Adaptation of conformable residual series algorithm for solving temporal fractional gas dynamics models

Abstract

In this paper, we introduced, discussed, and investigated analytical-approximate solutions for nonlinear time fractional gas dynamics equations in terms of conformable differential operator. The proposed algorithm relies upon the conformable power series method and residual error of the generalized Taylor series in terms of the conformable sense. This technique provides analytical solutions in the form of rapid and accurate convergent series in terms of the multiple fractional power series with easily computable components. In this direction, error estimation and convergence analysis for solutions of fractional gas dynamics equations are provided as well. Eventually, several physical examples are tested to justify the theoretical portion and give a clear explanation of dynamic systems for the proposed model for different orders of fractional case $\beta \in (\mathrm{ }0,1].$ The obtained numeric-analytic results indicate that the current algorithm is simple, effective, and profitably dealing with the complexity of many nonlinear fractional dispersion problems.

Keywords:

• Fractional gas dynamics equation
• conformable derivative
• analytical solution
• fractional residual series method
• multivariable series expansion

1. Introduction and motivation

Temporal fractional partial differential equations play a substantial role in converting several real-life systems into mathematical models. They are basically utilized to provide sufficient information about the historical states of the original systems and to understand the underlying physics that govern these systems, as well as to study the dynamics of predicted solutions. This leads to a better comprehension of real-life systems, interpretations of traveling wave propagation, reduces computational complexity, and facilitates console design without losing any of the genetic solution behaviors (Agarwal, Baleanu, Chen, Momani, & Machado, 2019; Al-Smadi, Abu Arqub, & Gaith, 2021; Hammad, Agarwal, Momani, & Alsharari, 2021; Miller & Ross, 1993; Shi et al., 2021). In recent years, fractional differential equations (FDEs) have received considerable attention because they are popularly used to explain many complex phenomena in various applications such as signal processing, information theory, control theory, finance, the fluid flow, and systems identification. The most significant advantage of using FDEs in these applications and others is their non-local property. Anyhow, many fractional differential operators are well known to be nonlocal, indicating that the forthcoming state of the physical model depends on its historical state, along with current state. On the other hand, several developed definitions are appeared in the recent literature of fractional operators of a local nature as well. Although non-local differential and integral operators have attracted a lot of attention due to the enormous influence of physical applications that depend on nonlocality properties and long-term memory. They are not without flaws in many places such as the Leibniz, chain, and quotient rules, as well as the semi-group features. In this orientation, local derivatives are proposed depending on the natural form of classical derivatives to preserve the local nature of derivatives, avoid violation of unusual features, and investigate the gradient properties of local derivatives (Al-Deiakeh, Abu Arqub, Al-Smadi, & Momani, 2021; Allahviranloo, Salahshour, & Abbasbandy, 2012; Al-Smadi, Abu Arqub, & Gaith, 2021; Al-Smadi & Abu Arqub, 2019; Arqub, 2019; Majeed, Kamran, Abbas, & Bin Misro, 2021). In fact, this is the main reason why fractional differential operators give an outstanding tool for depicting the hereditary and memory properties of various engineering, mathematical and physical problems. Moreover, there are many singular and non‐singular concepts of fractional operators with wide applications in various scientific fields, including Caputo, Hadamard, Machado, Caputo-Fabrizio, Grünwald-Letnikov, Riemann-Liouville, Atangana-Baleanu, and Kiryakova (Abu Arqub & Al-Smadi, 2020a, 2020b; Al-Smadi, Abu Arqub, & Gaith, 2021; Al-Smadi, Djeddi, et al., 2021; Atangana & Baleanu, 2016; Atangana & Nieto, 2015; Caputo & Fabrizio, 2015; Hasan et al., 2021).

The dynamic behavior and fundamental physics of many common physical phenomena can be predicted by numerical simulations of fractional partial differential equations, which generally behave in an unusual manner. To clarify further, consider the underlying conformable fractional gas dynamics system (Das & Kumar, 2011): (1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) with the underlying initial condition (2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) where $x\in [a,b],$ $t\ge 0,$ $T_t^{\beta }$ stands for the time-fractional conformable derivative of order $\beta ,$ $u\left(x,t\right)$ is a probability density function for spatiotemporal wave profile, and $u_0(x)$ is given smooth function of $x.$ Assume that the smooth solution of fractional gas dynamics model (1(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) ) and (2(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) ) is exist and unique in the interval of interest. Particularly, for $\beta =1,$ we get the nonlinear classical gas dynamics equations. The gas dynamics equation is defined as a mathematical expression in the light of the physical laws of conservation, including momentum energy, and mass conservation laws, and so forth. The nonlinear time-fractional gas dynamics model is applicable in many fantastic physical phenomena, such that contact discontinuities, shock fronts, and rarefactions (Kumar & Rashidi, 2014).

In the literature, there is no general classical way of dealing with nonlinear FPDEs and presenting their exact solution due to the complexity of fractional calculus involving such equations. Therefore, efficient, and appropriate numerical techniques are needed to determine the coefficients of the fractional power series solutions for these equations is needed. Several numeric-analytic techniques have been exploited in the recent years to deal with fractional order PDEs and their system. Some of these works, to name a few, have studied and analyzed approximate solutions of both nonlinear homogeneous and inhomogeneous gas dynamic equations by applying the differential transform approach (Biazar & Eslami, 2011). In Das and Kumar (2011), the homotopy analysis transform method has been used to solve the nonlinear homogeneous and inhomogeneous gas dynamic equations of the time-fractional model. Many other exciting applications in the medical, engineering, physical, and natural sciences can be found in Abu Arqub, Odibat, and Al-Smadi (2018), Atangana (2016), Atangana and Gómez-Aguilar (2018), Mousa, Agarwal, Alsharari, and Momani, (2021), Sunarto, Agarwal, Sulaiman, Chew, and Momani, (2021) and references therein.

The motivation of this paper is to design and implement a novel efficient analytical algorithm in view of the conformable residual series method (CRSM) for solving the homogeneous and inhomogeneous nonlinear conformable time-fractional gas dynamics equation. In this direction, the CRSM is an effective analytical technique for determining power series solutions for a wide range of systems of ordinary differential equations, partial differential equations, integro-differential, integral equations, and fractional fuzzy operator (Akram, Abbas, Riaz, Ismail, & Ali, 2020; Al-Smadi, 2021; Bira, Sekhar, & Zeidan, 2018; Khalid, Abbas, Iqbal, & Baleanu, 2020; Abbas, Iqbal, Singh, & Md, 2020). It is an advanced approximation technique that is directly applicable and flexible to create accurate solutions of both linear and nonlinear cases without imposing any linearization, transformation, and constraints on nature of the physical system. Unlike the traditional technique of PS, the CRSM approach does not require a comparison of the corresponding coefficients and no recursion relationship is needed. Besides, the series coefficients are calculated by determining the correlation of residual error with the desired conformable fractional derivatives, which produce a system of algebraic equations of one or more variables. Sometimes CRSM gives a closed form of a series solution, especially if the analytic solutions are polynomial.

