# Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials

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## Abstract

**:**

## 1. Introduction

- A novel operational matrix of fractional order is derived in the sense of the Liouville–Caputo fractional derivative for the MVP.
- The technique is a combination of the collocation technique with the Tau method.
- The method converts the nonlinear fractional differential equation into a system of algebraic equations that are solved easily.
- The convergence analysis is performed to prove the error bound for the technique.
- The proposed technique is adapted for solving various examples with the application including the Bagley–Torvik and Bratu models.
- The acquired results prove that the technique is better than the other methods in terms of error and computational cost.
- The proposed algorithm can be extended to more complex problems having real-life applications.

## 2. Preliminaries and Notations

**Definition 1**

**Theorem 1**

**Corollary 1.**

## 3. Morgan–Voyce Polynomials

**Definition 2.**

#### 3.1. Morgan–Voyce Polynomials Operational Matrices of Derivatives

#### 3.2. $MV\left(t\right)$ Polynomials Integer-Order Operational Matrix of Derivatives

#### 3.3. $MV\left(t\right)$ Polynomials Fractional-Order Operational Matrix of Derivatives

## 4. Proposed Methodology and Convergence Analysis

#### 4.1. Proposed Methodology

#### 4.2. Convergence of Morgan–Voyce Bases

**Theorem 2.**

**Proof.**

## 5. Numerical Simulations

**Example 1**

**Example 2**

**Example 3**

- Case I: ${f}_{0}=0,\phantom{\rule{4pt}{0ex}}{g}_{0}=1,$ and $\tau =1$. The analytical solution is $g\left(t\right)=cos\left(t\right)$.
- Case II: ${f}_{0}=0.01,\phantom{\rule{4pt}{0ex}}{g}_{0}=0,\phantom{\rule{4pt}{0ex}}\varrho =1,$ and $\tau =1$. The precise solution $g\left(t\right)=0.005\phantom{\rule{0.166667em}{0ex}}t\phantom{\rule{0.166667em}{0ex}}sin\left(t\right)$.
- Case III: ${f}_{0}=1,\phantom{\rule{4pt}{0ex}}{g}_{0}=0,$ and $\tau =1$. The true solution is $g\left(t\right)=\frac{1}{1-{\varrho}^{2}}\left(cos\left(\varrho t\right)-cos\left(t\right)\right)$,where ${\varrho}^{2}\ne 1$. Here, we have $\varrho =6$.

**Example 4**

**Example 5**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The absolute error (

**left**), while (

**right**) the numerical solution for different fractional-order cases of $\xi $ for Example 1 with $m=3$.

**Figure 2.**The exact and an approximate are shown on the right, while the absolute error is shown on the left for Example 2 where $t\in {I}_{1}$ and $m=3$.

**Figure 3.**The exact and an approximate are shown on the right, while the absolute error is shown on the left for Example 2 where $t\in {I}_{2}$ and $m=3$.

**Figure 4.**For Example 3, Case I with $m=6$: the absolute error is shown on (

**left**), the analytic and the approximation solutions are presented on (

**right**).

**Figure 5.**For Example 3, Case II with $m=6$: the absolute error is shown on (

**left**), and the analytic and an approximation solutions are presented on (

**right**).

**Figure 6.**For Example 3, Case III with $m=6$: the absolute error is shown on the (

**left**), and the analytic and an approximation solutions are presented on the (

**right**).

**Figure 7.**The absolute error (

**left**), while the numerical and approximate solutions (

**right**) for Example 4 with $m=3$.

**Figure 8.**The absolute error (

**left**) and the numerical solution (

**right**) for various fractional-order cases of $\xi $ for Example 5 with $m=6$.

t | LDG [51] | VIM [52] | FIM [52] | MVOMM | Exact |
---|---|---|---|---|---|

0.10 | $1.101000000$ | $1.183140356$ | $1.103763584$ | $1.100999999$ | $1.101000$ |

0.25 | $1.265624999$ | $1.438783940$ | $1.269040456$ | $1.265624999$ | $1.265625$ |

