An efficient hybridization scheme for time-fractional Cauchy equations with convergence analysis

1.   Introduction

The fractional calculus, and its application, have proved itself as an optimal method to study the real world problems. This branch of applied analysis has been used in various domains of science, engineering and technology ^{[1,2,3,4,5,6,7]}. In 2013, Bulut et al. ^[8] analyzed the time-fractional generalized Burger equation and trial equations for optimizing the wave equation. He ^[9] presented the compact solution in porous media for seepage flow equation in 1998. In 2020, Dubey et al. ^[10] examined the fractional order computer virus propagation model. Kumar et al. ^[11] presented the mathematical model for chemical system. Singh et al. ^[12] examined the fractional order multi-dimensional diffusion problems.

In 2015, Ramswaroop et al. ^[13] presented the new computational method for a biological system as fractional Lotka-Volterra application. Ghanbari and Kumar ^[14] studied a fractional order predator-prey model with Beddington-DeAngelis functional response by using a numerical scheme. Zhou ^[15] gave variation iteration method for solving Cauchy equation. A detailed analysis of Cauchy problems have been given in several works ^[16,17,18]. Dubey et al. ^[19] studied fractional order Black-Scholes European option pricing model. Recently, Maitama et al. ^[20,21] proposed an integrated fractional transformation for analyzing the steady heat transfer problem. In 2019, the Caputo-fractional differential equation is analyzed by an improved Shehu transformation by Belgacem et al. ^[22]. Bokhari et al. ^[23] presented a novel application of the Shehu transformation for solving Atangana-Baleanu derivatives in 2019. El-Tawil ^[24,25] introduced the q-homotopy analysis method whose mechanism is based on homotopy which is generalized of the homotopy analysis method introduced and applied by Liao ^{[26,27,28,29]}. In 2018, Noeiaghdam et al. ^[30] applied homotopy analysis on a modified epidemiological model of computer viruses. In 2021, Noeiaghdam et al. ^[31] approached homotopy analysis transform method for the nonlinear bio-mathematical model of malaria infection. In 2021, Noeiaghdam et al. ^[32] presented a nonlinear fractional order model of COVID-19. In 2016, Noeiaghdam et al. ^[33] applied homotopy analysis transform for solving Abel's integral equations of first kind. In 2017, Singh et al. ^[34] introduced an efficient method for solving the time fractional Rosenau-Hyman equation. In 2010, Keskin et al. ^[35] introduced a new method based on fractional PDE. This method is applied for reducing the domain of differential transformation. Gupta ^[36] analyzed fractional Bennery-Lin equation by fractional PDE in 2011. The key outcome of this approach is, the discreteness of approximate solutions. Srivastava et al. ^[37] defined the RDTM solutions over the Caputo-time fractional order hyperbolic telegraph equation. In recent years several methods and applied aspects for fractional calculus have been studied by many authors ^{[38,39,40,41,42]}.

In this paper, the time-fractional Cauchy problem is analyzed by the q-HASTA and the RDTA over the Caputo's sense fractional derivatives. The q-HASTA is a well coupling of the q-HAM. The homotopical approach comprises with differential topology, the Shehu transform and the Laplace-type integral transform ^{[20,21,22,23]}. The q-HAM ^[24,25] is an extension of homotopy analysis methods ^{[26,27,28,29,30]}. The RDTA technique is an optimized method developed by Keskin et al. ^[35]. There are 3 key outcomes of this paper stated as, small parameters, compact system of equation and approximate functional range. Thus, the proposed method is referred as the efficient and compact with respect to time and computations. The initial equation is introduced next.

The time fractional Cauchy differential equation as follows:

If $\delta (\zeta, \tau) = \delta$(constant), $\beta (\zeta) = 0$ and $\eta = 1$, Eq (1.1) is called as the transport equation ^[16,18] that can play crucial role in the moving of wind and the spread of AIDS. If we assume $\delta (\zeta, \tau) = w(\zeta, \tau),$ $\eta = 1$ the Eq (1.2) is formed as the inviscid Burgers' equation ^[17,18] approaches the one-dimensional stream of particles have zero viscosity. Next, the pre-requisites are given.

2.   Preliminaries

This section presents the applied notations, feature and definitions related to fractional calculus deals with generalization of integer order derivatives and integrations.

