# A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations

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## Abstract

This article mainly explores and applies a modified form of the analytical method, namely the homotopy analysis transform method (HATM) for solving time-fractional Cauchy reaction–diffusion equations (TFCRDEs). Then mainly we address the error norms \(L_{2}\) and \(L_{\infty }\) for a convergence study of the proposed method. We also find existence, uniqueness and convergence in the analysis for TFCRDEs. The projected method is illustrated by solving some numerical examples. The obtained numerical solutions by the HATM method show that it is simple to employ. An excellent conformity obtained between the solution got by the HATM method and the various well-known results available in the current literature. Also the existence and uniqueness of the solution have been demonstrated.

## Keywords

Homotopy analysis transform method Fractional Cauchy reaction–diffusion equation Mittag-Leffler function Optimal value## 1 Introduction

The beginning of fractional calculus is considered as 30 September 1695 when the derivative of arbitrary order was described by Leibniz [1]. After that many renowned mathematicians have studied the application of the fractional derivative and fractional differential equations (FDEs); some of them were Liouville, Grunwald, Letnikov and Riemann [2]. A lot of significant phenomena are well described by FDEs in electromagnetics, acoustics, viscoelasticity, electro chemistry and material science [3]. Moreover, some basic results associated to solving FDEs may be found in [4, 5, 6, 7].

*w*is the concentration,

*r*is the reaction parameter and \(D>0\) is the diffusion coefficient. The fractional derivative

*λ*considered in this paper is in the sense of Caputo.

In this paper, we have applied HATM for solving linear and nonlinear TFCRDEs. The HATM method provides excellent agreement between two powerful methods, one is the most popular and useful homotopy analysis method (HAM) and the other one is the Laplace transform method. The HAM was first proposed and applied by Liao in [17] to solve lots of nonlinear problems. The HAM has been successfully applied by many researchers for solving linear and nonlinear partial differential equations [18, 19].

But presently, concentration of diverse researchers is on finding the solution behavior of different nonlinear equations by means of different methods jointed with Laplace transform, among them the variation iteration transform method [20] and the homotopy analysis transform method [21, 22]. The advantage of HATM over HAM is that it gives rapidly convergent series solution only by taking a small number of terms and hence HATM is very powerful and efficient in finding approximate solutions as well as analytical solutions of many fractional physical models. Moreover, the analytical method of using the Laplace transform and its inverse is shown in [23, 24, 25]. The other work related to this can be found in [26, 27, 28, 29, 30, 31, 32, 33, 34]. The plan of this article is to find approximate analytical solutions of TFCRDEs with the time derivative *λ* (\(0 < \lambda\le1\)).

## 2 Existence and uniqueness

In this section, we establish the existence and uniqueness of a solution of differential equation (1.1). We first present a few necessary definitions.

*f*of order \(\lambda>0\) is defined by

*f*of order \(\lambda >0\) is defined by

*A*is the infinitesimal generator of an analytic semigroup \(\{ T(t) : t \ge0\}\) and is self-adjoint [35]. By introducing \(v(t)x = w(x, t)\) and \(\gamma(t)x=r(x,t)\), Eq. (1.1) can be written as

*v*of the above problem we mean that

*A*equipped with the graph norm \(\|v\|_{\mathcal{D}}=\|v\|+\|Av\|\). It is not difficult to check that \(f(t,v)=\gamma(t)v\) satisfies the Lipschitz condition. For any \(v_{1}, v_{2} \in D(A)\), we have

*A*is discrete with eigenvalues \(\mu _{n}=-n^{2}D\), \(n\in\mathbb{N,}\) and the eigenfunctions are of the form \(\psi _{n}(z)= (\frac{2}{\pi} )^{\frac{1}{2}} \sin n z\). Moreover, \(\{\psi_{n} : n \in\mathbb{N}\}\) is an orthonormal basis for

*X*, and

*θ*with \(\frac{\pi}{2}<\theta< \pi\) and the curve \(\gamma_{\theta}=\{re^{i\theta}: r \ge0\} \cup\{re^{-i\theta : r \ge0}\}\).

