# Solving Higher-Order Boundary and Initial Value Problems via Chebyshev–Spectral Method: Application in Elastic Foundation

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## Abstract

**:**

## 1. Introduction

## 2. Some Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

**Remark**

**1.**

**Proposition**

**1.**

## 3. Outline of the Method for Boundary and Initial Value Problems

#### 3.1. Operational Matrix of Derivative

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.2. Description of the Proposed Method for BVPs and IVPs

Algorithm 1: The construction of proposed method for the boundary value problems (BVPs) |

Step 1. Input:$N,n,{c}_{{}_{N-i}},{a}_{i},{l}_{j},(i=0,\dots ,n,j=0,\dots ,n-1)$; $R\left(x\right)$; Step 2. Compute: ${b}_{k}=\frac{2{c}_{k}}{\pi}{\int}_{-1}^{1}R\left(x\right){T}_{k}\left(x\right){(1-{x}^{2})}^{-\frac{1}{2}}dx$;for $k=0,\dots ,N-n$. Step 3. Compute the matrix ${\mathbf{E}}_{N}$ from matrix $\mathbf{E}$.Step 4. Compute:4.1. ${\mathbf{E}}_{N}^{i}$ from step 3, (i=2, …, n).4.2. ${\mathbf{X}}_{n}$ based on $\mathbf{X}$ from (22).Step 5. Set:
$$\mathbf{B}=\left[\begin{array}{cccc}{\beta}_{00}& {\beta}_{01}& \dots & {\beta}_{0,n-1}\\ {\beta}_{10}& {\beta}_{11}& \dots & {\beta}_{1,n-1}\\ \vdots & \vdots & \vdots & \vdots \\ {\beta}_{n-1,0}& {\beta}_{n-1,1}& \dots & {\beta}_{n-1,n-1}\end{array}\right]\left[\begin{array}{cccc}{T}_{0}(-1)& {T}_{1}(-1)& \dots & {T}_{{}_{N}}(-1)\\ 0& {T}_{1}^{{}^{\prime}}\left(1\right)& \dots & {T}_{{}_{N}}^{{}^{\prime}}\left(1\right)\\ \vdots & \vdots & \vdots & \vdots \\ 0& \dots & \dots & {T}_{{}_{N}}^{(n-1)}\left(1\right)\end{array}\right].$$
Step 6. Obtain the solution ${\mathbf{V}}_{N}$ from the system of (24) and set: ${u}_{N}\left(x\right)={\mathbf{T}}_{N,x}{\mathbf{V}}_{N}^{T}$. |

#### Construction of the Proposed Method for IVPs

**Remark**

**2.**

## 4. Numerical Results

**Example 1.**Consider the third-order boundary value problem

**Example 2.**Consider the first-order initial value problem:

**Example 3.**Consider an engineering problem of the fourth-order initial value problem which is arising in elastic foundation [20] as follows,

**Example 4.**Consider the following fifth-order boundary value problem,

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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N | Maximal Errors |
---|---|

Presented Method | |

7 | $9.12\times {10}^{-2}$ |

8 | $5.74\times {10}^{-3}$ |

10 | $1.22\times {10}^{-4}$ |

12 | $1.58\times {10}^{-6}$ |

15 | $1.49\times {10}^{-9}$ |

N | Maximal Errors | ||
---|---|---|---|

Presented Method | Method in [3] | Method in [9] | |

6 | $1.17\times {10}^{-2}$ | $1.45\times {10}^{-2}$ | $2.7\times {10}^{-2}$ |

8 | $9.09\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $3.8\times {10}^{-3}$ |

11 | $4.48\times {10}^{-5}$ | $2.03\times {10}^{-4}$ | $4.11\times {10}^{-4}$ |

15 | $1.66\times {10}^{-8}$ | $-$ | $-$ |

18 | $4.93\times {10}^{-12}$ | $-$ | $-$ |

N | Maximal Errors | |
---|---|---|

Exp. 3 | Exp. 4 | |

11 | $1.54\times {10}^{-9}$ | $2.63\times {10}^{-3}$ |

13 | $5.85\times {10}^{-14}$ | $4.11\times {10}^{-5}$ |

15 | $3.64\times {10}^{-15}$ | $5.15\times {10}^{-7}$ |

17 | $1.27\times {10}^{-16}$ | $5.01\times {10}^{-9}$ |

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**MDPI and ACS Style**

Agarwal, P.; Attary, M.; Maghasedi, M.; Kumam, P.
Solving Higher-Order Boundary and Initial Value Problems via Chebyshev–Spectral Method: Application in Elastic Foundation. *Symmetry* **2020**, *12*, 987.
https://doi.org/10.3390/sym12060987

**AMA Style**

Agarwal P, Attary M, Maghasedi M, Kumam P.
Solving Higher-Order Boundary and Initial Value Problems via Chebyshev–Spectral Method: Application in Elastic Foundation. *Symmetry*. 2020; 12(6):987.
https://doi.org/10.3390/sym12060987

**Chicago/Turabian Style**

Agarwal, Praveen, Maryam Attary, Mohammad Maghasedi, and Poom Kumam.
2020. "Solving Higher-Order Boundary and Initial Value Problems via Chebyshev–Spectral Method: Application in Elastic Foundation" *Symmetry* 12, no. 6: 987.
https://doi.org/10.3390/sym12060987