The primary impetus for this work is to design a novel iterative computational algorithm to generate analytical solution for temporal fractional gas dynamics models using a recent fractional operator in view of the conformable sense. The conformable fractional derivative is adopted for its simplicity in dealing specifically with nonlinear terms. To this end, the residual error functions are defined to obtain the unknown parameters in the originated conformable power series expansion. The acquired approximate solutions uniformly converge to the analytical solutions. Simultaneously, error estimation and convergence analysis are discussed. Finally, several numerical experiments are provided to illustrate the great flexibility and reliability of the developed algorithm. The remaining parts of this analysis are formulated as follows. Section 2 describes the elementary definitions and mathematical concepts of conformable fractional calculus, and the generalized power series. Building CRS algorithm is introduced in Section 3 to provide an accurate series solution for the nonlinear conformable fractional gas dynamics equation. Next, in Section 4, three numerical applications are tested to show capability and potentiality of the CRS technique. Section 5 is devoted to brief conclusions.

2. Basic mathematical concepts

This section provides some important introductory definitions that will be used mainly for the rest of the analysis. The fractional operator under the conformable sense is proposed based on the classical notion of limits. Using the conformable concept, many applications have emerged in different areas of applied mathematics, engineering, and physics, such as ocean sciences, electrodynamics, fractals, rheology, and atmosphere. For more details about fractional operators, we refer to Al-Smadi, Abu Arqub, and Hadid (2020a, 2020b), Al-Smadi, Abu Arqub, and Momani (2020), Amryeen, Harun, Al-Smadi, and Alias (2020), Arqub (2013) and Momani, Djeddi, Al-Smadi, and Al-Omari (2021). In the following, the basic results and definitions relating to conformable time-fractional derivative and the Taylor expansion formula in conformable sense are revisited.

Definition 2.1

(Khalil, Al Horani, Yousef, & Sababheh, 2014): Let $f$ be $n$-differentiable at $t>s.$ For $\beta >0,$ the fractional derivative of order $\beta$ in conformable sense of $f:[s,\mathbb{\infty })\to \mathbb{R}$ is given by (3) $$\frac{d^{\beta }f}{d{t}^{\beta }}=T^{\beta }f(t)=\underset{\epsilon \to 0}{\lim }\frac{f^{\left(⌈\beta ⌉-1\right)}\left(t+\epsilon {(t-s)}^{⌈\beta ⌉-\beta }\right)-f^{\left(⌈\beta ⌉-1\right)}\left(t\right)}{\epsilon },\mathrm{ }\beta \in n-1,n],\mathrm{ }t>s,$$(3) $$T^{\beta }f\left(s\right)=\underset{t\to s^{+}}{\lim }{T}^{\beta }f\left(t\right),$$ provided $f(t)$ is $\beta$-differintiable in some $(0,s),$ $s>0,$ and $\underset{t\to s^{+}}{\mathit{lim}}{T}^{\beta }f\left(t\right)$ exists, where $⌈\beta ⌉$ is the smallest integer greater than or equal $\beta .$

Theorem 2.1

(Anderson & Ulness, 2015): Let the functions $f(t)$ and $g(t)$ be $\beta$-differentiable of order $\beta \in (0, 1]$ at a point $t>s$. Then,

1. $\frac{d^{\beta }}{d{t}^{\beta }}(kf\mathrm{ }+hg)\mathrm{ }=\mathrm{ }k\mathrm{ }{f}^{\left(\beta \right)}+h\mathrm{ }{g}^{\left(\beta \right)},$ $\forall k,\mathrm{ }h\in \mathbb{R}.$

2. $\frac{d^{\beta }}{d{t}^{\beta }}\left(\lambda \right)=0,$ $\lambda$ is a constant.

3. $\frac{d^{\beta }}{d{t}^{\beta }}\left(\lambda f\right)=\lambda \frac{d^{\beta }}{d{t}^{\beta }}\left(f\right).$

4. $\frac{d^{\beta }}{d{t}^{\beta }}\left({\left(t-a\right)}^r\right)=\mathrm{ }r{\left(t-a\right)}^{r-\beta },$ $\forall r\in \mathbb{R}.$

5. $\frac{d^{\beta }}{d{t}^{\beta }}f(t)={\left(t-a\right)}^{1-\beta }\frac{d}{dt}f(t)$ whenever $f$ is differentiable function.

Corollary 2.1:

For$\beta \in (n-1,n]$, if $f:[0,\infty )\to \mathbb{R}$ is $\beta$-differentiable at $t>s$, then $f$ is continuous at $s.$

Definition 2.2

(Anderson & Ulness, 2015): For $\beta \in (n-1,n],$ the conformable fractional integral of a function $f(t)$ is defined as (4) $$I_s^{\beta }f\left(t\right)=\frac{1}{\left(n-1\right)!}{\int }_s^t\frac{{\left(t-\tau \right)}^{n-1}f\left(\tau \right)}{{\left(\tau -s\right)}^{n-\beta }}d\tau , t>\tau \ge s\ge 0.$$(4)

Theorem 2.2:

Let $\beta \in (n-1,n]$ and assume $f(t)$ be $n$-times differentiable function. Then,

1. $\mathrm{ }\frac{d^{\beta }}{d{t}^{\beta }}\left(I_s^{\beta }f\left(t\right)\right)=f(t).$

2. $\mathrm{ }{I}_s^{\beta }\left(\frac{d^{\beta }}{d{t}^{\beta }}f\left(t\right)\right)=f\left(t\right)-{\sum }_{k=0}^{n-1}\frac{f^{\left(k\right)}\left(s\right){\left(t-s\right)}^k}{k!}.$

Definition 2.3

(Abdeljawad, 2015): Let ${\partial }^{k}u/\partial t^k$ and ${\partial }^{k}u/\partial x^k$ defined on $I\times [s,\infty )$ for $k=1,2,\dots ,n-1.$ Then, the time-fractional conformable partial derivative of order $\beta$ of $u\left(x,t\right):I\times [s,\mathbb{\infty })\to \mathbb{R}$ is given as (5) $$T_t^{\beta }u\left(x,t\right)=\frac{{\partial }^{\beta }u\left(x,t\right)}{\partial t^{\beta }}=\underset{\epsilon \to 0}{\lim }\frac{u_t^{(n-1)}\left(x,t+\epsilon {\left(t-s\right)}^{n-\beta }\right)-u_t^{\left(n-1\right)}(x,t)}{\epsilon }, \beta \in n-1,n], t>s\ge 0.$$(5)

Definition 2.4

(Abdeljawad, 2015): For $\beta \in (n-1,n],$ the conformable fractional integral starting from $s$ of $u\left(x,t\right):I\times [s,\mathbb{\infty })\to \mathbb{R}$ is given as (6) $$I_s^{\beta }u\left(x,t\right)=\{\begin{array}{cc}\frac{1}{\left(n-1\right)!}{\int }_s^t\frac{{\left(t-\tau \right)}^{n-1}u\left(x,\tau \right)}{{\left(\tau -s\right)}^{n-\alpha }}d\tau ,& x\in I, t>\tau \ge s\ge 0, \\ u\left(x,t\right),\hfill & \beta =0.\end{array}$$(6)

Definition 2.5

(Al-Smadi, 2021): For $0\le n-1<\beta \le n,$ a power series (PS) of the form (7) $$\sum _{k=0}^{\infty }{h}_k\left(x\right){\left(t-t_0\right) }^{k\beta }=h_0\left(x\right)+h_1\left(x\right){\left(t-t_0\right)}^{\beta }+h_2\left(x\right){\left(t-t_0\right)}^{2\beta }+\dots , x\in I, t_0\le t<t_0+R^{1/\beta },R>0$$(7) is called a MFPS at $t=t_0,$ where$t$ is the independent temporal variable and $h_k(x)$ is unknown parameter of series expansion (7(7) $$\sum _{k=0}^{\infty }{h}_k\left(x\right){\left(t-t_0\right) }^{k\beta }=h_0\left(x\right)+h_1\left(x\right){\left(t-t_0\right)}^{\beta }+h_2\left(x\right){\left(t-t_0\right)}^{2\beta }+\dots , x\in I, t_0\le t<t_0+R^{1/\beta },R>0$$(7) ).

