9. Singular boundary integral equations of subsonic transport BVP

The following assertion provides a solution for the aforementioned boundary value problems.

Theorem 9.1. If the solution u(x;z) of subsonic transport BVP satisfies the Holder condition on D; namely,

$$\|\|u\_j(\mathbf{x}, z) - u\_j(y, t)\|\| \le \mathbb{C} \|(\mathbf{x}, z) - (y, t)\|\|^\beta, \ x \in \mathbb{S}, \ y \in \mathbb{S}\_\beta$$

then u(x;z) satisfies the singular boundary integral equation

$$\begin{aligned} 0, \mathsf{5}u\_i(\mathbf{x}, z) &= \widehat{\mathsf{g}}\_j^{\ast} \widehat{\mathsf{U}}\_i^j(\mathbf{x}, z) + \int\_D \mathsf{U}\_i^j(\mathbf{x}, y, z, \mathsf{τ}) \, p\_j(y, \mathsf{τ}) dD(y, \mathsf{τ}) - \\ &- V.P.\int\_D T\_i^j(\mathbf{x}, y, z, \mathsf{τ}, n(y, \mathsf{τ})) u\_j(y, \mathsf{τ}) dD(y, \mathsf{τ}) - i, j = 1, 2, 3 \end{aligned} \tag{45}$$

By transposing the last two terms to the left-hand side of the relation, we obtain the formula of

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

This theorem gives us resolving system of integral equations for defining unknown values of

Note also that the subsonic analog of the Somigliana formula was obtained for generalized functions. But since they are regular, from the Dubois-Reymond lemma ([3]: 97), the solution is classical. However, if the acting loads are described by singular generalized functions, which often takes place in physical problems, then one should use a representation of a generalized solution in the convolution form (Eq. (43)) with the evaluation of convolutions by the defini-

10. Supersonic green's tensor and its antiderivative with respect to z

θ1 V1 � θ2 V2 � �

θ1 V1 � <sup>θ</sup><sup>2</sup> V2 � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> <sup>j</sup> <sup>∥</sup>x∥<sup>2</sup> <sup>q</sup>

U xð Þ ; <sup>z</sup> <sup>∈</sup>f g <sup>z</sup> <sup>&</sup>gt; <sup>0</sup> , U<sup>k</sup>

One can readily see that its components are zero outside the sonic cones:

Kl

For solution of supersonic problems, we introduce the tensor W

� <sup>x</sup><sup>2</sup> 2 r<sup>4</sup>M<sup>2</sup> 2

� <sup>x</sup><sup>2</sup> 1 r<sup>4</sup>M<sup>2</sup> 2

> θ1 V1 � <sup>θ</sup><sup>2</sup> V2 � �

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mj <sup>2</sup> � <sup>1</sup>:

<sup>1</sup> have singularities of the type (z

ci j

þ ð Þ θ1V<sup>1</sup> � θ2V<sup>2</sup> � �, <sup>2</sup>πU<sup>3</sup>

> <sup>2</sup> ¼ � <sup>x</sup>2<sup>z</sup> r2

> > , mj ¼

<sup>þ</sup> ¼ f g ð Þ x; z : z > ml∥x∥ , l ¼ 1, 2:

<sup>i</sup> ! <sup>0</sup>, U<sup>k</sup> i 0

<sup>i</sup> in the supersonic case which has the

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

It satisfies the radiation condi-

<sup>2</sup> � <sup>m</sup><sup>2</sup> jr 2 ) �1/2 .

ð Þ x; z , which is the antideriv-

<sup>j</sup> ! 0 by x<sup>0</sup> ! ∞: (47)

(46)

143

ð Þ θ1V<sup>1</sup> � θ2V<sup>2</sup> ,

ð Þ θ1V<sup>1</sup> � θ2V<sup>2</sup> ,

<sup>3</sup> <sup>¼</sup> <sup>θ</sup><sup>1</sup> V1 þ θ2m<sup>2</sup> 2 V2 � �,

From Eq. (25), we get the regular representation of <sup>U</sup><sup>b</sup> <sup>j</sup>

<sup>1</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup> V2 þ z<sup>2</sup>x<sup>2</sup> 1 r<sup>4</sup>M<sup>2</sup> 2

<sup>2</sup> <sup>¼</sup> <sup>θ</sup><sup>2</sup> V<sup>2</sup> þ z<sup>2</sup>x<sup>2</sup> 2 r<sup>4</sup>M<sup>2</sup> 2

<sup>r</sup><sup>4</sup> <sup>z</sup><sup>2</sup> <sup>θ</sup><sup>1</sup> V1 � <sup>θ</sup><sup>2</sup> V2 � �

> θ1 V1 � <sup>θ</sup><sup>2</sup> V2 � �, <sup>2</sup>πU<sup>3</sup>

� ¼

2πU<sup>1</sup>

2πU<sup>2</sup>

<sup>1</sup> <sup>¼</sup> <sup>x</sup>1x<sup>2</sup>

<sup>1</sup> ¼ � <sup>x</sup>1<sup>z</sup> r2

> suppz

On the surfaces of the cones, the components U<sup>3</sup>

<sup>j</sup> with respect to z:

