# Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models

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## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Haar Wavelets and Their Integrals

#### Function Approximation

## 4. Description of the Method

#### Note

## 5. Stability

## 6. Illustrative Examples

#### 6.1. Problem 5.1

#### 6.2. Problem 5.2

#### 6.3. Problem 5.3

## 7. Initial Disturbance and Noisy Data

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**

**Solutions profile of Problem 5.1**. (

**a**) Exact and approximate solutions at $t=4,\phantom{\rule{3.33333pt}{0ex}}\tau =0.001.$ (

**b**) Absolute error in (

**a**). (

**c**) Exact 3D plot. (

**d**) Approximate 3D plot at $t=4,\phantom{\rule{3.33333pt}{0ex}}\tau =0.01,\phantom{\rule{3.33333pt}{0ex}}\lambda =4$.

**Figure 2.**

**Solutions profile of Problem 5.2**. (

**a**) Exact and approximate solutions at $t=4,\phantom{\rule{3.33333pt}{0ex}}\tau =0.001.$ (

**b**) Absolute error in (

**a**). (

**c**) Exact 3D plot. (

**d**) Approximate 3D plot at $t=4,\phantom{\rule{3.33333pt}{0ex}}\tau =0.01,\phantom{\rule{3.33333pt}{0ex}}\lambda =5$.

**Figure 3.**

**Solutions profile of problem 5.3.**(

**a**) Exact and approximate solutions at $t=0.2,\phantom{\rule{3.33333pt}{0ex}}\tau =0.001.$ (

**b**) Absolute error in (

**a**). (

**c**) Exact 3D plot. (

**d**) = Approximate 3D plot at $t=0.2,\phantom{\rule{3.33333pt}{0ex}}\tau =0.01,\phantom{\rule{3.33333pt}{0ex}}\lambda =5$.

**Figure 4.**

**Error profile of all problems at $\mathbf{t}=\mathbf{1}.\mathbf{0}$**. (

**a**) Error in Problem 5.1. (

**b**) Error in Problem 5.2. (

**c**) Error in Problem 5.3.

Methods | Points | t | $\mathit{x}=0.1$ | $\mathit{x}=0.2$ | $\mathit{x}=0.3$ | $\mathit{x}=0.4$ | $\mathit{x}=0.5$ |
---|---|---|---|---|---|---|---|

Present | 64 | 0.02 | 6.54 × 10${}^{-7}$ | 1.24 × 10${}^{-6}$ | 1.71 × 10${}^{-6}$ | 2.01 × 10${}^{-6}$ | 2.11 × 10${}^{-6}$ |

64 | 0.05 | 5.22 × 10${}^{-6}$ | 9.94 × 10${}^{-6}$ | 1.36 × 10${}^{-5}$ | 1.60 × 10${}^{-5}$ | 1.69 × 10${}^{-5}$ | |

64 | 1 | 7.14 × 10${}^{-4}$ | 1.35 × 10${}^{-3}$ | 1.87 × 10${}^{-3}$ | 2.20 × 10${}^{-3}$ | 2.31 × 10${}^{-3}$ | |

128 | 0.02 | 2.60 × 10${}^{-7}$ | 4.94 × 10${}^{-7}$ | 6.81 × 10${}^{-7}$ | 8.00 × 10${}^{-7}$ | 8.42 × 10${}^{-7}$ | |

128 | 0.05 | 2.59 × 10${}^{-6}$ | 4.94 × 10${}^{-6}$ | 6.80 × 10${}^{-6}$ | 7.99 × 10${}^{-6}$ | 8.40 × 10${}^{-6}$ | |

128 | 1 | 6.83 × 10${}^{-4}$ | 1.29 × 10${}^{-3}$ | 1.78 × 10${}^{-3}$ | 2.10 × 10${}^{-3}$ | 2.21 × 10${}^{-3}$ | |

Mittal [39] | 91 | 0.02 | 3.20 × 10${}^{-5}$ | 6.08 × 10${}^{-5}$ | 8.37 × 10${}^{-5}$ | 9.84 × 10${}^{-5}$ | 1.04 × 10${}^{-4}$ |

91 | 0.05 | 3.59 × 10${}^{-5}$ | 6.83 × 10${}^{-5}$ | 9.39 × 10${}^{-5}$ | 1.10 × 10${}^{-4}$ | 1.16 × 10${}^{-4}$ | |

91 | 1 | 6.32 × 10${}^{-5}$ | 1.20 × 10${}^{-4}$ | 1.65 × 10${}^{-4}$ | 1.94 × 10${}^{-4}$ | 2.04 × 10${}^{-4}$ | |