Theorem 2.3

(Al-Smadi, 2021): Let $u\left(x,t\right)$ be a function that has infinitely conformable time-fractional partial derivatives at any point $t$ on $[t_0,t_0+R^{1/\beta })$ such that $u\left(x,t\right)$ has the following MFPS about $t_0>0$: (8) $$u\left(x,t\right)=\sum _{k=0}^{\infty }{h}_k\left(x\right){\left(t-t_0\right) }^{k\beta }, x\in I, t_0\le t<t_0+R^{1/\beta },R>0.$$(8)

Then, the unknown functions $h_k(x)$ are given by $h_k(x)=\frac{T_t^{k\beta }u\left(x,t_0\right)}{{\beta }^k(k)!},$ where $T_t^{k\beta }$ refers to fractional derivative of order $k$ in the meaning of conformable concept such that $T^{k\beta }=\underset{k-\mathit{times}}{\underbrace{T^{\beta }·T^{\beta }···T^{\beta }}}.$

3. The CRSM for time-fractional gas dynamics equation

In this section, the procedure used to obtain an analytical solution of a temporal fractional gas dynamics model is described in terms of conformable derivatives fitted with certain initial conditions in a finite time-space domain (Abdeljawad, 2015; Abu Arqub et al., 2022; Acan & Baleanu, 2018; Anderson & Ulness, 2015; Khalil et al., 2014; Senol, Tasbozan, & Kurt, 2019; Thabet & Kendre, 2018). The solution methodology is based on compensating the multiple fractional PS expansion in the truncated residual functions and minimizing the error to zero. This aids to derive a recursion formula for computing the unknown parameters of the time-fractional series by employing the conformable differentiation on the residual error function many times. Towards this end, we suppose firstly the solution by CRPS technique for conformable time-fractional gas dynamic equation (1(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) )-(2(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) ) about the initial data $t_0=0$ has the following multiple fractional PS: (9) $$u\left(x,t\right)=\sum _{m=0}^{\infty }{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!},0<\beta \le 1,t\ge 0 , x\in \mathbb{R}.$$(9)

Using the initial condition (2)(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) so that $u\left(x,0\right)=u_0\left(x\right)=h_0(x),$ the series solution will be in the following form: (10) $$u\left(x,t\right)=u_0\left(x\right)+\sum _{m=1}^{\infty }{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(10)

For obtaining the kth-CRS approximate solutions, let us consider $u_k\left(x,t\right)$ can be given by (11) $$u_k\left(x,t\right)=u_0\left(x\right)+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(11)

Thus, to determine the unknown coefficients $h_m(x)$ for $m=1,2,\dots ,k,$ we define the following $\mathit{kth}$-residual function: (12) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right).$$(12) and the following residual function: $${\mathit{Res}}_u\left(x,t\right)=\underset{k\to \infty }{\mathit{lim}}{\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right).$$

It is convenient to observe that ${\mathit{Res}}_u\left(x,t\right)=0$ for every $x\in \mathbb{R}$ and $0<t<R^{1/\beta },$ in which $R^{1/\beta }$ is the convergence radius of the CRS (10). Meanwhile, $T_t^{k\beta }{\mathit{Res}}_u\left(x,t\right)=0,$ and $T_t^{(m-1)\beta }{\mathit{Res}}_u{\left(x,t\right)}_{│t=0}=T_t^{(k-1)\beta }{\mathit{Res}}_u^k{\left(x,t\right)}_{│t=0}=0$ for each $k=1,2,\dots ,$ due to the fact that fractional derivative of a constant in the conformable sense is zero.

Consequently, the following fractional equation that can be traditionally solved: (13) $$T_t^{(k-1)\beta }{\mathit{Res}}_u^k{\left(x,t\right)}_{│t=0}=0, k=1,2,3,\dots .$$(13)

By determining the value of the coefficients$\mathrm{ }{h}_m\left(x\right),m=1,2,\dots ,k,$ of Eq. (11)(11) $$u_k\left(x,t\right)=u_0\left(x\right)+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(11) , the solution of fractional models (1)(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) -(2)(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) can be completely constructed.

However, to determine the value of first unknown coefficient, $h_1(x),$ substitute $k=1$ in both sides of the 1st-residual equation of Eq. (12)(12) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right).$$(12) to get the following result: (14) $${\mathit{Res}}_u^1\left(x,t\right)=T_t^{\beta }{u}_1\left(x,t\right)+u_1\left(x,t\right){\left(u_1\left(x,t\right)\right)}_x-u_1\left(x,t\right)\left(1-u_1\left(x,t\right)\right)=h_1\left(x\right)+\left(\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left({\delta }^{\prime }\left(x\right)+h_1^{\prime }\left(x\right)\frac{t^{\beta }}{\beta }\right)-\left(\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left(1-\delta \left(x\right)-h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)$$(14)

.

Now, depending on the fractional formula (13)(13) $$T_t^{(k-1)\beta }{\mathit{Res}}_u^k{\left(x,t\right)}_{│t=0}=0, k=1,2,3,\dots .$$(13) , we get $h_1\left(x\right)=\delta \left(x\right)-{\delta }^2\left(x\right)-\begin{array}{c}\delta \left(x\right){\delta }^{\prime }\left(x\right)\end{array}.$

Likewise, the value of second unknown coefficient, $h_2(x),$ of Eq. (11)(11) $$u_k\left(x,t\right)=u_0\left(x\right)+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(11) can be obtained by substituting the $2\mathrm{n}\mathrm{d}$-truncated series, $u_2\left(x,t\right)=\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }+h_2\left(x\right)\frac{t^{2\beta }}{2{\beta }^2},$ into the $2\mathrm{n}\mathrm{d}$-residual function of Eq. (12)(12) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right).$$(12) as follows, (15) $${\mathit{Res}}_u^2\left(x,t\right)=T_t^{\beta }{u}_2\left(x,t\right)+u_2\left(x,t\right){\left(u_2\left(x,t\right)\right)}_x-u_2\left(x,t\right)\left(1-u_2\left(x,t\right)\right)=h_1\left(x\right)+h_2\left(x\right)\frac{t^{\beta }}{\beta }+\left(\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }+h_2\left(x\right)\frac{t^{2\beta }}{2{\beta }^2}\right)\left(\delta \prime \left(x\right)+h_1^{\prime }\left(x\right)\frac{t^{\beta }}{\beta }+h_2^{\prime }\left(x\right)\frac{t^{2\beta }}{2{\beta }^2}\right)-\left(\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }+h_2\left(x\right)\frac{t^{2\beta }}{2{\beta }^2}\right)\left(1-\delta \left(x\right)-h_1\left(x\right)\frac{t^{\beta }}{\beta }-h_2\left(x\right)\frac{t^{2\beta }}{2{\beta }^2}\right).$$(15)