0.30 | $1.326999999$ | − | − | $1.326999999$ | $1.327000$ |

0.40 | $1.463999999$ | − | − | $1.463999999$ | $1.464000$ |

0.50 | $1.625000000$ | $1.519844510$ | $1.623997167$ | $1.624999999$ | $1.625000$ |

0.60 | $1.816000000$ | − | − | $1.815999999$ | $1.816000$ |

0.75 | $2.171875000$ | $0.830835570$ | $2.166900262$ | $2.171875000$ | $2.171875$ |

0.80 | $2.312000000$ | − | − | $2.312000000$ | $2.312000$ |

0.90 | $2.629000000$ | − | − | $2.629000000$ | $2.629000$ |

**Table 2.**Comparison of the strategy used in the present study for Example 1 with distinct errors and the LWS introduced in [53].

Error Type | LWS [53] $(\mathit{k}=0,\phantom{\rule{4pt}{0ex}}\mathit{H}=4)$ | LWS [53] $(\mathit{k}=1,\phantom{\rule{4pt}{0ex}}\mathit{H}=4)$ | MVOMM $(\mathit{m}=3)$ |
---|---|---|---|

${L}^{2}$-error | $5.9\times {10}^{-15}$ | $4.9\times {10}^{-15}$ | $1.78\times {10}^{-15}$ |

${L}_{\infty}$-error | $9.3\times {10}^{-15}$ | $7.8\times {10}^{-15}$ | $8.98\times {10}^{-16}$ |

**Table 3.**Absolute error comparisons for the presented method for Example 2, $\xi =1.5,\phantom{\rule{0.166667em}{0ex}}m=3$.

$\mathit{t}\in {\mathit{I}}_{1}$ | Absolute Errors | $\mathit{t}\in {\mathit{I}}_{2}$ | Absolute Errors |
---|---|---|---|

0.0 | $4.44089\times {10}^{-16}$ | 1.0 | $8.88178\times {10}^{-15}$ |

0.1 | $4.37773\times {10}^{-16}$ | 1.1 | $8.70304\times {10}^{-15}$ |

0.2 | $4.20204\times {10}^{-16}$ | 1.2 | $8.51763\times {10}^{-15}$ |

0.3 | $3.93453\times {10}^{-16}$ | 1.3 | $8.31890\times {10}^{-15}$ |

0.4 | $3.59589\times {10}^{-16}$ | 1.4 | $8.10019\times {10}^{-15}$ |

0.5 | $3.20684\times {10}^{-16}$ | 1.5 | $7.85483\times {10}^{-15}$ |

0.6 | $2.78806\times {10}^{-16}$ | 1.6 | $7.57616\times {10}^{-15}$ |

0.7 | $2.36027\times {10}^{-16}$ | 1.7 | $7.25753\times {10}^{-15}$ |

0.8 | $1.94416\times {10}^{-16}$ | 1.8 | $6.89226\times {10}^{-15}$ |

0.9 | $1.56044\times {10}^{-16}$ | 1.9 | $6.47371\times {10}^{-15}$ |

1.0 | − | 2.0 | $5.99520\times {10}^{-15}$ |

Case I | Case II | Case III | ||||
---|---|---|---|---|---|---|

t | LDG [51] | MVOMM $(\mathit{m}=4)$ | LDG [51] | MVOMM $(\mathit{m}=4)$ | LDG [51] | MVOMM $(\mathit{m}=6)$ |

0.0 | − | $2.22\times {10}^{-16}$ | − | $2.42\times {10}^{-20}$ | − | $3.95\times {10}^{-20}$ |

0.1 | $1.95\times {10}^{-8}$ | $4.44\times {10}^{-16}$ | $7.76\times {10}^{-10}$ | $1.14\times {10}^{-15}$ | $1.12\times {10}^{-9}$ | $7.12\times {10}^{-17}$ |

0.2 | $1.36\times {10}^{-8}$ | $1.89\times {10}^{-15}$ | $5.38\times {10}^{-10}$ | $3.58\times {10}^{-15}$ | $5.91\times {10}^{-10}$ | $1.92\times {10}^{-16}$ |