181 | 0.02 | 3.55 × 10${}^{-6}$ | 6.76 × 10${}^{-6}$ | 9.30 × 10${}^{-6}$ | 1.09 × 10${}^{-5}$ | 1.15 × 10${}^{-5}$ | |

181 | 0.05 | 3.99 × 10${}^{-6}$ | 7.58 × 10${}^{-6}$ | 1.04 × 10${}^{-5}$ | 1.23 × 10${}^{-5}$ | 1.29 × 10${}^{-5}$ | |

181 | 1 | 7.00 × 10${}^{-6}$ | 1.33 × 10${}^{-5}$ | 1.83 × 10${}^{-5}$ | 2.16 × 10${}^{-5}$ | 2.27 × 10${}^{-5}$ | |

Caglar [40] | 121 | 0.02 | 4.80 × 10${}^{-6}$ | 9.70 × 10${}^{-6}$ | 1.40 × 10${}^{-5}$ | 1.90 × 10${}^{-5}$ | 2.40 × 10${}^{-5}$ |

191 | 0.02 | 5.20 × 10${}^{-6}$ | 2.10 × 10${}^{-6}$ | 3.10 × 10${}^{-6}$ | 4.20 × 10${}^{-6}$ | 5.20 × 10${}^{-6}$ | |

521 | 0.02 | 4.90 × 10${}^{-7}$ | 9.90 × 10${}^{-7}$ | 1.40 × 10${}^{-6}$ | 1.90 × 10${}^{-6}$ | 2.40 × 10${}^{-6}$ | |

Aziz et al. [12] | 20 | 0.05 | 9.30 × 10${}^{-6}$ | 8.00 × 10${}^{-6}$ | 2.80 × 10${}^{-6}$ | 1.00 × 10${}^{-6}$ | 2.70 × 10${}^{-6}$ |

Rashidinia [13] | 20 | 0.05 | 2.91 × 10${}^{-6}$ | 1.73 × 10${}^{-6}$ | 1.60 × 10${}^{-6}$ | 2.23 × 10${}^{-6}$ | 2.60 × 10${}^{-7}$ |

Mohammadi [41] | 40 | 0.05 | 2.96 × 10${}^{-6}$ | 1.77 × 10${}^{-6}$ | 1.64 × 10${}^{-6}$ | 2.28 × 10${}^{-6}$ | 2.65 × 10${}^{-7}$ |

Problem 4.1 | |||
---|---|---|---|

$\mathit{\lambda}$ | $\mathit{\tau}$ | ${\mathit{L}}_{\infty}$ | Rate |

2 | 1/100 | 90,568 × 10${}^{-3}$ | |

3 | 1/200 | 2.4428 × 10${}^{-3}$ | 1.8904 |

4 | 1/400 | 6.8223 × 10${}^{-4}$ | 1.8402 |

5 | 1/800 | 2.0398 × 10${}^{-4}$ | 1.7418 |

Methods | Points | $\mathit{\tau}$ | $\mathit{t}=0.2$ | $\mathit{t}=0.4$ | $\mathit{t}=0.8$ | $\mathit{t}=1$ | $\mathit{t}=2$ | $\mathit{t}=4$ |
---|---|---|---|---|---|---|---|---|

Present | 32 | 0.001 | 1.76 × 10${}^{-13}$ | 5.72 × 10${}^{-13}$ | 1.80 × 10${}^{-12}$ | 2.14 × 10${}^{-12}$ | 2.87 × 10${}^{-12}$ | 2.39 × 10${}^{-12}$ |

64 | 0.001 | 1.72 × 10${}^{-13}$ | 5.75 × 10${}^{-13}$ | 1.48 × 10${}^{-12}$ | 1.90 × 10${}^{-12}$ | 2.95 × 10${}^{-12}$ | 6.13 × 10${}^{-13}$ | |

[41] | 100 | 0.01 | 1.78 × 10${}^{-5}$ | 5.85 × 10${}^{-5}$ | 1.57 × 10${}^{-4}$ | 2.00 × 10${}^{-4}$ | 2.95 × 10${}^{-4}$ | 1.60 × 10${}^{-4}$ |

200 | 0.005 | 2.38 × 10${}^{-6}$ | 7.80 × 10${}^{-6}$ | 2.09 × 10${}^{-5}$ | 2.67 × 10${}^{-5}$ | 3.94 × 10${}^{-5}$ | 2.57 × 10${}^{-5}$ |

