# Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Developed FIM-SCP

**Definition**

**1**

**Lemma**

**1**

- (i)
- The zeros of shifted Chebyshev polynomial ${S}_{n}\left(x\right)$ for $x\in [0,L]$ are$${x}_{k}=\frac{L}{2}\left(\right)open="("\; close=")">cos\left(\right)open="("\; close=")">\frac{2k-1}{2n}\pi ,\phantom{\rule{4pt}{0ex}}k\in \{1,2,3,\cdots ,n\}.$$
- (ii)
- The vth order derivatives of shifted Chebyshev polynomial ${S}_{n}\left(x\right)$ at $x=0$ are$$\begin{array}{cc}\hfill \frac{{d}^{v}}{d{x}^{v}}{S}_{n}\left(x\right){|}_{x=0}& ={(-1)}^{v+n}\phantom{\rule{0.166667em}{0ex}}\prod _{k=0}^{v-1}\phantom{\rule{0.166667em}{0ex}}\frac{2}{L}\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">\frac{{n}^{2}-{k}^{2}}{2k+1}.\hfill \end{array}$$
- (iii)
- The single integrations of shifted Chebyshev polynomial ${S}_{n}\left(x\right)$ for $n\ge 2$ are$$\begin{array}{c}\hfill {\int}_{0}^{x}{S}_{n}\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi =\frac{L}{4}\left(\right)open="("\; close=")">\frac{{S}_{n+1}\left(x\right)}{n+1}-\frac{{S}_{n-1}\left(x\right)}{n-1}-\frac{2{(-1)}^{n}}{{n}^{2}-1},\end{array}$$
- (iv)
- The shifted Chebyshev matrix $\mathbf{S}$ at each node ${x}_{k}$ defined by (5) is$$\mathbf{S}=\left[\begin{array}{cccc}{S}_{0}\left({x}_{1}\right)& {S}_{1}\left({x}_{1}\right)& \cdots & {S}_{n-1}\left({x}_{1}\right)\\ {S}_{0}\left({x}_{2}\right)& {S}_{1}\left({x}_{2}\right)& \cdots & {S}_{n-1}\left({x}_{2}\right)\\ \vdots & \vdots & \ddots & \vdots \\ {S}_{0}\left({x}_{n}\right)& {S}_{1}\left({x}_{n}\right)& \cdots & {S}_{n-1}\left({x}_{n}\right)\end{array}\right].$$Then, it has the multiplicative inverse ${\mathbf{S}}^{-1}=\frac{1}{n}\mathrm{diag}\left(\right)open="\{"\; close="\}">1,2,2,\cdots ,2$.

**Remark**

**1**

## 3. Procedure for Solving System of FIDEs

**Definition**

**2.**

**Definition**

**3.**

#### 3.1. Operational Matrix of Fractional Integration

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Remark**

**2.**

#### 3.2. Numerical Solution for System of FIDEs

#### 3.3. Experimental Examples for System of FIDEs

**Example**

**1**

**Example**

**2**

**Example**

**3**

## 4. Procedure for Solving System of CIDEs

#### 4.1. Numerical Solution for System of CIDEs

#### 4.2. Experimental Examples for System of CIDEs

**Example**

**4**

**Example**

**5**

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The behaviors of the approximate solutions in Example 1. (

**a**,

**b**) depict the comparisons between analytical and numerical solutions ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ with $M=30$ and ${\alpha}_{1}={\alpha}_{2}=1$. The error analysis again verifies that at several values of $\alpha \in \{0.91,0.93,0.95,0.97,0.99,1\}$, the obtained numerical solutions converge to the integer order solutions which their behaviors are also displayed in (

**c**,

**d**).

**Figure 2.**The behaviors of the approximate solutions in Example 2. We plot the obtained approximate solutions ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ compared to their analytical solutions as depicted in Figure (

**a**,

**b**). (

**c**,

**d**) show the behavior of the obtained approximate solutions for the proposed system of FIDEs with the nodal point $M=30$ for different values of the fractional orders $\alpha \in \{1.91,1.93,1.95,1.97,1.99,2\}$.

**Figure 3.**The behaviors of the approximate solutions in Example 3. The comparing graphs between approximate and exact solutions as depicted in (

**a**–

**c**) and also the behaviors of the obtained solutions when the fractional order $\alpha \to 1$ are shown in (

**d**–

**f**), where ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}:=\alpha \in \{0.95,0.96,0.97,0.98,0.99,1\}$.

**Table 1.**Absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ when ${\alpha}_{1}={\alpha}_{2}=1$ for Example 1.

x | OMTF [24] | FIM-SCP | ||
---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.1 | $5.5\times {10}^{-4}$ | $5.5\times {10}^{-4}$ | $6.4393\times {10}^{-15}$ | $1.9984\times {10}^{-15}$ |

0.5 | $2.7\times {10}^{-4}$ | $2.7\times {10}^{-4}$ | $1.7764\times {10}^{-15}$ | $2.6645\times {10}^{-15}$ |