Definition 2.1. ^[43] A real function $£(\tau), \tau > 0$ is said to be in the space $C_{\upsilon }, \upsilon \in R$ if there exists a real number$\rho (>\upsilon),$ such that $£(\tau) = \tau ^{\rho } \beta (\tau),$ where $\beta (\tau)\in C\left[0, \infty \right),$ and its said to be in the space $C_{\upsilon }^{n}$ if and only if $£^{{\rm (n)}} \in C_{\upsilon }; n\in N.$

Definition 2.2. If $n\in N, n-1 < \eta \le n,$ the derivative property of the Laplace transform in the Caputo sense $D_{\tau }^{\eta } £(\tau)$ is obtained by the Caputo ^[1,2] and Kilbas et al. ^[3] in the form

Definition 2.3. For $\eta > {\rm 0, £ }(\tau)\in C_{\nu }, \nu \ge -1,$ the Riemann-Liouville fractional integral ${\rm I}^{\eta } £(\tau)$ is expressed as

Definition 2.4. For $n\in N, n-1 < \eta \le n$, $£(\tau)\in C_{\nu }^{n}, \nu \ge -1, \tau > 0,$ then the Caputo ^[1] fractional derivative of $£(\tau)$ is illustrated as

Definition 2.5. The fractional derivative property of Shehu transform in the Riemann-Liouville form, ^[20] if $n-1 < \eta \le n, \eta > 0,$$r = 1+[\eta]$and $£(\tau), I^{r-\eta } £(\tau), \frac{d}{d\tau } I^{r-\eta } £(\tau), \cdots,$ $\frac{d^{r} }{d\tau ^{r} } I^{r-\eta } £(\tau),$ $D_{\tau }^{\eta } £(\tau)\in A,$ was obtained by

And the Caputo form, ^[21] if ${\rm{ \mathsf{ η}}}$ $>$ 0, $r = 1+[\eta]$and $£(\tau), \frac{d}{dt} £(\tau), \, \cdots, \, \frac{d^{r} }{dt^{r} } £(\tau),$ $D^{\eta } £(t)\in A,$ was obtained by the Caputo ^[1,2] as

Definition 2.6. Let we consider $V(s, u)$ denote the Shehu transform of $v(\tau),$ and then the inverse Shehu transform is given by ^[21]; defined as

3.   Analysis of q-HASTA

We analysis the q-HASTA for a general fractional differential equation of order $\eta$ given in the following manners:

Where $D_{\tau }^{\eta } w(\zeta, \tau)$ denote the fractional Caputo derivative of $w(\zeta, \tau), \, R$ and $N$ are the linear and the nonlinear differential operators respectively, $\kappa (\zeta, \tau)$ denoted the source term.

Next, we employ the Shehu transform on Eq (3.1), we get

Using the result of Shehu transform for Caputo fractional derivative, we get

We involved the nonlinear operator as

Now we use the classical HAM and a homotopy equation is presented as

where S is the Shehu transform operator, $\rho \in [0, \, \frac{1}{n}], n\ge 1$ and $\hbar \ne 0$ are the embedding parameter and auxiliary parameter respectively, $H(\zeta, \tau)$ a non-zero auxiliary function, $w_{0} (\zeta, \tau)$ is an basis assumption of $w(\zeta, \tau)$ and $\chi (\zeta, \tau {\kern 1pt} {\kern 1pt}; \rho)$ is a unknown real function which construct the following result

Consequently, as $\rho$ goes from 0 to $\frac{1}{n}$, the solution $\chi (\zeta, \, \, \tau {\kern 1pt} {\kern 1pt}; \, \, \rho)$ converge from the initial guess $w_{0} (\zeta, \tau)$ to the solution $w(\zeta, \tau).$ Expressing of $\chi (\zeta, \tau {\kern 1pt} {\kern 1pt}; \, \, \rho)$ as the Taylor's series form with respect to $\rho,$ as

where

If we select appropriate values of the initial guess, auxiliary linear operator, the auxiliary parameter and the auxiliary function, the series (3.7) converges at $\rho = \frac{1}{n},$ then it provides the solutions of the given problem Eq (3.1), in the form

And the governing equation can be deduced from the zero deformation Eq (3.5).