Consequently, by computing the $\beta$ th-conformable derivative of ${\mathit{Res}}_u^2\left(x,t\right),$ it follows (16) $$T_t^{\beta }{\mathit{Res}}_u^2\left(x,t\right)=h_2\left(x\right)+h_1\left(x\right){\delta }^{\prime }\left(x\right)+\delta \left(x\right)h_1^{\prime }\left(x\right)+\frac{h_2\left(x\right){\delta }^{\prime }\left(x\right)t^{\beta }}{\beta }+\frac{2t^{\beta }{h}_1\left(x\right)h_1^{\prime }\left(x\right)}{\beta }+\frac{3t^{2\beta }{h}_2\left(x\right)h_1^{\prime }\left(x\right)}{2\beta }+\frac{t^{\beta }\delta \left(x\right)h_2^{\prime }\left(x\right)}{\beta }+\frac{3t^{2\beta }{h}_1\left(x\right)h_2^{\prime }\left(x\right)}{2{\beta }^2}+\frac{t^{3\beta }{h}_2\left(x\right)h_2^{\prime }\left(x\right)}{{\beta }^3}+h_1\left(x\right)-2\delta \left(x\right)h_1\left(x\right)+\frac{2t^{\beta }{h}_1^2\left(x\right)}{\beta }+\frac{t^{\beta }{h}_2\left(x\right)}{\beta }-\frac{2t^{\beta }\delta \left(x\right)h_2\left(x\right)}{\beta }+\frac{3t^{2\beta }{h}_1\left(x\right)h_2\left(x\right)}{{\beta }^2}+\frac{t^{3\beta }{h}_2^2\left(x\right)}{{\beta }^3}.$$(16)

Thus, by solving $T_t^{\beta }{{\mathit{Res}}_u^2\left(x,t\right)}_{│t=0}=0,$ the 2nd unknown coefficient can be directly obtained as follows $h_2\left(x\right)=2\delta \left(x\right)h_1\left(x\right)-h_1\left(x\right)-h_1\left(x\right){\delta }^{\mathrm{\prime }}\left(x\right)-\delta (x)h_1^{\mathrm{\prime }}\left(x\right).$

To obtain the $3\mathrm{r}\mathrm{d}$-CRS approximate solution, we find out the value of $h_3\left(x\right)$ by rewriting $u_3\left(x,t\right)=\delta \left(x\right)+h_1\left(x\right)\frac{t^{\beta }}{\beta }+h_2\left(x\right)\frac{t^{2\beta }}{2{\beta }^2}+h_3\left(x\right)\frac{t^{3\beta }}{3{!\beta }^3}$ through ${\mathit{Res}}_u^3\left(x,t\right),$ and then by applying the fractional operator $T_t^{2\beta }$ of the resulting equation, we get (17) $$T_t^{2\beta }{\mathit{Res}}_u^3\left(x,t\right)=T_t^{2\beta }(T_t^{\beta }{u}_3\left(x,t\right)+u_3\left(x,t\right){\left(u_2\left(x,t\right)\right)}_x-u_3\left(x,t\right)\left(1-u_3\left(x,t\right)\right))=\frac{h_3\left(x\right)}{\beta }+\frac{h_2\left(x\right){\delta }^{\mathrm{\prime }}\left(x\right)}{\beta }+\frac{t^{\beta }{h}_3\left(x\right){\delta }^{\mathrm{\prime }}\left(x\right)}{2{\beta }^2}+\frac{2h_1\left(x\right)h_1^{\mathrm{\prime }}\left(x\right)}{\beta }+\frac{3t^{\beta }{h}_2\left(x\right)h_1^{\mathrm{\prime }}\left(x\right)}{2{\beta }^2}+\frac{2t^{2\beta }{h}_3\left(x\right)h_1^{\mathrm{\prime }}\left(x\right)}{3{\beta }^3}+\frac{\delta \left(x\right)h_2^{\mathrm{\prime }}\left(x\right)}{\beta }+\frac{3t^{\beta }{h}_1\left(x\right)h_2^{\mathrm{\prime }}\left(x\right)}{2{\beta }^2}+\frac{t^{2\beta }{h}_2\left(x\right)h_2^{\mathrm{\prime }}\left(x\right)}{{\beta }^3}+\frac{5t^{3\beta }{h}_3\left(x\right)h_2^{\mathrm{\prime }}\left(x\right)}{12{\beta }^4}+\frac{t^{\beta }\delta \left(x\right)h_3^{\mathrm{\prime }}\left(x\right)}{2{\beta }^2}+\frac{2t^{2\beta }{h}_1\left(x\right)h_3^{\mathrm{\prime }}\left(x\right)}{3{\beta }^3}+\frac{5t^{3\beta }{h}_2\left(x\right)h_3^{\mathrm{\prime }}\left(x\right)}{12{\beta }^4}+\frac{t^{4\beta }{h}_3\left(x\right)h_3^{\mathrm{\prime }}\left(x\right)}{6{\beta }^5}-\frac{2h_1^2\left(x\right)}{\beta }-\frac{h_2\left(x\right)}{\beta }-\frac{3t^{\beta }{h}_1\left(x\right)h_2\left(x\right)}{{\beta }^2}-\frac{t^{2\beta }{h}_2^2\left(x\right)}{{\beta }^3}-\frac{t^{\beta }{h}_3\left(x\right)}{2{\beta }^2}+\frac{t^{\beta }\delta \left(x\right)h_3\left(x\right)}{{\beta }^2}+\frac{t^{4\beta }{h}_3^2\left(x\right)}{6{\beta }^5}.$$(17)

Using the fact that $T_t^{2\beta }{{\mathit{Res}}_u^3\left(x,t\right)}_{│t=0}=0,$ we can easily obtain $h_3\left(x\right)=h_2\left(x\right)+2h_1^2\left(x\right)-h_2\left(x\right){\delta }^{\mathrm{\prime }}\left(x\right)-2h_1\left(x\right)h_1^{\mathrm{\prime }}\left(x\right)-\delta \left(x\right)h_2^{\mathrm{\prime }}\left(x\right).$

In the same manner, by applying the same fashion for $k=4,$ and based on the fractional operator formula $T_t^{3\beta }{{\mathit{Res}}_u^4\left(x,t\right)}_{│t=0}=0,$ it yields $h_4\left(x\right)=\frac{1}{2}\left(h_3(x){\delta }^{\mathrm{\prime }}(x)+3h_2\left(x\right)h_1^{\mathrm{\prime }}\left(x\right)+3h_1(x)h_2^{\mathrm{\prime }}\left(x\right)+3\delta (x)h_3^{\mathrm{\prime }}\left(x\right)-h_3(x)\right)+3h_1(x)h_2(x)+\delta (x)h_3(x).$

However, depending on the forms of the achieved coefficients $h_m\left(x\right)$ for $m=0,1,2,3,4,$ the $4\mathrm{t}\mathrm{h}$-CRS approximate solution of fractional model (1)(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) -(2)(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) can be rewritten as, $$u_4(x,t)=\delta (x)+(\delta (x)-{\delta }^2(x)-\begin{array}{c}\delta (x){\delta }^{\prime }(x)\end{array})\frac{t^{\beta }}{\beta }+(2\delta (x)h_1(x)-h_1(x)-h_1(x){\delta }^{\prime }(x)-\delta \left(x\right)h_1^{\prime }(x))\frac{t^{2\beta }}{2{\beta }^2}+(h_2(x)+2h_1^2(x)-h_2(x){\delta }^{\prime }(x)-2h_1(x)h_1^{\prime }(x)-\delta (x)h_2^{\prime }(x))\frac{t^{3\beta }}{3{!\beta }^3}+(\frac{1}{2}(h_3\left(x\right){\delta }^{\prime }\left(x\right)+3h_2(x)h_1^{\prime }(x)+3h_1\left(x\right)h_2^{\prime }(x)+3\delta \left(x\right)h_3^{\prime }(x)-h_3\left(x\right))+3h_1\left(x\right)h_2\left(x\right)+\delta \left(x\right)h_3\left(x\right))\frac{t^{4\beta }}{4{!\beta }^4}.$$

Using similar argument and continuing in such approach up to arbitrary order $m,$ the CRS solution $u_m\left(x,t\right)$ of fractional gas dynamics model (1)(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) -(2)(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) will be given. Furthermore, highly accurate will be achieved by calculation more terms of the CRS solution.