0.9 | $1.9\times {10}^{-3}$ | $1.9\times {10}^{-3}$ | $1.7764\times {10}^{-15}$ | $1.7764\times {10}^{-15}$ |

**Table 2.**Absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ at different orders $\alpha $ for Example 1.

x | $\mathit{\alpha}=0.99$ | $\mathit{\alpha}=0.999$ | $\mathit{\alpha}=0.9999$ | |||
---|---|---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.1 | $3.8691\times {10}^{-2}$ | $3.8203\times {10}^{-2}$ | $3.9405\times {10}^{-3}$ | $3.8926\times {10}^{-3}$ | $3.9477\times {10}^{-4}$ | $3.8999\times {10}^{-4}$ |

0.5 | $6.2202\times {10}^{-2}$ | $4.0183\times {10}^{-2}$ | $6.3909\times {10}^{-3}$ | $4.1382\times {10}^{-3}$ | $6.4082\times {10}^{-4}$ | $4.1504\times {10}^{-4}$ |

0.9 | $7.7116\times {10}^{-2}$ | $3.9661\times {10}^{-2}$ | $7.9564\times {10}^{-3}$ | $4.0861\times {10}^{-3}$ | $7.9813\times {10}^{-4}$ | $4.0984\times {10}^{-4}$ |

**Table 3.**Absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ when ${\alpha}_{1}={\alpha}_{2}=2$ for Example 2.

x | OMTF [24] | FIM-SCP | ||
---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.1 | $4.6\times {10}^{-4}$ | $4.5\times {10}^{-5}$ | $1.1564\times {10}^{-12}$ | $1.0630\times {10}^{-14}$ |

0.5 | $3.2\times {10}^{-5}$ | $2.6\times {10}^{-4}$ | $6.2355\times {10}^{-12}$ | $8.7264\times {10}^{-14}$ |

0.9 | $2.2\times {10}^{-3}$ | $5.2\times {10}^{-3}$ | $1.1497\times {10}^{-11}$ | $4.2233\times {10}^{-13}$ |

**Table 4.**Absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ at different orders $\alpha $ for Example 2.

x | $\mathit{\alpha}=1.99$ | $\mathit{\alpha}=1.999$ | $\mathit{\alpha}=1.9999$ | |||
---|---|---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.1 | $1.6774\times {10}^{-1}$ | $4.7755\times {10}^{-3}$ | $1.7219\times {10}^{-2}$ | $4.8221\times {10}^{-4}$ | $1.7264\times {10}^{-3}$ | $4.8267\times {10}^{-5}$ |

0.5 | $9.6692\times {10}^{-1}$ | $3.4738\times {10}^{-2}$ | $1.0059\times {10}^{-1}$ | $3.5360\times {10}^{-3}$ | $1.0099\times {10}^{-2}$ | $3.5423\times {10}^{-4}$ |

0.9 | $1.8061\times {10}^{-0}$ | $2.8713\times {10}^{-2}$ | $1.8876\times {10}^{-1}$ | $2.9213\times {10}^{-3}$ | $1.8960\times {10}^{-2}$ | $2.9265\times {10}^{-4}$ |

**Table 5.**Absolute errors of ${u}_{1}\left(x\right)$, ${u}_{2}\left(x\right)$ and ${u}_{3}\left(x\right)$ at ${\alpha}_{1}={\alpha}_{2}={\alpha}_{3}=1$ for Example 3.

x | OMTF [24] | FIM-SCP | ||||
---|---|---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{3}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{3}}\left(\mathbf{x}\right)$ | |

0.1 | $5.4\times {10}^{-4}$ | $2.6\times {10}^{-3}$ | $6.6\times {10}^{-3}$ | $2.0206\times {10}^{-14}$ | $7.3275\times {10}^{-15}$ | $1.3323\times {10}^{-15}$ |

0.5 | $4.6\times {10}^{-4}$ | $4.9\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $1.1102\times {10}^{-14}$ | $1.7764\times {10}^{-15}$ | $2.5757\times {10}^{-14}$ |

0.9 | $2.6\times {10}^{-3}$ | $3.5\times {10}^{-2}$ | $9.8\times {10}^{-2}$ | $2.5313\times {10}^{-14}$ | $4.4409\times {10}^{-15}$ | $1.7764\times {10}^{-15}$ |

M | Example 1 | Example 2 | Example 3 | |||
---|---|---|---|---|---|---|

$\parallel {\mathit{u}}^{*}-{\mathit{u}}_{\mathit{M}}{\parallel}_{2}$ | Order p | $\parallel {\mathit{u}}^{*}-{\mathit{u}}_{\mathit{M}}{\parallel}_{2}$ | Order p | $\parallel {\mathit{u}}^{*}-{\mathit{u}}_{\mathit{M}}{\parallel}_{2}$ | Order p | |

4 | $1.7668\times {10}^{-3}$ | - | $1.2217\times {10}^{-2}$ | - | $3.7762\times {10}^{-1}$ | - |

