14. Dynamic analog of the Somigliana formula in supersonic case

For regularization of Eq. (61), we put W,z instead of U in the second term and use the property of differentiation of convolution:

<sup>r</sup>c<sup>2</sup>buk <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> <sup>k</sup>∗PjδSð Þx θð Þþ z W cj k , iz<sup>∗</sup> <sup>λ</sup>umnmδij <sup>þ</sup> <sup>μ</sup> uinj <sup>þ</sup> ujni � � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þ¼ <sup>z</sup> <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> <sup>k</sup>∗PjδSð Þx θð Þþz W ↼ j k , i <sup>∗</sup> <sup>λ</sup>um, znmδij <sup>þ</sup> <sup>μ</sup> ui, znj <sup>þ</sup> uj, zni � � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þþ<sup>z</sup> þW cj k, i <sup>∗</sup> <sup>λ</sup>umnmδij <sup>þ</sup> <sup>μ</sup> uinj <sup>þ</sup> ujni � � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>δ</sup>ð Þ¼ <sup>z</sup> <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> <sup>k</sup>∗PjδSð Þx θð Þþ z W cj <sup>k</sup>, <sup>m</sup><sup>∗</sup>Cil jmui, znlð Þ<sup>x</sup> <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þþ <sup>z</sup> <sup>C</sup>il jmW cj <sup>k</sup>, <sup>m</sup> ∗ <sup>x</sup> uið Þ <sup>x</sup>:<sup>0</sup> nlð Þ<sup>x</sup> <sup>δ</sup>Sð Þ<sup>x</sup> (63)

From here on, we use Eq. (57) we get the formula which can be written in integral form.

Theorem 14.1. The generalized solution of supersonic transport BVP can be presented in the form:

$$
\widehat{\boldsymbol{\mu}}\_{k} = \widehat{\boldsymbol{\mathcal{U}}}\_{k}^{\boldsymbol{j}} \ast \boldsymbol{p}\_{\boldsymbol{j}} \delta\_{\boldsymbol{S}}(\mathbf{x}) \boldsymbol{\theta}(\mathbf{z}) + \widehat{\boldsymbol{\mathcal{C}}}\_{\operatorname{jm}}^{\boldsymbol{i}l} \widehat{\boldsymbol{W}}\_{k^{\*}m}^{\boldsymbol{j}} \ast \boldsymbol{u}\_{l,z} \boldsymbol{n}\_{l}(\mathbf{x}) \boldsymbol{\delta}\_{\boldsymbol{S}}(\mathbf{x}) \boldsymbol{\theta}(\mathbf{z}) \tag{64}
$$

Note that

completed.

on D.

that

z�ð mkr

0

<sup>¼</sup> <sup>H</sup>js

fronts of the form <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup>

ments of boundary surface.

Hjs

ikð Þ <sup>x</sup> � <sup>y</sup>; n yð Þ uj, <sup>τ</sup>ð Þ <sup>y</sup>; <sup>τ</sup> <sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>H</sup>js

In virtue of this equity, we get from Eq. (66) the last formula of the theorem.

All integrals exist; indeed, the integrands are integrable everywhere, including the fronts of fundamental solutions, because the kernels of the integrands have weak singularities on the

This formula is a dynamic analog of Somigliana formula for supersonic loads. It defines the displacement in elastic medium by using boundary values of stresses and velocity of displace-

This formula also preserves its form for ð Þ x; z ∈ D with regard to the definition of HS

15. Singular boundary integral equations of supersonic transport BVP

� <sup>&</sup>lt; C xk k � <sup>y</sup> <sup>β</sup>

z�ð mkr

0

<sup>i</sup> ð Þ x � y; z; n yð Þ ujð Þ y; z � mkr dS yð Þ, r ¼ k k x � y

Uj i

> o dτ�

n

, x, y ∈S:

ð Þ x � y; z � τ pj

Тheorem15.1. If the classical solution of BVP satisfies the Holder's conditions at D+, i.e.,

θð Þ z � mkr dS yð Þ

<sup>i</sup> ð Þ x � y; z � τ; n yð Þ uj, <sup>z</sup>ð Þ y; τ

Proof. The desired assertion follows from Theorem 14.1 and Theorem 11.1 for tensor H by analogy of the proof of Theorem 12.1 about singular boundary integral equations in the

ujð Þ� <sup>x</sup>; <sup>z</sup> ujð Þ <sup>y</sup>; <sup>z</sup> � � �

then it satisfies the singular boundary integral equation at D+

2

ð

Sk <sup>z</sup> x<sup>0</sup> ð Þ

�Hjd

Hjs

subsonic case. Full proofs of these theorems can be found in [7].