For the sake of completeness and simplicity, the following algorithm summarizes the main steps for obtaining the CRSM solution to fractional gas dynamics model (1)(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) -(2)(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) :

Algorithm 3.1: To detect the unknown components $h_m\left(x\right),$ $m=1,2,3,\dots ,$ of the multivariable conformable expansion (10(10) $$u\left(x,t\right)=u_0\left(x\right)+\sum _{m=1}^{\infty }{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(10) ), perform the following basic steps:

• Step1: Consider the initial data $u\left(x,0\right)=u_0(x)$ as the initial CRS solution $u_0\left(x,t\right)=u_0(x)$ of fractional gas dynamics model (1(1) $$\begin{array}{c}{T}_t^{\beta }u(x,t)+u(x,t)u_x(x,t)-u(x,t)(1-u(x,t))=0\end{array},\mathrm{ }0<\beta \le 1,$$(1) )-(2(2) $$u\left(x,0\right)=u_0\left(x\right),$$(2) ).

• Step2: Substitute $u_k\left(x,t\right)=u_0\left(x\right)+{\sum }_{m=1}^k{h}_m(x)\frac{t^{m\beta }}{{\beta }^{m}m!}$ into the truncated residual function ${\mathit{Res}}_u^k\left(x,t\right)$ of formula (12).

• Step3: For $i=1,2,3,\dots ,n;$ Letting $k=i,$ and then compute $T_t^{\left(i-1\right)\beta }{\mathit{Res}}_u^i(x,t).$

• Step4: Simplify the resulting equation, then calculate $T_t^{\left(i-1\right)\beta }{\mathit{Res}}_u^i\left(x,0\right)=0.$

• Step5: Put the achieved coefficient $h_i(x)$ back into Step 2 to obtain $i$ th-term approximation.

• Step6: For $i=i+1,$ do Step 3 till to arbitrary number $n.$ Then, Stop.

Theorem 3.1:

For $k\in \mathbb{N},$ let $u_k(x,t)$ and $u(x,t)$ be recpectivly the approximate and exact solutions of the conformable fractional gas dynamics model (1)-(2). If there exists a positive constant $\gamma \in \left[0,1\right]$ so that $\Vert u_{k+1}(x,t)\Vert \le \gamma \Vert u_k\left(x,\tau \right)\Vert$ for all $\left(x,t\right)\in \left[a,b\right]\times [0,T]$ along with $\Vert u_0\left(x\right)\Vert <\infty$ for all $x\in \left[a,b\right]$. Then, $u_k(x,t)$ converges to $u(x,\tau )$ as soon as $k\to \infty .$

Proof.

For all $\left(x,t\right)\in \left[a,b\right]\times \left[0,T\right],$ it yields that $\Vert u_{k+1}(x,t)\Vert \le \gamma \Vert u_k\left(x,t\right)\Vert ,$ which gives $\Vert u_1(x,t)\Vert \le \gamma \Vert u_0\left(x,t\right)\Vert =\gamma \Vert u_0\left(x\right)\Vert .$ Thus, $\Vert u_2(x,t)\Vert \le {\gamma }^2\Vert u_0\left(x\right)\Vert ,$ and so $\Vert u_k(x,t)\Vert \le {\gamma }^k\Vert u_0\left(x\right)\Vert .$ Therefore, we get ${\sum }_{m=k+1}^{\mathrm{\infty }}\Vert u_m(x,t)\Vert \le {\sum }_{m=k+1}^{\mathrm{\infty }}{\gamma }^m\Vert u_0\left(x\right)\Vert =\Vert u_0\left(x\right)\Vert {\sum }_{m=k+1}^{\mathrm{\infty }}{\gamma }^k.$

Hence, we conclude that $$\Vert u\left(x,t\right)-u_k(x,t)\Vert =\Vert \sum _{m=k+1}^{\mathrm{\infty }}{u}_m(x,t)\Vert \le \sum _{m=k+1}^{\mathrm{\infty }}\Vert u_m\left(x,\tau \right)\Vert \mathrm{ }\le \Vert u_0\left(\mathcal{x}\right)\Vert \sum _{m=k+1}^{\mathrm{\infty }}{\gamma }^m=\frac{{\rho }_{\mathrm{ }}^{k+1}}{1-\rho }\Vert u_0\left(\mathcal{x}\right)\Vert \to 0,$$ as soon as $k\to \infty .$

4. Numerical experiments and discussion

In this section, three applications are numerically tested to illustrate the efficiency, and high performance of the present procedure. All symbolic and results are performed using programming packages of Mathematica 11.

Example 4.1.

The following nonlinear, homogeneous, conformable time-fractional gas dynamic equation (Acan & Baleanu, 2018; Das & Kumar, 2011) is considered (18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) along with the initial condition (19) $$u\left(x,0\right)=e^{-x},$$(19) in which the exact solution of the fractional model (18(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) )-(19(19) $$u\left(x,0\right)=e^{-x},$$(19) ) at $\beta =1$ can be given as $u\left(x,t\right)=e^{t-x}.$

Based on the aforementioned description of CRS algorithm, letting $h_0\left(x\right)=e^{-x},$ then the truncated residual function of the fractional gas dynamic model (18)(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) -(19)(19) $$u\left(x,0\right)=e^{-x},$$(19) can be given as (20) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right),$$(20) provided that $u_k\left(x,t\right)=e^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$

Consequently, by setting $k=1$ in Eq. (20)(20) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right),$$(20) , the first truncated residual function, ${\mathit{Res}}_u^1(x,t),$ will be ${\mathit{Res}}_u^1(x,t)=T_t^{\beta }(e^{-x}+h_1(x)\frac{t^{\beta }}{\beta })+(e^{-x}+h_1(x)\frac{t^{\beta }}{\beta })(-e^{-x}+h_1^{\mathrm{\prime }}(x)\frac{t^{\beta }}{\beta })-(e^{-x}+h_1(x)\frac{t^{\beta }}{\beta })(1-(e^{-x}+h_1(x)\frac{t^{\beta }}{\beta })).$ Thus, by utilizing the principal rule ${\mathit{Res}}_u^1(x,0)=0,$ we have ${\mathit{Res}}_{\mathrm{u}}^1(x,0)=h_1(x)+e^{-2x}-(e^{-x})(1-e^{-x})=0.$ That is, the first unknown coefficient can be given as $h_1(x)=e^{-x}.$ In this direction, the 1st-CRS solution is $u_1(x,t)=e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }.$