$\mathit{t}=0.2$ | $\mathit{t}=0.5$ | $\mathit{t}=1$ | $\mathit{t}=4$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{\lambda}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ |

4 | 6.10 × 10${}^{-5}$ | 3.38 × 10${}^{-4}$ | 3.86 × 10${}^{-4}$ | 2.11 × 10${}^{-3}$ | 1.02 × 10${}^{-3}$ | 5.58 × 10${}^{-3}$ | 8.64 × 10${}^{-4}$ | 4.72 × 10${}^{-3}$ |

5 | 1.11 × 10${}^{-5}$ | 7.36 × 10${}^{-5}$ | 5.85 × 10${}^{-5}$ | 3.29 × 10${}^{-4}$ | 2.24 × 10${}^{-4}$ | 1.23 × 10${}^{-3}$ | 2.29 × 10${}^{-4}$ | 1.25 × 10${}^{-3}$ |

$\mathit{\lambda}$ | $\mathit{\tau}$ | ${\mathit{L}}_{\mathit{\infty}}$ | Rate |
---|---|---|---|

2 | 1/100 | 1.5254 × 10${}^{-2}$ | |

3 | 1/200 | 3.9824 × 10${}^{-3}$ | 1.9374 |

4 | 1/400 | 9.5864 × 10${}^{-4}$ | 2.0545 |

5 | 1/800 | 2.1397 × 10${}^{-4}$ | 2.1635 |

$\mathit{t}=0.2$ | $\mathit{t}=0.5$ | $\mathit{t}=1$ | $\mathit{t}=4$ | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{\lambda}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ |

4 | 2.73 × 10${}^{-3}$ | 1.93 × 10${}^{-2}$ | 1.29 × 10${}^{-2}$ | 9.08 × 10${}^{-2}$ | 3.14 × 10${}^{-2}$ | 2.20 × 10${}^{-1}$ | 2.50 × 10${}^{-2}$ | 1.75 × 10${}^{-1}$ |

5 | 2.72 × 10${}^{-3}$ | 1.93 × 10${}^{-3}$ | 1.27 × 10${}^{-2}$ | 9.07 × 10${}^{-2}$ | 3.08 × 10${}^{-2}$ | 2.20 × 10${}^{-1}$ | 2.46 × 10${}^{-2}$ | 1.75 × 10${}^{-1}$ |

Methods | Points | $\mathit{\tau}$ | $\mathit{t}=0.2$ | $\mathit{t}=0.4$ | $\mathit{t}=0.8$ | $\mathit{t}=1$ | $\mathit{t}=2$ | $\mathit{t}=4$ |
---|---|---|---|---|---|---|---|---|

Present | 32 | 0.01 | 1.40 × 10${}^{-3}$ | 1.51 × 10${}^{-3}$ | 1.46 × 10${}^{-4}$ | 6.95 × 10${}^{-4}$ | 3.62 × 10${}^{-3}$ | 2.68 × 10${}^{-3}$ |

[41] | 100 | 0.01 | 7.82 × 10${}^{-3}$ | 2.59 × 10${}^{-2}$ | 7.27 × 10${}^{-2}$ | 9.43 × 10${}^{-2}$ | 1.44 × 10${}^{-1}$ | 7.65 × 10${}^{-2}$ |

$\mathit{\lambda}$ | Problem 5.1 | Problem 5.2 | Problem 5.3 |
---|---|---|---|

$\rho (\Xi )$ | $\rho (\Xi )$ | $\rho (\Xi )$ | |

1 | 0.99984 | 0.99937 | 0.99937 |

2 | 0.99984 | 0.99929 | 0.99929 |

3 | 0.99984 | 0.99927 | 0.99927 |

4 | 0.99984 | 0.99927 | 0.99927 |

$\mathit{\u03f5}={10}^{-2}$ | Problem 5.1 | Problem 5.2 | Problem 5.3 | |||

$\lambda $ | ${L}_{\infty}$ | ${L}_{2}$ | ${L}_{\infty}$ | ${L}_{2}$ | ${L}_{\infty}$ | ${L}_{2}$ |

1 | 4.53 × 10${}^{-2}$ | 2.02 × 10${}^{-1}$ | 4.82 × 10${}^{-1}$ | 3.16 × 10${}^{0}$ | 1.30 × 10${}^{-1}$ | 2.27 × 10${}^{-1}$ |

