# An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique

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## Abstract

**:**

## 1. Introduction

## 2. Research Basics Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. Mathematical Formulation

## 4. Discussions

#### 4.1. Linear Diffusivity and Nonlinear Conductivity

#### 4.2. Linear Diffusivity and Linear Conductivity

#### 4.3. Non-Linear Diffusivity and Linear Conductivity

## 5. The Basic Concept of the q-HATM

#### 5.1. Solution Procedure for the First Case of Richard’s Equation

#### 5.2. Solution Procedure for the Second Case of Richard’s Equation

#### 5.3. Solution Procedure for the Third Case of Richard’s Equation

## 6. Numerical Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The graph of $U\left(x,t\right)$ versus depth x at $n=1,\alpha =1$, and for $t=0.2,0.4,0.6,0.8,1$ for Equation (29).

**Figure 4.**The graph of $U\left(x,t\right)$ versus depth $x$ at $n=1,t=0.1$, and for $\alpha =1,0.75,0.50$ for Equation (29).

**Figure 5.**The sketch of $\hslash $-curve profiles for the $U$ against $\hslash $ for Equation (29) at (

**a**) $n=1$ and (

**b**) $n=2$ when $t=0.01,x=0.5$ for $\alpha =1,0.75,0.50$.

**Figure 6.**The graph of $U\left(x,t\right)$ versus time $t$ at $n=1,\text{}x=0.5$, and for $\alpha =1,\text{}0.75,\text{}0.50$ for Equation (29).

**Figure 9.**The graph of $U\left(x,t\right)$ versus depth x at $n=1,\alpha =1$, and for $t=0.2,0.4,0.6,0.8,1$ for Equation (37).

**Figure 10.**The graph of $U\left(x,t\right)$ versus depth $x$ at $n=1,\text{}t=0.1$, and for $\alpha =1,\text{}0.75,\text{}0.50$ for Equation (37).

**Figure 11.**The sketch of $\hslash $-curve profiles for the $U$ against $\hslash $ for Equation (37) at (

**a**) $n=1$ and (

**b**) $n=2$ when $t=0.01,x=0.5$ for diverse values of $\alpha $.

**Figure 12.**The graph of $U\left(x,t\right)$ versus time $t$ at $n=1,\text{}x=0.5$, and for $\alpha =1,\text{}0.75,\text{}0.50$ for Equation (37).

**Figure 15.**The graph of $U\left(x,t\right)$ versus depth x at $n=1,\text{}\alpha =1$, and for $t=0.2,\text{}0.4,\text{}0.6,\text{}0.8,\text{}1$ for Equation (45).

**Figure 16.**The graph of $U\left(x,t\right)$ versus depth $x$ at $n=1,\text{}t=0.1$, and for $\alpha =1,\text{}0.75,\text{}0.50$ for Equation (45).

**Figure 17.**The sketch of $\hslash $-curve for the $U$ against $\hslash $ for considered Equation (45) at (

**a**) $n=1$ and (

**b**) $n=2$ when $t=0.01,x=0.5$ for sundry values of $\alpha $.

**Figure 18.**The graph of $U\left(x,t\right)$ versus time $t$ at $n=1,\text{}x=0.5$, and for $\alpha =1,\text{}0.75,\text{}0.50$ for Equation (45).

$\mathit{t}$ | $\mathit{x}=0.1$ | $\mathit{x}=0.2$ | $\mathit{x}=0.3$ | $\mathit{x}=0.4$ | $\mathit{x}=0.5$ | $\mathit{x}=0.6$ | $\mathit{x}=0.7$ | $\mathit{x}=0.8$ | $\mathit{x}=0.9$ |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.1252 | 0.2394 | 0.3510 | 0.4590 | 0.5620 | 0.6590 | 0.7486 | 0.8299 | 0.9015 |

0.4 | 0.1399 | 0.2681 | 0.3915 | 0.5077 | 0.6145 | 0.7095 | 0.7905 | 0.8549 | 0.9007 |

