptualization; Methodology; Validation; Investigation; Resources; Data curation; Supervision; Funding acquisition; Nejmeddine Chorfi: Conceptualization; Software; Investigation; Writing-original draft preparation; Writing-review and editing; Project administration; Funding acquisition; Dumitru Baleanu: Investigation; Project administration; Funding acquisition; Writing-review and editing; Supervision. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

Researchers Supporting Project number (RSP2024R153), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare that they have no conflicts interests.