Because \((\mu I-A)^{-1}\) is compact, from the above representation one can deduce that \(\{S_{\lambda}(t): t >0\}\) is a compact operator.

### Theorem 2.1

*Let*\(\gamma\in L^{p}([0,T]:\mathbb{R}^{+})\)*for*\(p =\frac{1}{\lambda}\). *If*\(\frac{1}{\varGamma\lambda} \sup_{s \in[0,T]} (\int_{0}^{s} \frac{\gamma (t)}{(s-t)^{1-\alpha}}\,dt ) <1\), *then the abstract Cauchy problem has a unique mild solution*.

The proof is similar to Theorem 2.1 of [36].

## 3 Fundamental scheme of HATM

*m*th-order deformation equation \(w_{m}(x,t)\) and for \(m \ge1\), at

*M*th order, we have

In this section, we study the convergence of HATM through the following theorem.

### Theorem 3.1

### Proof

## 4 Function of HATM and mathematical results

Four examples of TFCRDEs are solved to exhibit the HATM method. In the whole article, MATHEMATICA 7 software package has been used for the figures’ computational processes.

### Example 1

*m*th-order deformation equation for \(w_{m}(x,t)\)

Again if we take the standard value of \(\lambda=1\), then the series solution is reduced to \(e^{-x}+ x e^{-t}\), this is an exact solution of standard CRDEs and hence the result is absolutely in conformity with the homotopy perturbation given by Yildirim [16] and the Adomian decomposition method by Lesnic [13].

*ħ*curve of Eq. (4.1). As pointed out by Liao [17], we can choose any values of

*ħ*, where \(\hbar\in(\hbar_{1}, \hbar_{2} )\) and \(\hbar_{1} \approx-1.80 \), \(\hbar_{2} \approx-0.2 \). In the particular case if \(\hbar=-1\) the speed of convergence is most advantageous.

\(E_{7}\) in the solution of TFCRDEs using HATM for \(\lambda=1\)

( | Exact solution | Approximation solution | \(|u_{\mathrm{exact}}-u_{\mathrm{MHATM}} | \) |
---|---|---|---|

(0.1,0.1) | 0.9953211598 | 0.9953211598 | 9.99201 × 10 |

(0.1,0.2) | 0.9867104933 | 0.9867104933 | 2.95319 × 10 |

(0.1,0.3) | 0.9789192401 | 0.9789192401 | 1.68754 × 10 |

(0.2,0.1) | 0.9996982366 | 0.9996982366 | 2.27596 × 10 |

(0.2,0.2) | 0.9824769036 | 0.9824769036 | 6.20615 × 10 |

(0.2,0.3) | 0.9668943972 | 0.9668943972 | 8.32667 × 10 |

(0.3,0.1) | 1.0122694460 | 1.0122694460 | 2.55351 × 10 |

(0.3,0.2) | 0.9864374466 | 0.9864374466 | 4.32987 × 10 |

(0.3,0.3) | 0.9630636868 | 0.9630636868 | 4.39648 × 10 |

\(L_{2} \) and \(L_{\infty} \) error norms for TFCRDEs by HATM for \(\lambda=1\)

| \(L_{2} \) error norm | \(L_{\infty}\) error norm |
---|---|---|

0.1 | 1.87998 × 10 | 1.68754 × 10 |

0.2 | 3.10492 × 10 | 8.32667 × 10 |

0.3 | 3.18634 × 10 | 4.39648 × 10 |

*m*th order of approximation, also we can define the exact square residual error for equation, where

*ħ*can be found by solving nonlinear algebraic equation \(\frac{dE_{m}}{d\hbar}=0\) [37]. The numerical results are elaborated in Tables 3 and 4.