k¼1

ð

Sk <sup>z</sup> ð Þx

<sup>0</sup>, <sup>5</sup>uið Þ¼ <sup>x</sup>; <sup>z</sup> <sup>X</sup>

�V:P:

<sup>j</sup>∥x∥<sup>2</sup> � ��1=<sup>2</sup> in virtue of the properties of kernels <sup>U</sup> and <sup>H</sup>. The proof is

ikð ÞÞ x � y; n yð Þ ujð Þ y; z � mkr

ikð ÞÞð x � y; n yð Þ ujð Þ� y; z � mkr ujð Þ¼ y; 0

http://dx.doi.org/10.5772/intechopen.74538

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

�ð Þx θð Þz

149

∃

ð Þ� y; τ

C > 0, β > 0

which for x∉S has the next integral presentation

$$u\_{l}H\_{S}^{-}(\mathbf{x})\theta(\mathbf{z}) = \sum\_{k=1}^{2} \left[\theta(\mathbf{z} - m\_{l}r)dS(\mathbf{y})\right] \int\_{0}^{\mathbf{z} - m\_{l}r} \left\{\mathcal{U}\_{l}^{j}(\mathbf{x} - \mathbf{y}, \mathbf{z} - \mathbf{\tau})\right\} p\_{j}(\mathbf{y}, \mathbf{\tau}) - \tag{6.105}$$

$$-H\_{i}^{jd}(\mathbf{x} - \mathbf{y}, \mathbf{z} - \mathbf{\tau}, n(\mathbf{y}))u\_{l^{\*}z}(\mathbf{y}, \mathbf{\tau})\right\} d\mathbf{\tau} - \int\_{\mathbf{S}} H\_{i}^{j\epsilon}(\mathbf{x} - \mathbf{y}, z, n(\mathbf{y}))u\_{l}(\mathbf{y}, z - m\_{l}r)dS(\mathbf{y}) \tag{6.106}$$

$$\dots \tag{65}$$

r ¼ k k x � y

Proof. Formula (65) follows from Eq. (64) in virtue of Eqs. (61) and (32). Its integral form is

$$\mu\_i H\_S^-(\mathbf{x})\theta(z) = \int\_0^z \left\{ \mathcal{U}\_i^\dagger(\mathbf{x}, y, z - \tau) \, p\_j(y, \tau) - H\_i^\dagger(\mathbf{x}, y, z - \tau, n(y)) u\_{j \cdot z}(y, \tau) \right\} d\tau$$

If we use Eq. (55) for H<sup>i</sup> j ð Þ <sup>x</sup>; <sup>z</sup> as the support of <sup>H</sup>is <sup>j</sup> ð Þ <sup>x</sup>; <sup>z</sup> , Hid <sup>j</sup> ð Þ x; z , we get

$$\begin{split} u\_i H\_S^-(\mathbf{x}) \theta(\mathbf{z}) &= \sum\_{k=1}^2 \int\_S \theta(\mathbf{z} - m\_k r) dS(\mathbf{y}) \int\_0^{\mathbf{z} - m\_k r} \{ \mathcal{U}\_i^j(\mathbf{x} - \mathbf{y}, z - \tau) \} p\_j(\mathbf{y}, \tau) - \\ & \quad - (H\_{ik}^{jd}(\mathbf{x} - \mathbf{y}, z - \tau, n(\mathbf{y}) + H\_{ik}^{js}(\mathbf{x} - \mathbf{y}, n(\mathbf{y}))) u\_{j \cdot z}(\mathbf{y}, \tau) \} d\tau \end{split} \tag{66}$$

Note that

14. Dynamic analog of the Somigliana formula in supersonic case

cj k

� � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>δ</sup>ð Þ¼ <sup>z</sup>

of differentiation of convolution:

148 Differential Equations - Theory and Current Research

<sup>k</sup>∗PjδSð Þx θð Þþ z W

<sup>∗</sup> <sup>λ</sup>umnmδij <sup>þ</sup> <sup>μ</sup> uinj <sup>þ</sup> ujni

<sup>b</sup>uk <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

which for x∉S has the next integral presentation

ð

S

2

k¼1

�Hjd

<sup>S</sup> ð Þx θð Þ¼ z

ðz

n

2

ð

S

� ðHjd

k¼1

Uj i

ð Þ x; y; z � τ pj

ð Þ <sup>x</sup>; <sup>z</sup> as the support of <sup>H</sup>is

θð Þ z � mkr dS yð Þ

ik <sup>ð</sup><sup>x</sup> � <sup>y</sup>; <sup>z</sup> � <sup>τ</sup>; n yð Þþ <sup>H</sup>js