0.3 | $9.66\times {10}^{-9}$ | $3.33\times {10}^{-15}$ | $3.79\times {10}^{-10}$ | $6.32\times {10}^{-15}$ | $1.92\times {10}^{-8}$ | $3.07\times {10}^{-16}$ |

0.4 | $9.96\times {10}^{-9}$ | $4.89\times {10}^{-15}$ | $3.94\times {10}^{-10}$ | $8.86\times {10}^{-15}$ | $3.11\times {10}^{-8}$ | $4.18\times {10}^{-16}$ |

0.5 | $1.02\times {10}^{-8}$ | $6.22\times {10}^{-15}$ | $4.01\times {10}^{-10}$ | $1.11\times {10}^{-14}$ | $1.92\times {10}^{-8}$ | $5.31\times {10}^{-16}$ |

0.6 | $5.06\times {10}^{-9}$ | $7.44\times {10}^{-15}$ | $2.01\times {10}^{-10}$ | $1.33\times {10}^{-14}$ | $5.55\times {10}^{-9}$ | $6.33\times {10}^{-16}$ |

0.7 | $9.44\times {10}^{-9}$ | $8.77\times {10}^{-15}$ | $3.72\times {10}^{-10}$ | $1.55\times {10}^{-14}$ | $1.88\times {10}^{-8}$ | $7.27\times {10}^{-16}$ |

0.8 | $3.58\times {10}^{-9}$ | $1.01\times {10}^{-14}$ | $1.42\times {10}^{-10}$ | $1.78\times {10}^{-14}$ | $1.62\times {10}^{-8}$ | $8.18\times {10}^{-16}$ |

0.9 | $4.79\times {10}^{-9}$ | $1.10\times {10}^{-14}$ | $1.89\times {10}^{-10}$ | $1.94\times {10}^{-14}$ | $9.71\times {10}^{-9}$ | $9.03\times {10}^{-16}$ |

1.0 | $3.18\times {10}^{-15}$ | $1.08\times {10}^{-14}$ | $1.19\times {10}^{-16}$ | $1.87\times {10}^{-14}$ | $1.12\times {10}^{-19}$ | $9.15\times {10}^{-16}$ |

t | Exact | RKHS [54] $(\mathit{n}=20)$ | RKHS [54] $(\mathit{n}=40)$ | LWS [53] $(\mathit{H}=4)$ | MVOMM $(\mathit{m}=3)$ |
---|---|---|---|---|---|

0.2 | $-0.192000$ | $1.890\times {10}^{-4}$ | $5.700\times {10}^{-5}$ | $9.575\times {10}^{-15}$ | $4.547\times {10}^{-16}$ |

0.4 | $-0.336000$ | $2.537\times {10}^{-4}$ | $7.131\times {10}^{-5}$ | $3.758\times {10}^{-14}$ | $1.080\times {10}^{-15}$ |

0.6 | $-0.384000$ | $2.168\times {10}^{-4}$ | $5.992\times {10}^{-5}$ | $8.426\times {10}^{-14}$ | $1.833\times {10}^{-15}$ |

0.8 | $-0.288000$ | $1.198\times {10}^{-4}$ | $3.312\times {10}^{-5}$ | $1.497\times {10}^{-13}$ | $2.672\times {10}^{-15}$ |

1.0 | 0 | 0 | 0 | 0 | 0 |

**Table 6.**Comparison between the LWS introduced in [53] and the current study for Example 4 with various errors.

Error Type | LWS [53] $(\mathit{k}=0,\phantom{\rule{4pt}{0ex}}\mathit{H}=4)$ | LWS [53] $(\mathit{k}=1,\phantom{\rule{4pt}{0ex}}\mathit{H}=4)$ | MVOMM $(\mathit{m}=3)$ |
---|---|---|---|

${L}^{2}$-error | $8.5\times {10}^{-14}$ | $2.2\times {10}^{-13}$ | $3.55\times {10}^{-15}$ |

