# Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Constrained Expression for Eighth-Order Boundary-Value Problems

## 3. Parameter Initialization for Nonlinear Problems

## 4. Numerical Solution

#### 4.1. Linear Eighth-Order Problems

#### 4.1.1. Problem #1

#### 4.1.2. Problem #2

#### 4.1.3. Problem #3

#### 4.2. Nonlinear Eighth-Order Problems

#### 4.2.1. Problem #4

#### 4.2.2. Problem #5

#### 4.2.3. Problem #6

## 5. Accuracy of the Derivatives

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BVP | boundary value problem |

ODE | ordinary differential equation |

TFC | Theory of Functional Connections |

## Appendix A. Support Functions for General Points x i and x f

## References

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**Figure 1.**Initialization error of the solution of Problem #4 by imposing ${\mathit{\xi}}_{0}=\mathbf{0}$.

x | TFC Absolute Error | Ref. [19] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | $2.2204\times {10}^{-16}$ | $6.3\times {10}^{-11}$ |

0.2 | $1.1102\times {10}^{-16}$ | $6.5\times {10}^{-10}$ |

0.3 | $1.1102\times {10}^{-16}$ | $2.0\times {10}^{-09}$ |

0.4 | $1.1102\times {10}^{-16}$ | $3.3\times {10}^{-09}$ |

0.5 | $1.1102\times {10}^{-16}$ | $3.9\times {10}^{-09}$ |

0.6 | $6.6613\times {10}^{-16}$ | $3.4\times {10}^{-09}$ |

0.7 | $2.7756\times {10}^{-15}$ | $2.0\times {10}^{-09}$ |

0.8 | $3.8858\times {10}^{-15}$ | $6.9\times {10}^{-10}$ |

0.9 | $8.4932\times {10}^{-15}$ | $7.6\times {10}^{-11}$ |

1 | 0 | 0 |

x | TFC Absolute Error | Ref. [13] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | 0 | $1.63\times {10}^{-10}$ |

0.2 | $8.3267\times {10}^{-17}$ | $1.63\times {10}^{-09}$ |

0.3 | 0 | $4.90\times {10}^{-09}$ |

0.4 | $1.1102\times {10}^{-16}$ | $8.46\times {10}^{-09}$ |

0.5 | $5.5511\times {10}^{-17}$ | $1.01\times {10}^{-08}$ |

0.6 | $3.8858\times {10}^{-16}$ | $8.68\times {10}^{-09}$ |

0.7 | $3.3307\times {10}^{-16}$ | $5.15\times {10}^{-09}$ |

0.8 | $3.3307\times {10}^{-16}$ | $1.76\times {10}^{-09}$ |

0.9 | $8.0769\times {10}^{-15}$ | Not reported |

1 | 0 | 0 |

x | TFC Absolute Error | Ref. [19] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | $2.7756\times {10}^{-17}$ | $6.6\times {10}^{-12}$ |

0.2 | $2.7756\times {10}^{-17}$ | $6.9\times {10}^{-11}$ |

0.3 | 0 | $2.1\times {10}^{-10}$ |

0.4 | $5.5511\times {10}^{-17}$ | $3.5\times {10}^{-10}$ |

0.5 | 0 | $4.1\times {10}^{-10}$ |

0.6 | $7.2164\times {10}^{-16}$ | $3.5\times {10}^{-10}$ |

0.7 | $1.3323\times {10}^{-15}$ | $2.1\times {10}^{-10}$ |

0.8 | $1.1102\times {10}^{-15}$ | $7.2\times {10}^{-11}$ |

0.9 | $3.4417\times {10}^{-15}$ | $8.0\times {10}^{-12}$ |

1 | 0 | 0 |

x | TFC Absolute Error | Ref. [18] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | $2.2204\times {10}^{-16}$ | $2.503395\times {10}^{-06}$ |

0.2 | 0 | $8.940697\times {10}^{-06}$ |

0.3 | $2.2204\times {10}^{-16}$ | $1.561642\times {10}^{-05}$ |

0.4 | $4.4409\times {10}^{-16}$ | $1.823902\times {10}^{-05}$ |

0.5 | $2.2204\times {10}^{-16}$ | $8.821487\times {10}^{-06}$ |

0.6 | $6.6613\times {10}^{-16}$ | $7.510185\times {10}^{-06}$ |

0.7 | $3.5527\times {10}^{-15}$ | $1.883507\times {10}^{-05}$ |

0.8 | $7.5495\times {10}^{-15}$ | $1.931190\times {10}^{-05}$ |

0.9 | $1.0214\times {10}^{-14}$ | $1.168251\times {10}^{-05}$ |

1 | 0 | 0 |

x | TFC Absolute Error | Ref. [15] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | $1.5266\times {10}^{-16}$ | $2.01\times {10}^{-07}$ |