Now, for finding the 2nd-CRS solution, set $k=2$ in Eq. (20)(20) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right),$$(20) and then apply the conformable operator $T_t^{\beta }$ to both sides of the resulting formula as follows $$T_t^{\beta }{\mathit{Res}}_u^2(x,t)=T_t^{2\beta }(e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2{\beta }^2})+T_t^{\beta }(e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2{\beta }^2})(-e^{-x}-e^{-x}\frac{t^{\beta }}{\beta }+h_2^{\mathrm{\prime }}(x)\frac{t^{2\beta }}{2{\beta }^2})-T_t^{\beta }(e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2{\beta }^2})(1-(e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2{\beta }^2}))=h_2(x)+(-2e^{-2x}-(2e^{-2x}-e^{-x}{h}_2^{\mathrm{\prime }}(x)+e^{-x}{h}_2(x))\frac{t^{\beta }}{\beta }+e^{-x}(h_2^{\mathrm{\prime }}(x)-h_2(x))\frac{3t^{2\beta }}{2{\beta }^2}+h_2(x)h_2^{\mathrm{\prime }}(x)\frac{2t^{3\beta }}{{\beta }^3})-T_t^{\beta }(e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2{\beta }^2})((1-e^{-x})-e^{-x}\frac{t^{\beta }}{\beta }-h_2(x)\frac{t^{2\beta }}{2{\beta }^2}).$$

Anyhow, the principal rule $T_t^{\beta }{\mathit{Res}}_u^2\left(x,0\right)=0$ leasds to $h_2\left(x\right)-2{\mathrm{e}}^{-2x}-\left(e^{-x}-2e^{-2x}\right)=0,$ that is, $h_2\left(x\right)=e^{-x}.$ Therefore, the 2nd-CRS solution is given as $u_2\left(x,t\right)=e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+e^{-x}\frac{t^{2\beta }}{2{\beta }^2}.$

Following the same manner in the last dicussion, the unknown coeffections $h_m\left(x\right)$ for $m=1,2,3,4$ and $5$ will be obtained. Now, to obtain the 5th-CRS approximate solution of the fractional model (18)(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) and (19)(19) $$u\left(x,0\right)=e^{-x},$$(19) , we set $k=5$ in Eq. (20)(20) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }{u}_k\left(x,t\right)+u_k\left(x,t\right){\left(u_k\left(x,t\right)\right)}_x-u_k\left(x,t\right)\left(1-u_k\left(x,t\right)\right),$$(20) and then substitute $u_5\left(x,t\right)=e^{-x}+{\sum }_{m=1}^5{h}_m(x)\frac{t^{m\beta }}{{\beta }^{m}m!}$ into the fifth truncated residual equation as follows, (21) $${\mathit{Res}}_u^5(x,t)=T_t^{\beta }(e^{-x}+\sum _{k=1}^5{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!})+(e^{-x}+\sum _{k=1}^5{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!})(-e^{-x}+\sum _{k=1}^5{h}_m^{\prime }\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!})-(e^{-x}+\sum _{k=1}^5{h}_m(x)\frac{t^{m\beta }}{{\beta }^{m}m!})(1-e^{-x}-\sum _{k=1}^5{h}_m(x)\frac{t^{m\beta }}{{\beta }^{m}m!}).$$(21)

Hence, the $5\mathrm{t}\mathrm{h}$-CRS approximate solution of the nonlinear conformable time-fractional IVP’s (18)(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) and (19)(19) $$u\left(x,0\right)=e^{-x},$$(19) can be expressed as (22) $$u_5\left(x,t\right)=e^{-x}+e^{-x}\frac{t^{\beta }}{\beta }+e^{-x}\frac{t^{2\beta }}{2{\beta }^2}+e^{-x}\frac{t^{3\beta }}{3{!\beta }^3}+e^{-x}\frac{t^{4\beta }}{4{!\beta }^4}+e^{-x}\frac{t^{5\beta }}{5{!\beta }^5}.$$(22)

Anyhow, the CRS approximate solution $u_n\left(x,t\right)$ of the fractional model (18)(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) and (19)(19) $$u\left(x,0\right)=e^{-x},$$(19) can be written as follows (23) $$u_n\left(x,t\right)=\left(e^{-x}+\frac{e^{-x}{t}^{\beta }}{\beta }+\frac{e^{-x}{t}^{2\beta }}{2!{\beta }^2}+\frac{e^{-x}{t}^{3\beta }}{3{!\beta }^3}+\dots +\frac{e^{-x}{t}^{n\beta }}{n{!\beta }^n}\right)=\sum _{m=0}^n{e}^{-x}\frac{t^{m\beta }}{m!{\beta }^m}.$$(23)

From Eq. (23(23) $$u_n\left(x,t\right)=\left(e^{-x}+\frac{e^{-x}{t}^{\beta }}{\beta }+\frac{e^{-x}{t}^{2\beta }}{2!{\beta }^2}+\frac{e^{-x}{t}^{3\beta }}{3{!\beta }^3}+\dots +\frac{e^{-x}{t}^{n\beta }}{n{!\beta }^n}\right)=\sum _{m=0}^n{e}^{-x}\frac{t^{m\beta }}{m!{\beta }^m}.$$(23) ), we obtain $u\left(x,t\right)=\underset{n\to \mathrm{\infty }}{\lim }{u}_n\left(x,t\right).$

In case $\beta =1,$ the fractional series (23)(23) $$u_n\left(x,t\right)=\left(e^{-x}+\frac{e^{-x}{t}^{\beta }}{\beta }+\frac{e^{-x}{t}^{2\beta }}{2!{\beta }^2}+\frac{e^{-x}{t}^{3\beta }}{3{!\beta }^3}+\dots +\frac{e^{-x}{t}^{n\beta }}{n{!\beta }^n}\right)=\sum _{m=0}^n{e}^{-x}\frac{t^{m\beta }}{m!{\beta }^m}.$$(23) reduces to $u\left(x,t\right)=e^{t-x},$ which represents the exact solution of the classical gas dynamic model, and is in complete agreement with the results in Acan and Baleanu (2018), Das and Kumar (2011) and Thabet and Kendre (2018).

Based on the achieved results and without losing generalization, the accuracy and efficiency of the proposed algorithm are illustrated. The absolute and relative errors of the tenth-CRS approximate solution are summarized in Table 1 over [0,1] for a fixed value $x=3,$ different values of $t,$ and classical order of $\beta =1.$ One can note from Table 1 that the results indicate a good harmony between the exact and CRS solutions. Further, the behavior of the three-dimensional plots of the sixth-CRS approximate solution of the temporal fractional gas dynamic model (18)(18) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=0,\mathrm{ }t\ge 0,\mathrm{ }x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(18) and (19)(19) $$u\left(x,0\right)=e^{-x},$$(19) is shown in Figure 1 at different values of fractional order $\beta \in \{1,0.75,0.5\}$ that compared with the exact solution on the interval $[0,1] \times [-\pi ,\pi ].$ From these graphics, it can be concluded that the dynamic behaviors roughly match, and are in good agreement with each other, especially when considering the derivative of the integer-order. Meanwhile, it can be noted that the solutions of the studied fractional gas model propagate along the $x$-axis over time while maintaining their shape and amplitude in all orders of fractional cases.

Figure 1. Surface plots of 6th-CRS solutions, $u_6\left(x,t\right),$ of Example 4.1 at different values of beta on $\left[0,1\right]\times [-\pi ,\pi ]:$ (a) Exact solution; (b) $\beta =1;$ (c) $\beta =0.75;$ (d) $\beta =0.5.$

Table 1. The CRS soluation for Example 4.1 at $x=3,$ $k=10,$ and $\beta =1.$

$t_i$	Exact soluion	CRS solution	Absolute error	Relative error
$0.0$	$0.0497871$	$0.0497871$	$0.0$	$0.0$
$0.2$	$0.0608101$	$0.0608101$	$2.77556\times {10}^{-17}$	$4.56431\times {10}^{-16}$
$0.4$	$0.0742736$	$0.0742736$	$5.41095\times {10}^{-14}$	$7.28516\times {10}^{-13}$
$0.6$	$0.0907180$	$0.0907180$	$4.76222\times {10}^{-12}$	$5.24948\times {10}^{-11}$
$0.8$	$0.1108030$	$0.1108030$	$1.14748\times {10}^{-10}$	$1.03561\times {10}^{-9}$
$1.0$	$0.1353350$	$0.1353350$	$1.35982\times {10}^{-9}$	$1.00478\times {10}^{-8}$

Example 4.2.