${L}_{\infty}$-error | $1.6\times {10}^{-13}$ | $5.0\times {10}^{-13}$ | $1.88\times {10}^{-15}$ |

t | CFDM [55] | RKM [56] | Bessel-QLM $(\mathit{M}=7)$ [57] | MVOMM $(\mathit{m}=6)$ |
---|---|---|---|---|

$0.1$ | $7.1\times {10}^{-6}$ | $1.67\times {10}^{-5}$ | $4.20\times {10}^{-8}$ | $2.75\times {10}^{-17}$ |

$0.2$ | $1.23\times {10}^{-5}$ | $3.10\times {10}^{-7}$ | $1.22\times {10}^{-7}$ | $3.016\times {10}^{-17}$ |

$0.3$ | $1.71\times {10}^{-5}$ | $1.13\times {10}^{-6}$ | $1.86\times {10}^{-7}$ | $1.65\times {10}^{-16}$ |

$0.4$ | $2.26\times {10}^{-5}$ | $2.12\times {10}^{-4}$ | $2.61\times {10}^{-7}$ | $1.87\times {10}^{-16}$ |

$0.5$ | $2.90\times {10}^{-5}$ | $2.90\times {10}^{-6}$ | $3.55\times {10}^{-7}$ | $8.05\times {10}^{-17}$ |

$0.6$ | $3.69\times {10}^{-5}$ | $4.10\times {10}^{-6}$ | $4.10\times {10}^{-7}$ | $1.32\times {10}^{-16}$ |

$0.7$ | $4.72\times {10}^{-5}$ | $6.50\times {10}^{-6}$ | $5.79\times {10}^{-7}$ | $2.19\times {10}^{-16}$ |

$0.8$ | $6.14\times {10}^{-5}$ | $7.50\times {10}^{-6}$ | $6.83\times {10}^{-7}$ | $1.67\times {10}^{-16}$ |

$0.9$ | $8.32\times {10}^{-5}$ | $3.35\times {10}^{-6}$ | $3.04\times {10}^{-7}$ | $2.79\times {10}^{-16}$ |

$1.0$ | $1.29\times {10}^{-5}$ | $4.37\times {10}^{-8}$ | $3.23\times {10}^{-5}$ | $2.52\times {10}^{-16}$ |

**Table 8.**Comparison of the highest absolute errors for Example 5 using $\xi =2$ and various values of m from [55] and our proposed approach.

CFDM [55] | MVOMM | |||
---|---|---|---|---|

$N=m$ | ${L}_{\infty}$-Error | $m$ | ${L}_{\infty}$-Error | CPU Time (s) |

5 | $1.67\times {10}^{-3}$ | 2 | $1.42\times {10}^{-8}$ | $0.185$ |

10 | $8.32\times {10}^{-5}$ | 4 | $5.61\times {10}^{-13}$ | $0.225$ |

20 | $4.43\times {10}^{-6}$ | 6 | $4.11\times {10}^{-16}$ | $0.282$ |

40 | $2.38\times {10}^{-7}$ | 8 | $8.34\times {10}^{-16}$ | $0.586$ |

80 | $1.36\times {10}^{-8}$ | 10 | $1.47\times {10}^{-16}$ | $1.592$ |

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Srivastava, H.M.; Adel, W.; Izadi, M.; El-Sayed, A.A. Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials. *Fractal Fract.* **2023**, *7*, 301.
https://doi.org/10.3390/fractalfract7040301

**AMA Style**

Srivastava HM, Adel W, Izadi M, El-Sayed AA. Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials. *Fractal and Fractional*. 2023; 7(4):301.
https://doi.org/10.3390/fractalfract7040301

**Chicago/Turabian Style**

Srivastava, Hari Mohan, Waleed Adel, Mohammad Izadi, and Adel A. El-Sayed. 2023. "Solving Some Physics Problems Involving Fractional-Order Differential Equations with the Morgan-Voyce Polynomials" *Fractal and Fractional* 7, no. 4: 301.
https://doi.org/10.3390/fractalfract7040301