0

j

<sup>S</sup> ð Þ<sup>x</sup> <sup>θ</sup>ð Þ¼ <sup>z</sup> <sup>X</sup>

<sup>k</sup>∗pj

θð Þ z � mkr dS yð Þ

<sup>i</sup> ðx � y; z � τ; n yð ÞÞuj, <sup>z</sup>ðy; τÞ

↼ j k , i

cj <sup>k</sup>, <sup>m</sup><sup>∗</sup>Cil

<sup>k</sup>∗PjδSð Þx θð Þþz W

<sup>k</sup>∗PjδSð Þx θð Þþ z W

<sup>r</sup>c<sup>2</sup>buk <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

<sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

þW cj k, i

<sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

uiH�

<sup>S</sup> ð Þ<sup>x</sup> <sup>θ</sup>ð Þ¼ <sup>z</sup> <sup>X</sup>

uiH�

If we use Eq. (55) for H<sup>i</sup>

uiH�

For regularization of Eq. (61), we put W,z instead of U in the second term and use the property

<sup>∗</sup> <sup>λ</sup>um, znmδij <sup>þ</sup> <sup>μ</sup> ui, znj <sup>þ</sup> uj, zni

jmui, znlð Þ<sup>x</sup> <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þþ <sup>z</sup> <sup>C</sup>il

From here on, we use Eq. (57) we get the formula which can be written in integral form.

<sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þþ <sup>z</sup> <sup>C</sup>~il

z�ð mk r

0

Uj i

n

Theorem 14.1. The generalized solution of supersonic transport BVP can be presented in the form:

jmW cj

> o dτ � ð

r ¼ k k x � y

ð Þ� <sup>y</sup>; <sup>τ</sup> <sup>H</sup><sup>j</sup>

z�ð mkr

0

i

<sup>j</sup> ð Þ <sup>x</sup>; <sup>z</sup> , Hid

n

Uj i

Proof. Formula (65) follows from Eq. (64) in virtue of Eqs. (61) and (32). Its integral form is

ð Þ x � y; z � τ pj

S Hjs

, iz<sup>∗</sup> <sup>λ</sup>umnmδij <sup>þ</sup> <sup>μ</sup> uinj <sup>þ</sup> ujni

� � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þþ<sup>z</sup>

� � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þ¼ <sup>z</sup>

jmW cj <sup>k</sup>, <sup>m</sup> ∗

ðy; τÞ�

ð Þ x; y; z � τ; n yð Þ uj, <sup>z</sup>ð Þ y; τ

<sup>j</sup> ð Þ x; z , we get

ð Þ x � y; z � τ pj

ikðx � y; n yð ÞÞÞuj, <sup>z</sup>ðy; τÞ

<sup>x</sup> uið Þ <sup>x</sup>:<sup>0</sup> nlð Þ<sup>x</sup> <sup>δ</sup>Sð Þ<sup>x</sup>

<sup>k</sup>, <sup>m</sup><sup>∗</sup>ui, znlð Þ<sup>x</sup> <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þ<sup>z</sup> (64)

<sup>i</sup> ð Þ x � y; z; n yð Þ ujð Þ y; z � mkr dS yð Þ

ðy; τÞ�

o dτ

o dτ (63)

(65)

(66)

$$\begin{aligned} &\int\_0^{z-m\_{k'}} H^{\dot{s}}\_{ik}(x-y,n(y))u\_{j\cdot\tau}(y,\tau)d\tau = H^{\dot{s}}\_{ik}(x-y,n(y))(u\_j(y,z-m\_kr)-u\_j(y,0)=0) \\ &= H^{\dot{s}}\_{ik}(x-y,n(y))(u\_j(y,z-m\_kr)) \end{aligned}$$

In virtue of this equity, we get from Eq. (66) the last formula of the theorem.

All integrals exist; indeed, the integrands are integrable everywhere, including the fronts of fundamental solutions, because the kernels of the integrands have weak singularities on the fronts of the form <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> <sup>j</sup>∥x∥<sup>2</sup> � ��1=<sup>2</sup> in virtue of the properties of kernels <sup>U</sup> and <sup>H</sup>. The proof is completed.

This formula is a dynamic analog of Somigliana formula for supersonic loads. It defines the displacement in elastic medium by using boundary values of stresses and velocity of displacements of boundary surface.

This formula also preserves its form for ð Þ x; z ∈ D with regard to the definition of HS �ð Þx θð Þz on D.
