# Polynomial Least Squares Method for Fractional Lane–Emden Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Polynomial Least Squares Method

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Proof.**

- (1)
- (2)
- We attach to problem (1) and (2) the real functional:$$\mathcal{J}({d}_{2},{d}_{3}\cdots ,{d}_{n})=\underset{0}{\overset{1}{\int}}{\mathcal{D}}^{2}\left(\tilde{y}\left(x\right)\right)dx$$
- (3)
- We compute ${d}_{2}^{0},{d}_{3}^{0},\cdots {d}_{m}^{0}$ as values which give the minimum of the functional (9) and the value of ${d}_{0}^{0}$, ${d}_{1}^{0}$ as functions of ${d}_{2}^{0},{d}_{3}^{0},\cdots ,{d}_{m}^{0}$ using the initial conditions.
- (4)
- Using the constants ${d}_{0}^{0},{d}_{1}^{0},\cdots ,{d}_{m}^{0}$ thus determined, we construct the polynomial:$${T}_{n}\left(x\right)=\sum _{k=0}^{n}{d}_{k}^{0}{x}^{k}.$$

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

#### Error Estimation

## 3. Applications

#### 3.1. Application 1

#### 3.2. Application 2

#### 3.3. Application 3

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The absolute maximum error using Polynomial Least Squares Method (PLSM) for Application 3.

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**MDPI and ACS Style**

Căruntu, B.; Bota, C.; Lăpădat, M.; Paşca, M.S.
Polynomial Least Squares Method for Fractional Lane–Emden Equations. *Symmetry* **2019**, *11*, 479.
https://doi.org/10.3390/sym11040479

**AMA Style**

Căruntu B, Bota C, Lăpădat M, Paşca MS.
Polynomial Least Squares Method for Fractional Lane–Emden Equations. *Symmetry*. 2019; 11(4):479.
https://doi.org/10.3390/sym11040479

**Chicago/Turabian Style**

Căruntu, Bogdan, Constantin Bota, Marioara Lăpădat, and Mădălina Sofia Paşca.
2019. "Polynomial Least Squares Method for Fractional Lane–Emden Equations" *Symmetry* 11, no. 4: 479.
https://doi.org/10.3390/sym11040479