# On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. Operator Method for Solving VFIDE

**Theorem**

**1.**

**Proof.**

## 3. The Two Phase Nonlocal Integral Euler–Bernoulli Beam Model

## 4. Formulation and Solution of the Problem

#### 4.1. Solution of DBVP

#### 4.2. Solution of FIDBVP

#### 4.3. Algorithm

Algorithm 1 Algorithm for solving the fourth order boundary value problem (23), (25) with the Helmholtz type kernel (22). |

input $L,I,E,\tau ,{\xi}_{1},q\left(x\right)$ compute ${\xi}_{2}=1-{\xi}_{1}$ $g\left(x\right)=-\frac{{\xi}_{2}}{2\tau {\xi}_{1}}{e}^{\frac{x}{\tau}}$ $Q\left(s\right)=\frac{{\xi}_{1}({\tau}^{2}{s}^{2}-1)}{{s}^{2}({\xi}_{1}{\tau}^{2}{s}^{2}-1)}$ $G\left(s\right)=\mathcal{L}\left\{g\right(x\left)\right\}$ $\widehat{g}\left(x\right)={\mathcal{L}}^{-1}\left\{G\left(s\right)Q\left(s\right)\right\}$ ${D}^{-1}g\left(x\right)=\widehat{g}\left(x\right)-\frac{x}{L}\widehat{g}\left(L\right)$ $A{D}^{-1}g\left(x\right)=\frac{{d}^{2}}{d{x}^{2}}\left({D}^{-1}g\left(x\right)\right)$ $\mathsf{\Psi}\left(A{D}^{-1}g\left(x\right)\right)={\int}_{0}^{L}{e}^{-\frac{t}{\tau}}A{D}^{-1}g\left(t\right)dt$ $W=1-\mathsf{\Psi}\left(A{D}^{-1}g\left(x\right)\right)$ if $detW\ne 0$ compute $f\left(x\right)=\frac{1}{EI{\xi}_{1}}\left({\int}_{0}^{x}(x-t)q\left(t\right)dt-\frac{x}{L}{\int}_{0}^{L}(L-t)q\left(t\right)dt\right)$ $F\left(s\right)=\mathcal{L}\left\{f\right(x\left)\right\}$ $\widehat{f}\left(x\right)={\mathcal{L}}^{-1}\left\{F\left(s\right)Q\left(s\right)\right\}$ ${D}^{-1}f\left(x\right)=\widehat{f}\left(x\right)-\frac{x}{L}\widehat{f}\left(L\right)$ $A{D}^{-1}f\left(x\right)=\frac{{d}^{2}}{d{x}^{2}}\left({D}^{-1}f\left(x\right)\right)$ $\mathsf{\Psi}\left(A{D}^{-1}f\left(x\right)\right)={\int}_{0}^{L}{e}^{-\frac{t}{\tau}}A{D}^{-1}f\left(t\right)dt$ $w\left(x\right)={D}^{-1}f\left(x\right)+{D}^{-1}g\left(x\right){W}^{-1}\mathsf{\Psi}\left(A{D}^{-1}f\left(x\right)\right)$ print
$w\left(x\right)$else print ‘There is no unique solution’end |

## 5. Examples and Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Deflectionof simply supported beam subject to variable load and different values of $\tau $.

L (nm) | b (nm) | h (nm) | ${\mathit{q}}_{0}$ (nN/nm) | E (TPa) | $\mathit{\tau}={\mathit{e}}_{0}\mathit{a}$ (nm) | ${\mathit{\xi}}_{1}$ |
---|---|---|---|---|---|---|

10 | 1 | 1 | ${10}^{-4}$ | 5.5 | $[1.0,2.0]$ | $[0.1,1]$ |

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**MDPI and ACS Style**

Providas, E.
On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations. *AppliedMath* **2022**, *2*, 269-283.
https://doi.org/10.3390/appliedmath2020017

**AMA Style**

Providas E.
On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations. *AppliedMath*. 2022; 2(2):269-283.
https://doi.org/10.3390/appliedmath2020017

**Chicago/Turabian Style**

Providas, Efthimios.
2022. "On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations" *AppliedMath* 2, no. 2: 269-283.
https://doi.org/10.3390/appliedmath2020017