The following nonlinear, homogeneous, conformable time-fractional gas dynamic equation is considered (24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) with the initial condition (25) $$u\left(x,0\right)=b^{-x},$$(25) where $b>0,$ $b\ne 1,$ $t\ge 0,$ $x\in \mathbb{R},$ and $0<\beta \le 1.$ The exact solution of the fractional model (24) and (25) is $u\left(x,t\right)=b^{t-x}.$

To apply the CRPS approach, let $h_0\left(x\right)=b^{-x}$ in the truncated CRS solution of the time-fractional gas model (24)-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) so that (26) $$u_k\left(x,t\right)=b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}.$$(26)

Herein, the functions $h_m(x)$ can be found through defining the truncated residual function of the fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) as follows, (27) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)+\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(-b^{-x}\hspace{0.17em}\text{log}\hspace{0.17em}\left(b\right)+\sum _{m=1}^k{h}_m^{\prime }\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)-\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(1-b^{-x}-\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\mathrm{l}\mathrm{og}\left(b\right).$$(27)

Consequently, put $k=1$ in the$\mathit{kth}$-residual equation (27)(27) $${\mathit{Res}}_u^k\left(x,t\right)=T_t^{\beta }\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)+\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(-b^{-x}\hspace{0.17em}\text{log}\hspace{0.17em}\left(b\right)+\sum _{m=1}^k{h}_m^{\prime }\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)-\left(b^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(1-b^{-x}-\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\mathrm{l}\mathrm{og}\left(b\right).$$(27) to get (28) $${\mathit{Res}}_u^1\left(x,t\right)=T_t^{\beta }\left(b^{-x}+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)+\left(b^{-x}+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left(-b^{-x}\hspace{0.17em}\text{log}\hspace{0.17em}\left(b\right)+h_1^{\prime }\left(x\right)\frac{t^{\beta }}{\beta }\right)-\left(b^{-x}+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left(1-b^{-x}-h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\mathrm{l}\mathrm{og}\left(b\right)=h_1\left(x\right)+\left(b^{-x}+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left(-b^{-x}\mathit{log}\left(b\right)+h_1^{\prime }\left(x\right)\frac{t^{\beta }}{\beta }\right)-\left(b^{-x}+h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\left(1-b^{-x}-h_1\left(x\right)\frac{t^{\beta }}{\beta }\right)\mathrm{l}\mathrm{og}\left(b\right).$$(28)

Based uopn on the fact $T_t^{(k-1)\beta }{\mathit{Res}}_u^k{\left(x,t\right)}_{│t=0}=0$ for $k=1,$ the first term $h_1(x)$ can be obtained so that $h_1\left(x\right)=b^{-x}\mathrm{l}\mathrm{og}(b).$ Therefore, the first-CRS approximate solution of the time-fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) will be $u_1\left(x,t\right)=b^{-x}+b^{-x}\mathrm{l}\mathrm{og}(b)\frac{t^{\beta }}{\beta }.$

Using the same fashion for $k=2,$ and applying the $\beta$ th-conformable derivative on both sides of the resulting equation, it yields (29) $$T_t^{\beta }{\mathit{Res}}_u^2(x,t)=h_2(x)+T_t^{\beta }\{(b^{-x}+h_1(x)\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})(-b^{-x}\hspace{0.17em}\text{log}\hspace{0.17em}\left(b\right)+h_1^{\prime }(x)\frac{t^{\beta }}{\beta }+h_2^{\prime }(x)\frac{t^{2\beta }}{2!{\beta }^2})\}-T_t^{\beta }\{(b^{-x}+h_1(x)\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})(1-b^{-x}-h_1(x)\frac{t^{\beta }}{\beta }-h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})\hspace{0.17em}\text{log}\hspace{0.17em}(b)\}.$$(29)

Then, by solving Eq. (29)(29) $$T_t^{\beta }{\mathit{Res}}_u^2(x,t)=h_2(x)+T_t^{\beta }\{(b^{-x}+h_1(x)\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})(-b^{-x}\hspace{0.17em}\text{log}\hspace{0.17em}\left(b\right)+h_1^{\prime }(x)\frac{t^{\beta }}{\beta }+h_2^{\prime }(x)\frac{t^{2\beta }}{2!{\beta }^2})\}-T_t^{\beta }\{(b^{-x}+h_1(x)\frac{t^{\beta }}{\beta }+h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})(1-b^{-x}-h_1(x)\frac{t^{\beta }}{\beta }-h_2(x)\frac{t^{2\beta }}{2!{\beta }^2})\hspace{0.17em}\text{log}\hspace{0.17em}(b)\}.$$(29) for the unknown function $h_2\left(x\right)$ whenever $t=0,$ the value of $h_2\left(x\right)$ will be given such that $h_2\left(x\right)=b^{-x}{\left(\log (b)\right)}^2.$ Thus, the 2nd-CRS approximate solution of the fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) is given by (30) $$u_2\left(x,t\right)=b^{-x}+b^{-x}\mathrm{l}\mathrm{og}\left(b\right)\frac{t^{\beta }}{\beta }+b^{-x}{\left(\log \left(b\right)\right)}^2\frac{t^{2\beta }}{2!{\beta }^2}.$$(30)

Continue in the same manner for $k=3,4,$ and 5, one can obtain the fifth-CRS approximate solution of the fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) as follows, (31) $$u_5\left(x,t\right)=b^{-x}+b^{-x}\mathrm{l}\mathrm{og}\left(b\right)\frac{t^{\beta }}{\beta }+b^{-x}{\left(\log \left(b\right)\right)}^2\frac{t^{2\beta }}{2!{\beta }^2}+b^{-x}{\left(\log \left(b\right)\right)}^3\frac{t^{3\beta }}{3!{\beta }^3}+b^{-x}{\left(\log \left(b\right)\right)}^4\frac{t^{4\beta }}{4!{\beta }^4}+b^{-x}{\left(\log \left(b\right)\right)}^5\frac{t^{5\beta }}{5!{\beta }^5}.$$(31)

Furthermore, the CRS approximate solution $u\left(x,t\right)$ of fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) can be reformulatted in an infinite expansion as follows, (32) $$u(x,t)=\underset{k\to \infty }{\mathit{lim}}{u}_k(x,t)=b^{-x}(1+\frac{t^{\beta }}{\beta }+{(\mathit{log}\left(b\right))}^2\frac{t^{2\beta }}{2!{\beta }^2}+{(\mathit{log}\left(b\right))}^3\frac{t^{3\beta }}{3!{\beta }^3}+\dots +{(\mathit{log}\left(b\right))}^k\frac{t^{k\beta }}{k!{\beta }^k}+\dots )=b^{-x}\sum _{m=0}^{\infty }{(\mathit{log}\left(b\right))}^m\frac{t^{m\beta }}{m!{\beta }^m},$$(32) which is consistent with the results contained in Das and Kumar (2011).

