# Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem

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## Abstract

**:**

## 1. Introduction

- Introducing a free parameter in the existing or newly created model of the iterative scheme.
- Inserting a combination function into two iterative schemes; then, the parameter or combination function is optimized to raise the order of convergence.
- Several ideas presented here are novel and have not yet appeared in the literature, which can promote the development of fourth-order one-step iterative schemes while saving the computational cost.
- For the application of the derivative-free one-step iterative schemes, we developed a powerful Lie symmetry method to solve a second-order nonlinear boundary-value problem.

## 2. Preliminaries

## 3. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Numerical Experiments

## 5. Updating Three Parameters by Memory Method

## 6. A New Fourth-Order Iterative Scheme with Memory Updating

#### 6.1. A New Third-Order Result

**Theorem**

**4.**

**Proof.**

#### 6.2. Updating Two Parameters

#### 6.3. Updating Three Parameters

## 7. Improvement of D$\stackrel{\u02c7}{\mathrm{z}}$uni$\stackrel{\xb4}{\mathrm{c}}$’s Method and Optimal Combination

#### 7.1. Improvement of D$\stackrel{\u02c7}{\mathrm{z}}$uni$\stackrel{\xb4}{\mathrm{c}}$’s Memory Method

**Theorem**

**5.**

**Proof.**

#### 7.2. Optimal Combination of D$\stackrel{\u02c7}{\mathrm{z}}$uni$\stackrel{\xb4}{\mathrm{c}}$’s and Wu’s Iterative Methods

**Theorem**

**6.**

**Proof.**

#### 7.3. Optimal Combination of D$\stackrel{\u02c7}{\mathrm{z}}$uni$\stackrel{\xb4}{\mathrm{c}}$’s and Liu’s Iterative Methods

**Theorem**

**7.**

**Proof.**

## 8. Modification of D$\stackrel{\u02c7}{\mathrm{z}}$uni$\stackrel{\xb4}{\mathrm{c}}$’s Method

**Theorem**

**8.**

**Proof.**

## 9. A Lie Symmetry Method

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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n | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|

NM | 1.594 | 1.841 | 1.978 | 1.999 | × | × |

Algorithm 1 | 2.779 | 3.046 | × | × | ||

Algorithm 2 | 2.634 | 2.968 | × | × | ||

Algorithm 3 | 4.899 | × | × |

Functions | ${\mathit{x}}_{0}$ | NM | JM | TM | KM | CM | Algorithm 3 |
---|---|---|---|---|---|---|---|

${g}_{1}$ | −0.3 | 55 | 46 | 46 | 49 | 9 | 5 |

${g}_{2}$ | 0 | 5 | 3 | 3 | 3 | 3 | 3 |

${g}_{3}$ | 3 | 7 | 4 | 4 | 4 | 4 | 4 |

${g}_{4}$ | 3.5 | 11 | 6 | 6 | 7 | 7 | 5 |

${g}_{5}$ | 1 | 7 | 4 | 4 | 8 | 4 | 4 |

**Table 3.**The NI, COC, and E.I. for the method by updating three parameters, A, B, and Q, in the first memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | $[{\mathit{A}}_{0},{\mathit{B}}_{0}]$ | ${\mathit{Q}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/3}$ |
---|---|---|---|---|---|---|

${g}_{1}$ | 1.3 | $[17.09,0.48]$ | −0.1 | 3 | 4.207 | 1.614 |

${g}_{2}$ | 0.2 | $[-3.94,-0.027]$ | −0.5 | 3 | 5.433 | 1.758 |

${g}_{3}$ | 2.1 | $[3.99,0.87]$ | 0 | 3 | 5.090 | 1.720 |

${g}_{4}$ | −0.5 | $[2.503,0.627]$ | 0.3 | 3 | 6.095 | 1.827 |

${g}_{5}$ | 1.3 | $[-1.926,0.927]$ | 0 | 3 | 6.795 | 1.894 |

$\mathit{\eta}$ | −0.9 | −0.6 | −0.3 | −0.2 | 0.1 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|---|---|---|

NI | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

COC | 3.413 | 3.114 | 3.233 | 3.443 | 3.081 | 2.974 | 2.870 | 2.795 |

E.I. = (COC)${}^{1/2}$ | 1.847 | 1.765 | 1.798 | 1.855 | 1.755 | 1.724 | 1.694 | 1.672 |

**Table 5.**The NI, COC, and E.I. for the method by updating three parameters, A, B, and $\eta $, in the second memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | $[{\mathit{A}}_{0},{\mathit{B}}_{0}]$ | ${\mathit{\eta}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/2}$ |
---|---|---|---|---|---|---|

${f}_{1}$ | 1 | $[5.25,0.9048]$ | −0.3 | 4 | 4.680 | 2.163 |

${f}_{2}$ | 0.9 | $[14.44,-0.33]$ | 4 | 4 | 3.535 | 1.880 |

${f}_{3}$ | 2 | $[3.25,-0.231]$ | −0.3 | 4 | 3.992 | 1.998 |

${f}_{4}$ | −0.5 | $[6.13,0.54]$ | 10 | 4 | 4.237 | 2.058 |

${f}_{5}$ | 1.5 | $[0.548,-1.318]$ | 0.3 | 5 | 3.610 | 1.900 |

**Table 6.**The NI, COC, and E.I. for the method by updating two parameters, A and p, in the third memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{p}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/2}$ |
---|---|---|---|---|---|---|