Now differentiate the zero-order deformation Eq (3.5) m-times with respect to $\rho$ and taking $\rho = 0$ and finally dividing them by $m\, !,$ it yields

We define the vectors as

Taking the inverse Shehu transform on Eq (3.10), we have

where $\Re _{m} (\vec{w}_{m} {}_{-1})$ is given as

and

Finally, we solve the Eq (3.12) and we compute $w_{m} (\zeta, \tau)$ for $m\ge 1,$ selecting appropriate parameter $\hbar$ and $n,$ we deduce the q-HASTA convergence solution series as

Theorem 3.1. If appropriate selection of convergences control parameters $\hbar \ne 0, n\ge 1,$and also suitable selection of $H(\zeta, \tau)\ne 0, w_{0} (\zeta, \tau),$ in Eq (3.12) such that $\left\| \left. w_{m+1} \right\| \right. \le \left(\frac{\Phi }{n} \right)\left\| \left. w_{m} \right\| \right.,$ $0 < \Phi < n,$ then the solution series $\sum _{m = 0}^{+\infty }w_{m} (\zeta, \tau) (1/n)^{m},$uniformly convergence, where $\left. \left\| \cdot \right. \right\| _{\infty }$ presented the appropriate norm.

Proof. We prove that the sequence $\langle \vartheta _{m} \rangle _{m = 0}^{\infty },$ is the Cauchy sequence. Let's assume that $\vartheta _{m},$is the sequence of partially sums of $\vartheta _{m} = \sum _{k = 0}^{m}w_{k} (\zeta, \tau)(1/n)^{k}.$

By observation, we have

For $\forall \, m\ge l, \, m, l\in N,$ we deduced from the above equation

For $m, l\to \infty,$ and then $\left\| \vartheta _{m} -\vartheta _{l} \right\| \to 0.$ Therefore, the sequence $\langle \vartheta _{m} \rangle _{m = 0}^{\infty },$ is a Cauchy sequence, hence convergence.

Table 1 gives the basic results of the Sehu transform.

4.   Basic idea of the reduced differential transforms algorithm (RDTA)

To demonstrate the basis analysis of the RDTA, we take a function $j(\zeta, \tau)$ such as $j(\zeta, \tau) = \varepsilon (\alpha)\, \nu (\beta).$ According to the one dimension differential transform, we can be defined $j(\zeta, \tau)$ such as:

where $J(\alpha, \beta) = \varepsilon (\alpha)\, \nu (\beta)$ represent the spectrum of $j(\zeta, \tau)$.

The basis preliminaries and operations of the RDTA are defined below.

Definition 4.1. If $j(\zeta, \tau)$ is analytical and differentiable about the $\zeta$ and time scale $\tau$ in the interest domain, then the spectrum function ^[36,37]

Which presented the fractional reduced transformed of $j(\zeta, \tau)$ and its inverse transform of $J_{\mu } (\zeta)$ defined as

where $R_{D}$ is the reduced differential transform operator and it's inverse operator denoted by $R_{D}^{-1},$ ^[35].

From Eqs (4.2) and (4.3), we can see that

if we set $\tau = 0,$in above Eq (4.4), it convert to

From the above, we noted that the reduced differential transform solution can be deduced by expansion of the power series.

Definition 4.2. If $w(\zeta, \tau) = R_{D}^{-1} [W_{\mu } (\zeta)], p(\zeta, \tau) = R_{D}^{-1} [P_{\mu } (\zeta)],$ and $*$ denote the convolution, indicate form of the multiplication of the RDTA in fractional form. The mathematical fractional operation of the RDTA is shown in the Table 2.

5.   Numerical examples

Here we analysis the validities of both the proposed algorithm by series solution and graphical representations, taking auxiliary function$H(\zeta, \tau) = 1$. We use $w = w(\zeta, \tau)$is the function of space coordinate $\zeta (\zeta \in R)$and time scale $\tau \, (\tau > 0), \, \, a$ is arbitrary constant.