Optimal value of *ħ* for \(\lambda=1\)

Order of approximation | Optimal value of | Value of \(E_{m}\) |
---|---|---|

2 | −0.826476 | 8.9631 × 10 |

4 | −0.939232 | 8.90047 × 10 |

6 | −0.964903 | 8.8997 × 10 |

Optimal value of *ħ* for \(\lambda=0.9\)

Order of approximation | Optimal value of | Value of \(E_{m}\) |
---|---|---|

2 | −0.79381 | 8.84405 × 10 |

4 | −0.918672 | 8.73643 × 10 |

6 | −0.950037 | 8.73404 × 10 |

It is clear from Tables 3 and 4 that the optimal value of *ħ* are −0.826476, −0.939232, −0.964903 and −0.79381, −0.918672, −0.950037, respectively, in the case of different orders of approximations.

### Example 2

*m*th-order deformation equation (4.5) is

Next for the standard value of \(\lambda=1\), the above series solution reduced to \(e^{-x}+ x e^{-t}\), this is an exact solution of standard CRDEs and hence the result is absolutely conformity with that the homotopy perturbation given by Yildirim [16] and the Adomian decomposition method by Lesnic [13].

*λ*.

Again, the convergence of the above method for Eq. (4.5) is shown by drawing the absolute error curve.

*ħ*curve for Eq. (4.5) is shown. It is clear from Fig. 8 that the perfect range of

*ħ*is from −1.60 to −0.3.

*x*and

*t*by using the seventh-order approximate solution. Again, to show the validity and exactness of the proposed method the error norms \(L_{2} \) and \(L_{\infty} \) are presented in Table 6.

\(E_{7}\) in the solution of TFCRDEs by HATM for \(\lambda=1\)

( | Exact solution | Approximation solution | \(|u_{\mathrm{exact}}-u_{\mathrm{MHATM}} | \) |
---|---|---|---|

(0.1,0.1) | 1.1162780704 | 1.1162780704 | 4.62963 × 10 |

(0.1,0.2) | 1.2336780599 | 1.2336780599 | 7.89258 × 10 |

(0.1,0.3) | 1.3634251141 | 1.3634251141 | 6.866828 × 10 |

(0.2,0.1) | 1.1502737988 | 1.1502737988 | 2.49134 × 10 |

(0.2,0.2) | 1.2712491580 | 1.2712491580 | 3.77920 × 10 |

(0.2,0.3) | 1.4049475905 | 1.4049475905 | 4.90874 × 10 |

(0.3,0.1) | 1.0122694460 | 1.0122694460 | 7.84484 × 10 |

(0.3,0.2) | 1.3364274882 | 1.3364274882 | 6.75904 × 10 |

(0.3,0.3) | 1.4769807938 | 1.4769807938 | 3.81584 × 10 |

The error norm in the solution of TFCRDEs by HATM for \(\lambda=1\)

| \(L_{2} \) error norm | \(L_{\infty} \) error norm |
---|---|---|

0.1 | 6.4635 × 10 | 6.866828 × 10 |

0.2 | 1.84526 × 10 | 4.90874 × 10 |

0.3 | 1.75874 × 10 | 3.81584 × 10 |

### Example 3

*m*th-order deformation equations is as follows:

For \(\lambda=1\), this series is reduced to the closed form \(\frac {1}{10} e^{\operatorname{cos}x-t-11}\), which is an exact solution of the classical CRDEs and hence the result is absolutely in conformity with the variation iteration method given by Dehghan [14].

*λ*.

*ħ*curve. Here we can choose any values of

*ħ*, where \(\hbar\in(\hbar_{1}, \hbar_{2} )\) and \(\hbar_{1} \approx-1.70 \), \(\hbar_{2} \approx-0.5\).

### Example 4

*m*th-order deformation equations is as follows:

## Notes

### Acknowledgements

All authors would like to express their sincere thanks to the respected editors for their time and comments as regards the review process. The first author Dr. Sunil Kumar would like to acknowledge the financial support received from the National Board for Higher Mathematics, Department of Atomic Energy, Government of India (Approval No. 2/48(20)/2016/NBHM(R.P.)/R and D II/1014).

### Authors’ contributions

All authors contributed equally and significantly in writing this paper and typed, read, and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

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