# Integral Transform Method to Solve the Problem of Porous Slider without Velocity Slip

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## Abstract

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**great symmetry**between proposed method and variation iteration method, Adomian decomposition method but in integral transform method all the boundary conditions are applied, then a recursive scheme is used for the analytical solutions, which is unlike the Variational Iteration Method, Adomian Decomposition Method, and other existing analytical methods. Solutions are obtained for much larger Reynolds numbers, and they are compared with analytical and numerical methods. Effects of Reynolds number on velocity components are presented.

## 1. Introduction

## 2. Problem Formulation

## 3. Integral Transform Method

## 4. Graphs and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Table 1.**Comparison with homotopy Padé approximation method [12] for larger Reynolds numbers.

HAM [12] | Present | Error | |
---|---|---|---|

R | $-{h}^{///}(0)$ | ||

0.2 | 12.4653341665 | 12.465334166 | −5E-10 |

1 | 14.365346654 | 14.365346654 | −0E+0 |

2 | 16.8235673555 | 16.8235673555 | −0E+0 |

3 | 19.3565458558 | 19.3565458558 | −0E+0 |

4 | 21.9473510772 | 21.9473510772 | −0E+0 |

5 | 24.5830707891 | 24.55234070789 | −5E-10 |

6 | 27.2533109825 | 27.2533109818 | −7E-10 |

7 | 29.9476951163 | 29.9476951136 | −2.7E-9 |

8 | 32.6611632416 | 32.6611632329 | −8.7E-9 |

9 | 35.3924710097 | 35.3924709848 | −2.49E-8 |

10 | 38.1449125507 | 38.1449125256 | −2.51E-8 |

51.6 | 149.6714726105 | 149.6714725836 | −2.69E-8 |

70 | 197.9636925814 | 197.9636925523 | −2.91E-8 |

100 | 275.9321654987 | 275.9321651777 | −3.21E-7 |

300 | 788.4398765432 | 788.4398762142 | −3.29E-7 |

500 | 1291.1159263487 | 1289.1236547891 | −1.99 |

1000 | 2526.4 | 2398.1243 | −128.27 |

**Table 2.**Comparison with homotopy Padé approximation method [12] for larger Reynolds numbers.

HAM [12] | Present | Error | |
---|---|---|---|

R | $-{f}^{/}(0)$ | ||

0.2 | 1.0880258369 | 1.0880258369 | -0E-0 |

1 | 1.4053692581 | 1.4053692581 | -0E+0 |

2 | 1.7445321654 | 1.7445321654 | -0E+0 |

3 | 2.0361159263 | 2.0361159263 | -0E+0 |

4 | 2.2866753421 | 2.2866753421 | -0E+0 |

5 | 2.5287418529 | 2.5287418529 | -0E-0 |

6 | 2.9136985214 | 2.9136985214 | -0E-0 |

7 | 3.172839654 | 3.172839654 | -0E-0 |

8 | 3.6543219871 | 3.6543219866 | -5E-10 |

9 | 3.8293711234 | 3.8293711226 | -8E-10 |

10 | 4.1125836917 | 4.1125836759 | -1.58E-8 |

51.6 | 7.5531472583 | 7.5531472314 | -2.69E-8 |

70 | 8.7573572419 | 8.7573572108 | -3.11E-8 |

100 | 10.40147311 | 10.401472583 | -5.27E-7 |

300 | 17.81537007 | 17.815369258 | -8.12E-7 |

500 | 22.670 | 22.661 | -0.009 |

1000 | 30.432 | 30.429 | -3.00 |

**Table 3.**Comparison with homotopy Padé approximation method [12] for larger Reynolds numbers.

HAM [12] | Present | Error | |