Example 1. We consider the following transpose equation ^[16,17,18] in fractional form

and the initial conditions

Case 1: q-HASTA solution

Computing Shehu transform on Eq (5.1) with the initial condition (5.2) as

We presented the nonlinear operator as

and we have

The mth-order deformed equation is defined as

By using the inverse Shehu transform of Eq (5.6)

Simplifying the Eq (5.7), for $m = 1, 2, 3, \cdots,$ and we get

And so on, the components of $w_{m} (\zeta, \tau), m\ge 4,$ can be easily obtained. The series solution is given as

For $\hbar = -1, \, n = 1$ and $\eta = 1$ then clearly, the solution series (5.9) provides the solution and converges to the exact solution $w(\zeta, \tau) = \zeta ^{2} -2a\zeta \tau +a^{2} \tau ^{2},$ which is coincident to VIM ^[15].

Case 2: RDTA solution

We apply the RDTA to Eq (5.1), to obtain

Applying the RDTA with initial condition (5.2), we have

and so on, in this way, we found the exact solution in second terms iteration, selecting $\eta = 1.$ Using (5.11) we get the following approximations for the RDTA series solution

Which is coincident obtained by the q-HASTA$(\hbar = -1, n = 1)$ and VIM ^[15] at $\eta = 1,$ $\hbar = -1, n = 1,$ given as

The numerical analysis for the approximate series solution of Eq (5.1) obtained by using the q-HASTA & the RDTA with the exact solution (5.13) at $a = 0.01$ in Figures 1–5 are computed by using Maple package.

Example 2. The fractional nonlinear equation with variable coefficients is given as

and the initial conditions

Case 1: q-HASTA solution

To solve the Eqs (5.14) and (5.15) by using the Shehu transform with the initial condition, as

The nonlinear operator defined by

and thus

The mth-order deformed equation is given as

Computing the inverse Shehu transform, we get

Simplifying the Eq (5.20), for $m = 1, 2, 3, \cdots,$ we get

And so on, the components of $w_{m} (\zeta, \tau), m\ge 4,$ can be easily obtained. Therefore, the approximate solution series given as

If we select $\hbar = -1, \, n = 1$ and $\eta = 1$ then clearly, we can observe that the solution series (5.22), provide solution which is converges to the exact solution $w(\zeta, \tau) = \zeta ^{2} e^{-2\tau }.$

Case 2: RDTA solution

We apply the RDTA to Eq (5.14), we obtained the following relation

Using the RDTA with initial condition (5.15), we get

Solve the relation (5.23) and using Eq (5.24), we deduce the following components of $W_{m} (\zeta),$ for $m = 0, 1, 2, 3, \cdots,$ as

and so on, the components of $w_{m} (\zeta, t)$ for $m > 4$ can be obtained and the RDTA approximate series solution deduced as

When we select $\eta = 1$ then the series (5.26), rapidly converges to the exact solution

The above result (5.26) is in complete agreement with the q-HASTA $(\hbar = -1, n = 1)$ and VIM ^[15] at $\eta = 1$.

The numerical analysis for the approximate solution of Eq (5.14) obtained by using the q-HASTA & the RDTA and the original solution (5.13) in Figures 6–10 are computed by using maple package.

Example 3. Next, we take the fractional non-homogeneous equation

and the initial conditions

Case 1: q-HASTA solution

Taking the Shehu transform of Eq (5.28) and using condition (5.29), as

The nonlinear operator is

and we have

The mth-order deformed equation is defined as

The inverse Shehu transform of (5.33), we have

For $m = 1, 2, 3, \cdots,$ simplifying the Eq (5.34), we deduce

and so on, the components of $w_{m} (\zeta, \tau), m > 3,$ can be easily computed and the series solution is given by

If we select $\hbar = -1, \, n = 1$ and $\eta = 1$ then clearly, the series solution (5.36) converges to the exact solution $w(\zeta, \tau) = \tau (\zeta -\frac{t}{2})+e^{\zeta -\tau }.$

Case 2: RDTA solution

We use the RDTA to Eq (5.28), we obtained the following relation

the RDTA to the initial condition (5.37), we have

Apply Eq (5.38) in Eq (5.37), we get the following components of $W_{m} (\zeta),$ for $m = 0, 1, 2, 3, \cdots,$

And so on, the components of $w_{m} (\zeta, \tau)$ can be easily obtained and the RDTA approximate series solution deduced as