${f}_{1}$ | 1 | 5.25 | −0.3 | 3 | 5.169 | 2.274 |

${f}_{2}$ | 1.3 | 4.44 | −0.1 | 4 | 4.022 | 2.005 |

${f}_{3}$ | 2 | 5.11 | −3 | 3 | 3.126 | 2.031 |

${f}_{4}$ | −0.5 | 4.83 | −10 | 3 | 7.227 | 2.688 |

${f}_{5}$ | 1.3 | −1.926 | −10 | 3 | 6.945 | 2.635 |

**Table 7.**The NI, COC, and E.I. for the method by updating three parameters, A, B, and Q, in the fourth memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{B}}_{0}$ | ${\mathit{Q}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/3}$ |
---|---|---|---|---|---|---|---|

${f}_{1}$ | 1.2 | 23.75 | 0.39 | 5 | 3 | 5.156 | 1.728 |

${f}_{2}$ | 1.4 | −1.97 | 8.34 | 0.5 | 5 | 6.643 | 1.880 |

${f}_{3}$ | 2.2 | 3.25 | −0.23 | 2 | 3 | 9.127 | 2.090 |

${f}_{4}$ | −0.5 | 4.83 | 0.61 | 0.5 | 3 | 7.517 | 1.959 |

${f}_{5}$ | 1.3 | −3.733 | 0.475 | 0.1 | 3 | 7.091 | 1.921 |

**Table 8.**The NI, COC, and E.I. for the method by updating three parameters, A, B and Q, in the fifth memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{B}}_{0}$ | ${\mathit{Q}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/2}$ |
---|---|---|---|---|---|---|---|

${f}_{1}$ | 1.2 | 9.99 | 0.67 | 2 | 3 | 5.028 | 2.242 |

${f}_{2}$ | 1.3 | 5.06 | 469,683 | 1 | 6 | 5.013 | 2.239 |

${f}_{3}$ | 2.2 | 3.25 | −0.231 | 2 | 3 | 9.127 | 3.021 |

${f}_{4}$ | −0.5 | 4.83 | 0.61 | 5 | 3 | 7.343 | 2.742 |

${f}_{5}$ | 1.3 | −3.733 | 0.475 | 0.1 | 3 | 7.091 | 2.663 |

**Table 9.**The NI, COC, and E.I. for the method by updating three parameters, A, B, and $\beta $, in the sixth memory-accelerating technique.

Functions | ${\mathit{x}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{B}}_{0}$ | ${\mathit{\beta}}_{0}$ | NI | COC | E.I. = (COC)${}^{1/2}$ |
---|---|---|---|---|---|---|---|

${f}_{1}$ | 1.2 | 17.09 | 0.48 | 2 | 3 | 5.028 | 2.423 |

${f}_{2}$ | 1.3 | −13.79 | 0.721 | 2 | 5 | 8.526 | 2.920 |

${f}_{3}$ | 2.2 | 3.25 | −0.23 | 2 | 3 | 7.879 | 2.807 |

${f}_{4}$ | −0.5 | 22.29 | 0.394 | 1 | 3 | 5.502 | 2.346 |

${f}_{5}$ | 1.3 | −3.733 | 0.475 | 1 | 3 | 5.081 | 2.254 |

**Table 10.**The values of the COC for D$\stackrel{\u02c7}{z}$uni$\stackrel{\xb4}{c}$’s memory method and the sixth memory accelerating technique.

Functions | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{f}}_{3}$ | ${\mathit{f}}_{4}$ | ${\mathit{f}}_{5}$ |
---|---|---|---|---|---|

D$\stackrel{\u02c7}{z}$uni$\stackrel{\xb4}{\mathrm{c}}$’s method | 4.819 | 4.990 | 5.852 | 4.496 | 4.469 |

Equation (113) | 5.028 | 8.526 | 7.879 | 5.502 | 5.081 |

**Table 11.**For Equations (125) and (126), comparing the performances of the second, third, fifth, and sixth memory-accelerating techniques in the solution of a second-order nonlinear boundary-value problem by the Lie symmetry method.

Methods | Second | Third | Fifth | Sixth |
---|---|---|---|---|

NI | 4 | 3 | 3 | 4 |

Error of ${u}^{\prime}\left(0\right)$ | $2.736\times {10}^{-14}$ | $2.736\times {10}^{-14}$ | $2.736\times {10}^{-14}$ | $2.736\times {10}^{-14}$ |

Maximum error of u | $3.997\times {10}^{-14}$ | $3.997\times {10}^{-14}$ | $3.997\times {10}^{-14}$ | $3.997\times {10}^{-14}$ |

Methods | Second | Third | Fifth | Sixth |
---|---|---|---|---|

NI | 5 | 3 | 3 | 3 |

Error of ${u}^{\prime}\left(0\right)$ | $2.665\times {10}^{-13}$ | $2.647\times {10}^{-13}$ | $2.647\times {10}^{-13}$ | $2.647\times {10}^{-13}$ |

Maximum error of u | $4.352\times {10}^{-14}$ | $4.397\times {10}^{-14}$ | $4.397\times {10}^{-14}$ | $4.397\times {10}^{-14}$ |

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**MDPI and ACS Style**

Liu, C.-S.; Chang, C.-W.; Kuo, C.-L.
Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem. *Symmetry* **2024**, *16*, 120.
https://doi.org/10.3390/sym16010120

**AMA Style**

Liu C-S, Chang C-W, Kuo C-L.
Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem. *Symmetry*. 2024; 16(1):120.
https://doi.org/10.3390/sym16010120

**Chicago/Turabian Style**

Liu, Chein-Shan, Chih-Wen Chang, and Chung-Lun Kuo.
2024. "Memory-Accelerating Methods for One-Step Iterative Schemes with Lie Symmetry Method Solving Nonlinear Boundary-Value Problem" *Symmetry* 16, no. 1: 120.
https://doi.org/10.3390/sym16010120