**9. Generalized solutions of Biot's equations by non-stationary forces**

Using the properties of Green tensor, we obtain generalized solutions of nonstationary Biot's equations under the action of arbitrary mass forces in the Biot's medium, which satisfy the radiation condition at infinity. They have the form of tensor functional convolution:

$$u\_j(\mathbf{x}, t) = U\_j^k(\mathbf{x}, t) \* F\_k(\mathbf{x}, t), \quad j, k = 1, \ldots, 2N \tag{28}$$

It's taken according to the rules of convolution of generalized functions depending on the type of mass forces [10].

In order to get the classic solution, we must present formulas (28) in regular integral forms. For this, let us present matrix of Green tensor as sum of regular functions and singular functions, which contain delta-function:

$$U(\mathbf{x}, t) = U\_{\text{reg}}(\mathbf{x}, t) + U\_{\text{sing}}(t)\delta(\mathbf{x}).$$

Then also write:

$$u(\mathbf{x},t) = u\mathbf{1}(\mathbf{x},t) + u\mathbf{2}(\mathbf{x},t) \tag{29}$$

Here *u*1ð Þ *x*, *t* is representable by regular mass forces in the integral form:

$$u\mathbf{1}(\mathbf{x},t) = H(t)\int\_{\boldsymbol{\varrho}}^{t}d\tau \int\_{\mathbb{R}^{N}} U\_{\text{reg}}(\mathbf{x}-\boldsymbol{\jmath},\boldsymbol{\tau}) \times \begin{pmatrix} F\_{\boldsymbol{s}}(\boldsymbol{\jmath},t-\boldsymbol{\tau}) \\ F\_{\boldsymbol{f}}(\boldsymbol{\jmath},t-\boldsymbol{\tau}) \end{pmatrix} \,d\boldsymbol{\jmath}$$

The convolution with singular part is equal to:

$$\mu \mathcal{Q}(\mathbf{x}, t) = H(t) \int\_{\boldsymbol{\sigma}}^{t} U\_{\text{sing}}(\boldsymbol{\tau}) \times \begin{pmatrix} F\_s(\mathbf{x}, t - \boldsymbol{\tau}) \\ F\_f(\mathbf{x}, t - \boldsymbol{\tau}) \end{pmatrix} d\boldsymbol{\tau}$$

In 3D space, there are convolutions with simple layers on sound cones (see (27)). To construct their integral presentation, use this rule:

$$\begin{aligned} &a(\boldsymbol{x},t)\delta(c\_{m}t-\boldsymbol{r})\*F(\boldsymbol{x},t)=\\ &\boldsymbol{x}=H(t)\int\_{0}^{t}d\boldsymbol{\tau}\int\_{||\boldsymbol{y}-\boldsymbol{x}||=c\_{m}\boldsymbol{\tau}}a(\boldsymbol{x}-\boldsymbol{y},\boldsymbol{\tau})F\Big(\boldsymbol{y},t-\frac{||\boldsymbol{y}-\boldsymbol{x}||}{c\_{m}}\Big)d\boldsymbol{S}(\boldsymbol{y}),\end{aligned}$$

Here the internal integral is taken over sphere with center in the point *x*, and its radius is equal to *cmτ*.

*for k* ¼ *N* þ 1, … , 2*N*

problems (4).

**11. Conclusion**

using boundary element method.

**Acknowledgements**

**Author details**

**35**

Lyudmila Alexeyeva<sup>1</sup>

*∂ jU<sup>k</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92064*

*∂ jU<sup>k</sup>*

the pressure in the liquid of Biot's medium.

*j* ¼ 1, … , *N:*

*<sup>j</sup>* ¼ �*D*1ð Þ *<sup>d</sup>*31*∂<sup>k</sup>*�*<sup>N</sup>***Φ**<sup>01</sup> � *<sup>d</sup>*32*∂<sup>k</sup>*�*<sup>N</sup>***Φ**<sup>02</sup>

*Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot's Medium*

Substituting these formulas in (30), we define the stresses in the skeleton and

The obtained solutions give possibility to study the dynamics of porous waterand gas-saturated media and rods under actions of disturbance sources of different forms and can be used for solutions of boundary value problems in porous media by

These solutions can be used for describing wave processes by explosions and earthquakes. In these cases mass forces are described by using singular generalized

This work was financially supported by the Ministry of Education and Science of

function, such as multipoles, simple and double layers, and others.

\* and Yergali Kurmanov1,2

2 Al Farabi Kazakh National University, Almaty, Kazakhstan

\*Address all correspondence to: alexeeva@math.kz

provided the original work is properly cited.

1 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

the Republic of Kazakhstan (Grant AP05132272).

If we paste Φ02ð Þ *x*, *ω* instead of **Φ**02ð Þ *x*, *t* in formulas of this theorem, then formula (30) expresses complex amplitudes of stress tensor and pressure by periodic oscillations. It is used to determine stresses and pressure by solving the periodic

*<sup>j</sup>*þ*<sup>N</sup>* ¼ �*D*1ð Þ *bs*1*∂<sup>k</sup>*�*<sup>N</sup>***Φ**<sup>01</sup> � *bs*2*∂<sup>k</sup>*�*<sup>N</sup>***Φ**<sup>02</sup>

If components of acting forces *F x*ð Þ , *t* are double differentiable vector function, it is convenient to use the property of differentiation of convolution [10]:

$$\frac{\partial^2 \Phi\_{2m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} \ast F(\mathbf{x}, t) = \frac{\partial^2}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} (\Phi\_{2m} \ast F(\mathbf{x}, t)) = \Phi\_{2m}(\mathbf{x}, t) \ast \frac{\partial^2 F}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k}$$

Substituting the formulas of Theorem 4 into (29), we obtain displacements and stresses of skeleton and liquid in Biot's medium in spaces of dimension *N* = 1, 2, 3. Calculation of these convolutions by using these formulas essentially depends on the form of acting forces and gives possibility to construct regular presentation of generalized solution for wide class of acting forces, which are the classic solution of Biot's equation.