0.6 | 0.1542 | 0.2960 | 0.4307 | 0.5549 | 0.6653 | 0.7583 | 0.8307 | 0.8789 | 0.8997 |

0.8 | 0.1682 | 0.3232 | 0.4689 | 0.6009 | 0.7147 | 0.8056 | 0.8697 | 0.9021 | 0.8984 |

1 | 0.1820 | 0.3500 | 0.5065 | 0.6420 | 0.7632 | 0.8523 | 0.9079 | 0.9246 | 0.8970 |

$\mathit{t}$ | $\mathit{x}=0.1$ | $\mathit{x}=0.2$ | $\mathit{x}=0.3$ | $\mathit{x}=0.4$ | $\mathit{x}=0.5$ | $\mathit{x}=0.6$ | $\mathit{x}=0.7$ | $\mathit{x}=0.8$ | $\mathit{x}=0.9$ |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.1538 | 0.2705 | 0.3838 | 0.4925 | 0.5954 | 0.6914 | 0.7794 | 0.8582 | 0.9267 |

0.4 | 0.2088 | 0.3388 | 0.4605 | 0.5717 | 0.6704 | 0.7545 | 0.8221 | 0.8712 | 0.9000 |

0.6 | 0.2748 | 0.4130 | 0.5363 | 0.6422 | 0.7281 | 0.7919 | 0.8311 | 0.8438 | 0.8280 |

0.8 | 0.3512 | 0.4903 | 0.6069 | 0.6989 | 0.7642 | 0.8010 | 0.8075 | 0.7819 | 0.7222 |

1 | 0.4366 | 0.5667 | 0.6667 | 0.7359 | 0.7742 | 0.7808 | 0.7550 | 0.6955 | 0.5988 |

$\mathit{t}$ | $\mathit{x}=0.1$ | $\mathit{x}=0.2$ | $\mathit{x}=0.3$ | $\mathit{x}=0.4$ | $\mathit{x}=0.5$ | $\mathit{x}=0.6$ | $\mathit{x}=0.7$ | $\mathit{x}=0.8$ | $\mathit{x}=0.9$ |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.1195 | 0.2476 | 0.3720 | 0.4910 | 0.6030 | 0.7062 | 0.7992 | 0.8806 | 0.9489 |

0.4 | 0.1308 | 0.2924 | 0.4442 | 0.5817 | 0.7003 | 0.7962 | 0.8655 | 0.9049 | 0.9119 |

0.6 | 0.1446 | 0.3459 | 0.5278 | 0.6813 | 0.7985 | 0.8719 | 0.8940 | 0.8585 | 0.7634 |

0.8 | 0.1619 | 0.4101 | 0.6239 | 0.7900 | 0.8968 | 0.9317 | 0.8790 | 0.7264 | 0.4690 |

1 | 0.1848 | 0.4872 | 0.7326 | 0.9054 | 0.9939 | 0.9784 | 0.8214 | 0.4826 | −0.0152 |

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**MDPI and ACS Style**

Javare Gowda, R.; Singh, S.; Padmarajaiah, S.S.; Khan, U.; Zaib, A.; Weera, W.
An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique. *Fractal Fract.* **2022**, *6*, 249.
https://doi.org/10.3390/fractalfract6050249

**AMA Style**

Javare Gowda R, Singh S, Padmarajaiah SS, Khan U, Zaib A, Weera W.
An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique. *Fractal and Fractional*. 2022; 6(5):249.
https://doi.org/10.3390/fractalfract6050249

**Chicago/Turabian Style**

Javare Gowda, Rekha, Sandeep Singh, Suma Seethakal Padmarajaiah, Umair Khan, Aurang Zaib, and Wajaree Weera.
2022. "An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique" *Fractal and Fractional* 6, no. 5: 249.
https://doi.org/10.3390/fractalfract6050249