5 | $9.5749\times {10}^{-5}$ | 13.064 | $1.0315\times {10}^{-3}$ | 11.077 | $5.9667\times {10}^{-2}$ | 8.268 |

6 | $4.3033\times {10}^{-6}$ | 17.016 | $7.3615\times {10}^{-5}$ | 14.480 | $7.9496\times {10}^{-3}$ | 11.056 |

7 | $1.6451\times {10}^{-7}$ | 21.175 | $3.6184\times {10}^{-6}$ | 19.545 | $9.0269\times {10}^{-4}$ | 14.113 |

**Table 7.**A comparison of absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ for Example 4.

x | GPM [32] | BSA [33] | FIM-SCP | |||
---|---|---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.2 | $1.1926\times {10}^{-8}$ | $7.5681\times {10}^{-9}$ | $4.9477\times {10}^{-8}$ | $3.4781\times {10}^{-6}$ | $9.0571\times {10}^{-10}$ | $4.5561\times {10}^{-10}$ |

0.6 | $1.2158\times {10}^{-8}$ | $3.7151\times {10}^{-9}$ | $8.9823\times {10}^{-7}$ | $3.7114\times {10}^{-5}$ | $1.7562\times {10}^{-9\phantom{\rule{4pt}{0ex}}}$ | $7.7135\times {10}^{-10}$ |

1.0 | $2.5729\times {10}^{-8}$ | $1.9506\times {10}^{-8}$ | $1.5028\times {10}^{-5}$ | $1.2451\times {10}^{-4}$ | $6.3656\times {10}^{-10}$ | $3.5808\times {10}^{-10}$ |

**Table 8.**A comparison of absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ for Example 5.

x | STWS [34] | FIM-SCP | ||
---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.2 | $8.59\times {10}^{-7}$ | $2.25\times {10}^{-7}$ | $1.4956\times {10}^{-10}$ | $2.6680\times {10}^{-10}$ |

0.6 | $4.70\times {10}^{-6}$ | $1.06\times {10}^{-6}$ | $5.0321\times {10}^{-10}$ | $5.3544\times {10}^{-10}$ |

1.0 | $1.11\times {10}^{-5}$ | $4.69\times {10}^{-6}$ | $4.4410\times {10}^{-15}$ | $2.3993\times {10}^{-15}$ |

M | Example 4 | Example 5 | ||
---|---|---|---|---|

$\parallel {\mathit{u}}_{\mathit{M}}^{*}-{\mathit{u}}_{\mathit{M}}{\parallel}_{2}$ | Order p | $\parallel {\mathit{u}}_{\mathit{M}}^{*}-{\mathit{u}}_{\mathit{M}}{\parallel}_{2}$ | Order p | |

4 | $1.7668\times {10}^{-3}$ | - | $3.2209\times {10}^{-2}$ | - |

5 | $9.5749\times {10}^{-5}$ | 13.064 | $3.0215\times {10}^{-3}$ | 10.605 |

6 | $4.3033\times {10}^{-6}$ | 17.016 | $2.1610\times {10}^{-4}$ | 14.468 |

7 | $1.6451\times {10}^{-7}$ | 21.175 | $1.0621\times {10}^{-5}$ | 19.545 |

**Table 10.**Absolute errors of ${u}_{1}\left(x\right)$ and ${u}_{2}\left(x\right)$ at different x close to zero for Example 6.

x | RK4 | FIM-SCP | ||
---|---|---|---|---|

${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{1}}\left(\mathbf{x}\right)$ | ${\mathbf{Eu}}_{\mathbf{2}}\left(\mathbf{x}\right)$ | |

0.0025 | $8.5486\times {10}^{-8}$ | $8.5486\times {10}^{-8}$ | $2.3999\times {10}^{-15}$ | $1.1102\times {10}^{-15}$ |

0.0050 | $1.3998\times {10}^{-7}$ | $1.3998\times {10}^{-7}$ | $3.2335\times {10}^{-15}$ | $7.7716\times {10}^{-16}$ |

0.0075 | $1.8766\times {10}^{-7}$ | $1.8766\times {10}^{-7}$ | $3.8858\times {10}^{-15}$ | $9.9920\times {10}^{-16}$ |

0.0100 | $1.9206\times {10}^{-7}$ | $1.9206\times {10}^{-7}$ | $3.3584\times {10}^{-15}$ | $7.7716\times {10}^{-16}$ |

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**MDPI and ACS Style**

Duangpan, A.; Boonklurb, R.; Juytai, M.
Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials. *Fractal Fract.* **2021**, *5*, 103.
https://doi.org/10.3390/fractalfract5030103

**AMA Style**

Duangpan A, Boonklurb R, Juytai M.
Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials. *Fractal and Fractional*. 2021; 5(3):103.
https://doi.org/10.3390/fractalfract5030103

**Chicago/Turabian Style**

Duangpan, Ampol, Ratinan Boonklurb, and Matinee Juytai.
2021. "Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials" *Fractal and Fractional* 5, no. 3: 103.
https://doi.org/10.3390/fractalfract5030103