$w(\zeta, \tau) = \sum _{m = 0}^{\infty }W_{m} (\zeta) \, \tau ^{m\eta }. = W_{0} (\zeta)+W_{1} (\zeta)\, \tau ^{\eta } +W_{2} (\zeta)\, \tau ^{2\eta } +W_{3} (\zeta)\, \tau ^{3\eta } +W_{4} (\zeta)\, \tau ^{4\eta } +\cdots$

Which is the same as deduced by the q-HASTA $(\hbar = -1, n = 1)$ and VIM ^[15] at $\eta = 1$. We observed that the solution series (5.40) at $\eta = 1$, converges to the exact solution

The numerical simulations of the approximate solution (5.28) deduced by using the q-HASTA & the RDTA and the exact solution (5.41) are depicted in Figures 11–16.

Example 4. Now, we consider the inviscid Bugers' equation in fractional form as

and the initial conditions

Case 1: q-HASTA solution

Taking the Shehu transform of Eq (5.42) with the initial condition (5.43), we have

The nonlinear operator is

and thus

The mth-order deformed equation is given as

The inverse Shehu transform of (5.47), we get

For $m = 1, 2, 3, \cdots,$ simplifying the above Eq (5.48), we get

Iterating in this procedure, the components of $w_{m} (\zeta, \tau), m > 3,$ can be easily computed and the series solutions are determined by

If we select $\, \hbar \, -1, n = 1$ and $\eta = 1$, then the solution series (5.53) convert to $w(\zeta, \tau) = \zeta -\tau \zeta +\tau ^{2} \zeta -\tau ^{3} \zeta +\tau ^{4} \zeta -\cdots +(-1)^{r} \tau ^{r} \zeta +\cdots = \zeta \, (1+\tau)^{-1},$ which is exactly same as obtained by VIM ^[15].

Case 2: RDTA solution

Now we apply the RDTA to Eq (5.42), we get the following recurrence relation

apply the RDTA with the initial condition (5.43), we get

Apply Eq (5.52) in Eq (5.51), we get the components of $W_{m} (\zeta),$ for $m = 0, 1, 2, 3, \cdots,$ as

In this processes, the components of $w_{m} (\zeta, \tau), m > 4,$ can be easily found and the RDTA approximate solution deduced as

Which is the coincident with the solution series deduced by q-HASTA $(\hbar = -1, n = 1)$ and obtained by VIM ^[15] at $\eta = 1.$ The solution series (5.54), $\sum _{m = 0}^{\infty }W_{m} (\zeta) \, \tau ^{m\eta }$ convert to exact solution at $\eta = 1,$ given as

The numerical results of (5.42) deduced from the q-HASTA and the RDTA, and the exact solution (5.55) are depicted in Figures 17–22 and in Table 3.

6.   Results and discussion

We observed from the numerical results and from graphically representations (Figures 1–22) of the fractional Cauchy problems v/s space variable $\zeta$and time variable $\tau$ with order of fractional derivatives. The q-HASTA control the convergence of the solution series by using the appropriate values of auxiliary parameters $\hbar$ and $n$ at large admissible domain. $\hbar -curves$ show that the validated convergence range and also the horizontal line segments denoted the range for convergence of solution series. We observe that the convergence region directly proportional to auxiliary parameter $n$ have the advantage of the q-HASTA to the HAM, which is depicted in Figures 16 and 22. We also observed that the HAM (q-HASTA, $n = 1$) and RDTA, VIM (q-HASTA, $\hbar = -n$) are special cases of q-HASTA. Maple plotting represents the comparative approximate solutions behavior of the RDTA and the q-HASTA v/s exact and existing solutions.

7.   Conclusions

In this paper, we successfully applied computational algorithm namely q-HASTA and compared with the RDTA for solving the time fractional Cauchy equations and found the approximate solution series, closed to the exact solution. We observed that the RDTA is special condition of the q-HASTA. The q-HASTA is provide many more options for the interest of the convergence region by taking the number of parameters and auxiliary functions, whereas the RDTA algorithm provides solution series components from recursive relation, easy computational work, coming complex calculations and reducing the time and size of calculations.

Acknowledgments

The authors extend their appreciation to the Deputyship for research and innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number IF-PSAU-2021/01/17698.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.