To demonstrate the efficiency of the CRPS method for Example 4.2, the 3 D plots of the 7th-CRS solution of the temporal fractional gas dynamic model (24(24) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)\mathrm{l}\mathrm{og}\left(b\right)=0,$$(24) )-(25)(25) $$u\left(x,0\right)=b^{-x},$$(25) are given in Figure 2 for different orders of fractional case $\beta =1,$ $0.75,$ and $0.5$ that compared with the exact solution on the interval $[-1,1] \times [-4,4].$ While, Table 2 shows some numerical results for Example 4.2 with step size $0.16$ and $b=5.$ From these results, it can be noted that the approximate solutions are in good agreement with each other and with exact solution. Meanwhile, it can be indicated that the solutions of the studied fractional gas model propagate along the $x$-axis over time while maintaining their shape and amplitude in all orders of fractional cases.

Figure 2. Surface plots of 7th CRS solutions $u_7\left(x,t\right)$ of Example 4.2 for different orders of fractional levels on $\left[-1,1\right]\times [-4,4]$ at $b=4:$ (a) Exact solution; (b) $\beta =1;$ (c) $\beta =0.75;$ (d) $\beta =0.5.$

Table 2. The CRSM soluation for Example 4.2 at $x=1$ and $b=5.$

$t_i$	Exact Soluion	7^th-CRS solution			Absolute Error
		$\beta =1$	$\beta =0.9$	$\beta =0.8$
$0.16$	$0.258741$	$0.258741$	$0.260666$	$0.282022$	$9.87264\times {10}^{-11}$
$0.32$	$0.334734$	$0.334734$	$0.338492$	$0.379792$	$2.60338\times {10}^{-8}$
$0.48$	$0.433048$	$0.433048$	$0.438955$	$0.493735$	$7.77746\times {10}^{-8}$
$0.64$	$0.560236$	$0.560236$	$0.568728$	$0.611827$	$2.06580\times {10}^{-7}$
$0.80$	$0.724780$	$0.724778$	$0.736356$	$0.803564$	$1.58820\times {10}^{-6}$
$0.96$	$0.937651$	$0.937649$	$0.952795$	$0.998063$	$1.93707\times {10}^{-6}$

Example 4.3.

The following nonlinear, nonhomogeneous, conformable time-fractional gas dynamic equation is considered (33) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=-e^{t-x}, t\ge 0, x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(33) along with initial condition (34) $$u\left(x,0\right)=1-e^{-x}.$$(34)

According to the CRS approach, the truncated CRS approximate solution for nonlinear time-fractional gas dynamic model (33)(33) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=-e^{t-x}, t\ge 0, x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(33) -(34(34) $$u\left(x,0\right)=1-e^{-x}.$$(34) ), in light of the initial condition (34), can be written as $u_m\left(x,t\right)=1-e^{-x}+{\sum }_{m=1}^k{h}_m(x)\frac{t^{m\beta }}{{\beta }^{m}m!},$ in which the values of the unknown coefficients $h_m(x)$ for $m=1,2,\dots ,k$ can be obtained through constructing the truncated residual function ${\mathit{Res}}_u^k\left(x,t\right)$ as follows, (35) $$u\left(x,0\right)={\mathit{Res}}_u^m\left(x,t\right)=T_t^{\beta }\left(1-e^{-x}+\sum _{m=1}^k{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)+\left(1-e^{-x}+\sum _{m=1}^m{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(e^{-x}+\sum _{m=1}^m{h}_m^{\prime }\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)-\left(1-e^{-x}+\sum _{m=1}^m{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)\left(e^{-x}-\sum _{m=1}^m{h}_m\left(x\right)\frac{t^{m\beta }}{{\beta }^{m}m!}\right)+e^{t-x}.$$(35)

Following the argument of CRSM and based upon the fact $T_t^{(k-1)\beta }{{\mathit{Res}}_u^k\left(x,t\right)}_{│t=0}=0,$ $k=1,2,\dots ,5,$ the 5th-CRS approximate solution for nonhomogeneous time-fractional gas dynamic model (33)(33) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=-e^{t-x}, t\ge 0, x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(33) -(34(34) $$u\left(x,0\right)=1-e^{-x}.$$(34) ) can be written by $$u_5\left(x,t\right)=1-e^{-x}-e^{-x}\frac{t^{\beta }}{\beta }-e^{-x}\frac{t^{2\beta }}{2!{\beta }^2}-e^{-x}\frac{t^{3\beta }}{3!{\beta }^3}-e^{-x}\frac{t^{4\beta }}{4!{\beta }^4}--e^{-x}\frac{t^{5\beta }}{5!{\beta }^5}.$$

Therefore, the CRS solution of the time-fractional gas dynamic model (33)(33) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=-e^{t-x}, t\ge 0, x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(33) -(34(34) $$u\left(x,0\right)=1-e^{-x}.$$(34) ) can be written by the form: (36) $$u\left(x,t\right)=\underset{k\to \infty }{\mathit{lim}}{u}_k\left(x,t\right)=1-\sum _{j=0}^{\infty }{e}^{-x}\frac{t^{j\beta }}{j!{\beta }^j}.$$(36)

Particularly, at $\beta =1,$ the obtained CRS approximate solution coincides well to the series expansion of the exact solution $u\left(x,t\right)=1-e^{t-x}$ presented in Das and Kumar (2011).

In the following, the 3 D plots of 5th-CRS solution of the nonhomogeneous temporal fractional gas dynamic model (33)(33) $$T_t^{\beta }u\left(x,t\right)+u\left(x,t\right)u_x\left(x,t\right)-u\left(x,t\right)\left(1-u\left(x,t\right)\right)=-e^{t-x}, t\ge 0, x\in \mathbb{R},\mathbb{ }0<\beta \le 1$$(33) -(34) are displayed in Figure 3 over the interval $\left[0,2\right]\times [-2,2]$ for different orders of conformable derivative when $\beta =1,$ $\beta =0.75,$ and $\beta =0.5$ to verify the effect of the conformable derivative on the dynamics of the solution.

Figure 3. Surface plots of 5th CRS solutions $u_5\left(x,t\right)$ of Example 4.3 for different orders of conformable derivative on $\left[0,2\right]\times [-2,2]:$ (a) Exact solution; (b) $\beta =1;$ (c) $\beta =0.75;$ (d) $\beta =0.5.$

5. Conclusions

In this novel analysis, a modern and efficient analytic algorithm, the conformable residual series method, was designed and employed to investigate the approximate solutions of the nonlinear time-fractional gas dynamics model in terms of the conformable derivative. The CRS algorithm has been profitably and straightforwardly applied to obtain such solutions in a fast-converging multiple FPS without assuming any undue constraints or limitations on the nature of the given fractional problems. For this purpose and to support the theoretical framework, three illustrative applications are provided to demonstrate the potential of the proposed algorithm. With the aid of the Mathematica software, all simulations and mathematical calculations have been performed. The solutions acquired using the CRS algorithm agree well with the exact solutions in the classical case of beta value and in continuous harmony with each other for various orders of fractional cases. It can be concluded that the CRS method is a convenient and attractive tool for creating analytical solutions for many real world systems of broad interest in the scientific community. Certainly, investigating accurate approximate solutions for different types of temporal fractional models in the conformable sense is not easy, which prompts us to do more future work, including but not limited to, fractional integral and dynamical systems under uncertainty in the light of the conformable meaning of order $\beta > 1.$

Conflicts of interest

The authors declare no conflict of interest regarding the publication of this paper.