---|---|---|---|

R | $-{g}^{/}(0)$ | ||

0.2 | 1.0301258369 | 1.0301258369 | -0E-0 |

1 | 1.1531951623 | 1.1531951623 | -0E+0 |

2 | 1.3096314785 | 1.3096314785 | -0E+0 |

3 | 1.4658951623 | 1.4658951623 | -0E+0 |

4 | 1.6187258369 | 1.6187258369 | -0E+0 |

5 | 1.766159741 | 1.766159741 | -0E-0 |

6 | 1.9086456321 | 1.9086456321 | -0E-0 |

7 | 2.0464158231 | 2.0464158231 | -0E-0 |

8 | 2.1850474125 | 2.1850474125 | -0E-0 |

9 | 2.3368115987 | 2.3368115987 | -0E-0 |

10 | 2.5264314755 | 2.5264314755 | -0E-0 |

51.6 | 5.3011463295 | 5.30114632825 | -1.25E-9 |

70 | 6.1449236681 | 6.14492366789 | 2.1E-8 |

100 | 7.2881583541 | 7.2881580341 | -3.27E-7 |

300 | 12.5161258 | 12.5160446 | -8.12E-5 |

500 | 15.368124 | 15.258154 | -010997 |

1000 | 20.062 | 19.1583 | -0.9037 |

**Table 4.**Comparison between HPM, Numerical (Num.), and the presented technique for different Reynolds numbers.

HAM [12] | Num. [9] | Present | HPM [12] | Num. [9] | Present | HPM [12] | Num. [9] | Present | |
---|---|---|---|---|---|---|---|---|---|

R | $-{h}^{///}(0)$ | $-{f}^{/}(0)$ | $-{g}^{/}(0)$ | ||||||

0.2 | 12.465 | 12.465 | 12.465 | 1.0880 | 1.088 | 1.088 | 1.0301 | 1.0301 | 1.030 |

1 | 14.365 | 14.365 | 14.365 | 1.405 | 1.405 | 1.405 | 1.1531 | 1.153 | 1.153 |

2 | - | - | 16.8235 | - | - | 1.7445 | - | - | 1.30963 |

3 | - | - | 19.356 | - | - | 2.0361 | - | - | 1.4658 |

4 | - | - | 21.9473 | - | - | 2.2866 | - | - | 1.6187 |

5 | 24.586 | 24.584 | 24.584 | 2.4417 | 2.528 | 2.528 | - | - | 1.766 |

6 | - | - | 27.253 | - | - | - | - | - | 1.9086 |

7 | - | - | 29.9476 | - | - | - | - | - | 2.04641 |

8 | - | - | 32.6611 | - | - | - | - | - | 2.18504 |

9 | - | - | 35.3924 | - | - | - | - | - | 2.33681 |

10 | - | - | 38.1449 | - | - | - | - | - | 2.52643 |

11 | - | - | 40.9267 | - | - | - | - | - | 2.79796 |

12 | - | - | 43.7471 | - | - | - | - | - | 3.2218 |

13.8 | 48.9043 | 48.484 | 48.484 | 4.0229 | 4.022 | 4.022 | - | - | 4.70844 |

15 | - | - | 52.2461 | - | - | - | - | - | 6.53822 |

16 | - | - | 54.6612 | - | - | - | - | - | 8.72019 |

51.6 | 149.67 | 149.67 | 149.67 | 7.553 | 7.553 | 7.553 | 5.301 | 5.301 | 5.301 |

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## Share and Cite

**MDPI and ACS Style**

Faraz, N.; Khan, Y.; Lu, D.C.; Goodarzi, M.
Integral Transform Method to Solve the Problem of Porous Slider without Velocity Slip. *Symmetry* **2019**, *11*, 791.
https://doi.org/10.3390/sym11060791

**AMA Style**

Faraz N, Khan Y, Lu DC, Goodarzi M.
Integral Transform Method to Solve the Problem of Porous Slider without Velocity Slip. *Symmetry*. 2019; 11(6):791.
https://doi.org/10.3390/sym11060791

**Chicago/Turabian Style**

Faraz, Naeem, Yasir Khan, Dian Chen Lu, and Marjan Goodarzi.
2019. "Integral Transform Method to Solve the Problem of Porous Slider without Velocity Slip" *Symmetry* 11, no. 6: 791.
https://doi.org/10.3390/sym11060791