0.2 | $1.5821\times {10}^{-15}$ | $4.54\times {10}^{-07}$ |

0.3 | $7.0083\times {10}^{-14}$ | $1.52\times {10}^{-06}$ |

0.4 | $2.5846\times {10}^{-13}$ | $4.07\times {10}^{-06}$ |

0.5 | $3.2330\times {10}^{-13}$ | $6.71\times {10}^{-06}$ |

0.6 | $1.3139\times {10}^{-13}$ | $9.06\times {10}^{-06}$ |

0.7 | $2.1261\times {10}^{-14}$ | $1.00\times {10}^{-05}$ |

0.8 | $2.0539\times {10}^{-14}$ | $5.45\times {10}^{-06}$ |

0.9 | $3.3307\times {10}^{-16}$ | $2.59\times {10}^{-06}$ |

1 | 0 | 0 |

x | TFC Absolute Error | Ref. [19] Absolute Error |
---|---|---|

0 | 0 | 0 |

0.1 | $1.1102\times {10}^{-16}$ | $2.9\times {10}^{-12}$ |

0.2 | $1.1102\times {10}^{-16}$ | $2.7\times {10}^{-11}$ |

0.3 | 0 | $7.6\times {10}^{-11}$ |

0.4 | 0 | $1.3\times {10}^{-10}$ |

0.5 | $1.1102\times {10}^{-16}$ | $1.5\times {10}^{-10}$ |

0.6 | $1.1102\times {10}^{-16}$ | $1.3\times {10}^{-10}$ |

0.7 | $2.2204\times {10}^{-16}$ | $7.6\times {10}^{-11}$ |

0.8 | $3.2196\times {10}^{-15}$ | $2.5\times {10}^{-11}$ |

0.9 | $9.9920\times {10}^{-16}$ | $2.4\times {10}^{-12}$ |

1 | 0 | 0 |

Function | Mean Absolute Error: 10 Basis Functions | Mean Absolute Error: 30 Basis Functions |
---|---|---|

y | $7.5585\times {10}^{-14}$ | $9.6866\times {10}^{-16}$ |

${y}^{\prime}$ | $1.0534\times {10}^{-12}$ | $7.5884\times {10}^{-15}$ |

${y}^{\prime \prime}$ | $2.0202\times {10}^{-11}$ | $5.0360\times {10}^{-14}$ |

${y}^{\left(3\right)}$ | $4.9228\times {10}^{-10}$ | $4.0456\times {10}^{-13}$ |

${y}^{\left(4\right)}$ | $1.3318\times {10}^{-08}$ | $2.8079\times {10}^{-12}$ |

${y}^{\left(5\right)}$ | $3.8469\times {10}^{-07}$ | $1.3927\times {10}^{-11}$ |

${y}^{\left(6\right)}$ | $1.3150\times {10}^{-05}$ | $5.5250\times {10}^{-11}$ |

${y}^{\left(7\right)}$ | $3.9359\times {10}^{-04}$ | $2.0221\times {10}^{-10}$ |

${y}^{\left(8\right)}$ | $1.9399\times {10}^{-02}$ | $1.4765\times {10}^{-12}$ |

Function | Mean Absolute Error: 10 Basis Functions | Mean Absolute Error: 30 Basis Functions |
---|---|---|

y | $4.8255\times {10}^{-15}$ | $5.8919\times {10}^{-15}$ |

${y}^{\prime}$ | $8.3368\times {10}^{-15}$ | $9.9755\times {10}^{-15}$ |

${y}^{\prime \prime}$ | $7.1054\times {10}^{-14}$ | $5.9525\times {10}^{-14}$ |

${y}^{\left(3\right)}$ | $4.8760\times {10}^{-13}$ | $4.8352\times {10}^{-13}$ |

${y}^{\left(4\right)}$ | $1.5118\times {10}^{-12}$ | $1.6443\times {10}^{-12}$ |

${y}^{\left(5\right)}$ | $4.1244\times {10}^{-12}$ | $4.4041\times {10}^{-12}$ |

${y}^{\left(6\right)}$ | $3.1934\times {10}^{-12}$ | $3.2911\times {10}^{-12}$ |

${y}^{\left(7\right)}$ | $8.0532\times {10}^{-12}$ | $7.3956\times {10}^{-12}$ |

${y}^{\left(8\right)}$ | $8.1927\times {10}^{-11}$ | $8.7722\times {10}^{-12}$ |

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**MDPI and ACS Style**

Johnston, H.; Leake, C.; Mortari, D.
Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections. *Mathematics* **2020**, *8*, 397.
https://doi.org/10.3390/math8030397

**AMA Style**

Johnston H, Leake C, Mortari D.
Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections. *Mathematics*. 2020; 8(3):397.
https://doi.org/10.3390/math8030397

**Chicago/Turabian Style**

Johnston, Hunter, Carl Leake, and Daniele Mortari.
2020. "Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections" *Mathematics* 8, no. 3: 397.
https://doi.org/10.3390/math8030397