2 | 2.10 × 10${}^{-2}$ | 9.41 × 10${}^{-2}$ | 1.67 × 10${}^{-2}$ | 8.62 × 10${}^{-2}$ | 4.90 × 10${}^{-2}$ | 3.09 × 10${}^{-1}$ |

3 | 1.46 × 10${}^{-2}$ | 6.55 × 10${}^{-2}$ | 4.14 × 10${}^{-3}$ | 2.18 × 10${}^{-2}$ | 3.40 × 10${}^{-2}$ | 2.26 × 10${}^{-1}$ |

4 | 1.30 × 10${}^{-2}$ | 5.82 × 10${}^{-2}$ | 1.34 × 10${}^{-3}$ | 5.94 × 10${}^{-3}$ | 3.15 × 10${}^{-2}$ | 2.20 × 10${}^{-1}$ |

$\u03f5={10}^{-3}$ | Problem 5.1 | Problem 5.2 | Problem 5.3 | |||

$\lambda $ | ${L}_{\infty}$ | ${L}_{2}$ | ${L}_{\infty}$ | ${L}_{2}$ | ${L}_{\infty}$ | ${L}_{2}$ |

1 | 3.64 × 10${}^{-2}$ | 1.63 × 10${}^{-1}$ | 4.78 × 10${}^{-1}$ | 3.16 × 10${}^{0}$ | 1.36 × 10${}^{-1}$ | 7.63 × 10${}^{-1}$ |

2 | 1.18 × 10${}^{-2}$ | 5.30 × 10${}^{-2}$ | 1.55 × 10${}^{-2}$ | 8.56 × 10${}^{-2}$ | 5.02 × 10${}^{-2}$ | 3.16 × 10${}^{-1}$ |

3 | 5.38 × 10${}^{-3}$ | 2.40 × 10${}^{-2}$ | 3.99 × 10${}^{-3}$ | 2.18 × 10${}^{-2}$ | 3.39 × 10${}^{-2}$ | 2.26 × 10${}^{-1}$ |

4 | 3.75 × 10${}^{-3}$ | 1.67 × 10${}^{-2}$ | 9.02 × 10${}^{-4}$ | 4.70 × 10${}^{-3}$ | 3.13 × 10${}^{-2}$ | 2.20 × 10${}^{-1}$ |

**Table 10.**Maximum error norms of all problems for $t=1.0\phantom{\rule{3.33333pt}{0ex}}\u03f5={10}^{-2},$ with noisy initial data.

Noise = 1% | Problem 5.1 | Problem 5.2 | Problem 5.3 | |||
---|---|---|---|---|---|---|

$\mathit{\lambda}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ | ${\mathit{L}}_{\infty}$ | ${\mathit{L}}_{\mathbf{2}}$ |

1 | 3.55 × 10${}^{-2}$ | 1.58 × 10${}^{-1}$ | 4.77 × 10${}^{-1}$ | 3.16 × 10${}^{0}$ | 1.37 × 10${}^{-1}$ | 7.67 × 10${}^{-1}$ |

2 | 1.08 × 10${}^{-2}$ | 4.84 × 10${}^{-2}$ | 1.54 × 10${}^{-2}$ | 8.56 × 10${}^{-2}$ | 5.03 × 10${}^{-2}$ | 3.16 × 10${}^{-1}$ |

3 | 4.35 × 10${}^{-3}$ | 1.94 × 10${}^{-2}$ | 3.98 × 10${}^{-3}$ | 2.18 × 10${}^{-2}$ | 3.39 × 10${}^{-2}$ | 2.26 × 10${}^{-1}$ |

4 | 2.72 × 10${}^{-3}$ | 1.21 × 10${}^{-2}$ | 8.47 × 10${}^{-4}$ | 4.69 × 10${}^{-3}$ | 3.13 × 10${}^{-2}$ | 2.20 × 10${}^{-1}$ |

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**MDPI and ACS Style**

Ghafoor, A.; Haq, S.; Hussain, M.; Abdeljawad, T.; Alqudah, M.A.
Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models. *Entropy* **2022**, *24*, 567.
https://doi.org/10.3390/e24040567

**AMA Style**

Ghafoor A, Haq S, Hussain M, Abdeljawad T, Alqudah MA.
Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models. *Entropy*. 2022; 24(4):567.
https://doi.org/10.3390/e24040567

**Chicago/Turabian Style**

Ghafoor, Abdul, Sirajul Haq, Manzoor Hussain, Thabet Abdeljawad, and Manar A. Alqudah.
2022. "Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models" *Entropy* 24, no. 4: 567.
https://doi.org/10.3390/e24040567