# Modeling of Strength Characteristics of Polymer Concrete Via the Wave Equation with a Fractional Derivative

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}(see [20]).

## 3. Results

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 4. Discussion

- -
- The solution to problems (1)–(4) is presented.
- -
- The first seven eigenvalues of problems (10)–(11) are found in the case α = 1.47; c = 1.8; $\mathfrak{X}=1$, which gives us an opportunity to model the deformation-strength characteristics of polymer concrete (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under the influence of the gravity force, with an accuracy of two decimal places.
- -
- The functions from the system ${\left\{{\tilde{X}}_{m}\left(x\right)\right\}}_{m=1;2;\dots}$, which is biorthogonal to the system of eigenfunctions ${\left\{{X}_{m}\left(x\right)\right\}}_{m=1;2;\dots}$ of problems (10)–(11), in the case α = 1.47; c = 1.8; $\mathfrak{X}=1$, are found numerically and their graphs are plotted.
- -
- The inner products of the eigenfunctions ${\left\{{X}_{m}\left(x\right)\right\}}_{m=1;2;\dots}$ of problems (10)–(11) and functions from the biorthogonal system ${\left\{{\tilde{X}}_{m}\left(x\right)\right\}}_{m=1;2;\dots},$ in the case α = 1.47; c = 1.8; $\mathfrak{X}=1$, are calculated and the obtained result confirms the correctness of replacing series (13) and (17) with partial sums in the calculations.
- -
- Four numerical examples of the application of the solution to problems (1)–(4) to modeling changes in the deformation-strength characteristics of polymer concrete (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under the influence of the gravity force are considered.
- -
- The rate of decrease in terms (14) corresponding to the considered examples is obtained:$$\left|{X}_{m}\left(x\right){T}_{m}\left(t\right)\right|\le C\xb7{m}^{\gamma},$$$$0<C<0.3;\text{}-3\gamma -1.5$$
- -
- In the considered-above examples, we have established that the seventh (last) term contributes to the sum from 0.5% to 6%.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The first four functions, of the system ${\left\{{X}_{m}\left(x\right)\right\}}_{m=1;2;\dots}$, (at the top) and ${\left\{{\tilde{X}}_{m}\left(x\right)\right\}}_{m=1;2;\dots}$ (at the bottom), corresponding to the case α = 1.47 and c = 1.8.

**Figure 2.**The graph of the approximate solution ${u}_{\left(7\right)}\left(x,t\right)$ of problems (1)–(4) under the imposed conditions in Example 1.

**Figure 3.**A graph of the approximate solution ${u}_{\left(7\right)}\left(x,t\right)$ of problems (1)–(4) under the imposed conditions in Example 2.

**Figure 4.**A graph of the approximate solution ${u}_{\left(7\right)}\left(x,t\right)$ of problems (1)–(4) under the imposed conditions in Example 3.

**Figure 5.**A graph of the approximate solution ${u}_{\left(7\right)}\left(x,t\right)$ of problems (1)–(4) under the imposed conditions in Example 4.

**Table 1.**The first seven eigenvalues of boundary value problems (10)–(11) for α = 1.47, c = 1.8 and $\mathfrak{X}=1$.

λ_{1} | λ_{2} | λ_{3} | λ_{4} | λ_{5} | λ_{6} | λ_{7} |
---|---|---|---|---|---|---|

−16.51 | −59.49 | −125.13 | −213.33 | −323.27 | −455.09 | −607.31 |

**Table 2.**Inner product of the first seven eigenfunctions of boundary value problems (10)–(11) and functions of the biorthogonal system; α = 1.47, c = 1.8 and $\mathfrak{X}=1$.

$\langle {\tilde{\mathit{X}}}_{\mathit{k}},{\mathit{X}}_{\mathit{m}}\rangle $ | ${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | ${\mathit{X}}_{3}$ | ${\mathit{X}}_{4}$ | ${\mathit{X}}_{5}$ | ${\mathit{X}}_{6}$ | ${\mathit{X}}_{7}$ |
---|---|---|---|---|---|---|---|

${\tilde{X}}_{1}$ | 0.01046 | 0 | 0 | 0 | 0 | 0 | 0 |

${\tilde{X}}_{2}$ | 0 | −0.00213 | 0 | 0 | 0 | 0 | 0 |

${\tilde{X}}_{3}$ | 0 | 0 | 0.00076 | 0 | 0 | 0 | 0 |

${\tilde{X}}_{4}$ | 0 | 0 | 0 | −0.00035 | 0 | 0 | 0 |

${\tilde{X}}_{5}$ | 0 | 0 | 0 | 0 | 0.00018 | 0 | 0 |

${\tilde{X}}_{6}$ | 0 | 0 | 0 | 0 | 0 | −0.00011 | 0 |

${\tilde{X}}_{7}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0.00007 |

**Table 3.**The values of the coefficients ${A}_{m}$ of the solution ${u}_{\left(7\right)}\left(x,t\right)$ (see Example 1).

$\mathit{m}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${A}_{m}$ | 0.20862 | 0.05574 | −0.01698 | 0.01228 | −0.00812 | 0.00586 | −0.00503 |

**Table 4.**The values of coefficients ${A}_{m}\text{}$ and ${B}_{m}$ of the solution ${u}_{\left(7\right)}\left(x,t\right)$ (see Example 2).

$\mathit{m}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${A}_{m}$ | 0.14149 | −0.13295 | 0.07009 | −0.04226 | 0.02925 | −0.02173 | 0.01719 |

**Table 5.**The values of the coefficients ${A}_{m}$ and ${B}_{m}$ of the solution ${u}_{\left(7\right)}\left(x,t\right)$ (see Example 3).

$\mathit{m}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${A}_{m}$ | 0.06758 | 0.05946 | 0.01514 | 0.00500 | 0.00236 | 0.00120 | 0.00049 |

${B}_{m}$ | 1.81104 | −0.70356 | 0.38444 | −0.33961 | 0.23810 | −0.24218 | 0.18670 |

**Table 6.**The values of the coefficients ${A}_{m}$ and ${B}_{m}$ of the solution ${u}_{\left(7\right)}\left(x,t\right)$ (see Example 4).

$\mathit{m}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${A}_{m}$ | 0.14149 | −0.13295 | 0.07009 | −0.04226 | 0.02925 | −0.02173 | 0.01719 |

${B}_{m}$ | 0.57490 | −1.02545 | 0.78405 | −0.61730 | 0.52588 | −0.46347 | 0.42359 |

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Kirianova, L.
Modeling of Strength Characteristics of Polymer Concrete Via the Wave Equation with a Fractional Derivative. *Mathematics* **2020**, *8*, 1843.
https://doi.org/10.3390/math8101843

**AMA Style**

Kirianova L.
Modeling of Strength Characteristics of Polymer Concrete Via the Wave Equation with a Fractional Derivative. *Mathematics*. 2020; 8(10):1843.
https://doi.org/10.3390/math8101843

**Chicago/Turabian Style**

Kirianova, Ludmila.
2020. "Modeling of Strength Characteristics of Polymer Concrete Via the Wave Equation with a Fractional Derivative" *Mathematics* 8, no. 10: 1843.
https://doi.org/10.3390/math8101843