2πU<sup>2</sup>

2πU<sup>3</sup>

Here <sup>θ</sup><sup>j</sup> <sup>¼</sup> <sup>θ</sup> <sup>z</sup> � mj∥x<sup>∥</sup> � �, Vj

the theorem for the boundary points. The proof of the theorem is complete.

boundary displacements.

tion (see [3]: 133).

form

tions:

ative of <sup>U</sup><sup>b</sup> <sup>i</sup>

Proof. Let consider Eq. (45) for ð Þ <sup>x</sup>; <sup>z</sup> <sup>∈</sup> <sup>D</sup>�. Let (x<sup>∗</sup> ,z∗ ) ∈ D, x 0 ! (x<sup>∗</sup> ,z∗ ). Then, using Theorem 6.1, we have

lim ð Þ! x;z x∗;;z<sup>∗</sup> ð Þ uið Þ¼ <sup>x</sup>; <sup>z</sup> ui <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ¼ <sup>b</sup>gj ∗ Ub j i <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ lim ð Þ! <sup>x</sup>;<sup>z</sup> <sup>x</sup>∗;z<sup>∗</sup> ð Þ <sup>ð</sup> D Uj i ð Þ x; y; z; τ pj ð Þ y; τ dD yð Þ� ; τ � lim ð Þ! <sup>x</sup>;<sup>z</sup> <sup>x</sup>∗;z<sup>∗</sup> ð Þ <sup>ð</sup> D Tj i ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup>; <sup>z</sup><sup>∗</sup><sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup> <sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ lim ð Þ! <sup>x</sup>;<sup>z</sup> <sup>x</sup>∗;z<sup>∗</sup> ð Þ <sup>ð</sup> D Tj i ð Þ x; y; z; τ; n yð Þ ; τ dD yð Þ¼ ; τ <sup>¼</sup> <sup>b</sup>gj ∗ Ub j i <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ <sup>ð</sup> D Uj i ð Þ x; y; z; τ pj ð Þ y; τ dD yð Þ� ; τ �V:P: ð D Tj i ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup> ; z<sup>∗</sup> <sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup> <sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ lim ð Þ! x;z x∗;z<sup>∗</sup> ð Þ δ j <sup>i</sup> � ð D Uj i , <sup>z</sup>ð ÞÞ x; y; z; τ nzð Þy dDðy; τÞ 0 @ 1 A ¼ <sup>¼</sup> <sup>b</sup>gj ∗ Ub j i <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ <sup>ð</sup> D Uj i ð Þ x; y; z; τ pj ð Þ y; τ dD yð Þ� ; τ �V:P: ð D Tj i ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup>; <sup>z</sup><sup>∗</sup><sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup> <sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ<sup>δ</sup> j <sup>i</sup> <sup>¼</sup> <sup>b</sup>gj ∗ Ub j i <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ <sup>ð</sup> D Uj i ð Þ x; y; z; τ pj ð Þ y; τ dD yð Þ� ; τ �V:P: ð D Tj i ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ <sup>y</sup>; <sup>τ</sup> dD yð Þ� ; <sup>τ</sup> <sup>0</sup>, <sup>5</sup>ui <sup>x</sup><sup>∗</sup> ; <sup>z</sup><sup>∗</sup> ð Þþ ui <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ:

In the last relation, we have used the obvious properties: integrals with U<sup>j</sup> <sup>i</sup> exist by virtue of the Holder property of u on D and weak singularity U<sup>j</sup> <sup>i</sup> at D. Then if the surface integral exists, its value coincides with the principal value; the principal value of the integral containing the difference of integrated functions is equal to the difference of the principal values of integrals corresponding to each of these functions if they exist.

By transposing the last two terms to the left-hand side of the relation, we obtain the formula of the theorem for the boundary points. The proof of the theorem is complete.

then u(x;z) satisfies the singular boundary integral equation

∗ Ub j i ð Þþ x; z

> ∗ Ub j i

ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup>; <sup>z</sup><sup>∗</sup><sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup>

ð Þ x; y; z; τ; n yð Þ ; τ dD yð Þ¼ ; τ

ð Þ y; τ dD yð Þ� ; τ

; <sup>z</sup><sup>∗</sup><sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup>

, <sup>z</sup>ð ÞÞ x; y; z; τ nzð Þy dDðy; τÞ

ð Þ y; τ dD yð Þ� ; τ

ð Þ x; y; z; τ pj

ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ <sup>y</sup>; <sup>τ</sup> dD yð Þ� ; <sup>τ</sup> <sup>0</sup>, <sup>5</sup>ui <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ ui <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ:

value coincides with the principal value; the principal value of the integral containing the difference of integrated functions is equal to the difference of the principal values of integrals

ð

ð Þ x; y; z; τ pj

ð Þ x; y; z; τ; n yð Þ ; τ ujð Þ y; τ dD yð Þ� ; τ i, j ¼ 1, 2, 3

,z∗

<sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ lim

) ∈ D, x 0 ! (x<sup>∗</sup> ,z∗

ð

D Uj i

1 A ¼

ð Þ y; τ dD yð Þ� ; τ

ð Þ! x;z x∗;z<sup>∗</sup> ð Þ

ð Þ y; τ dD yð Þ� ; τ

ð Þ x; y; z; τ pj

(45)

). Then, using Theorem

ð Þ y; τ dD yð Þ� ; τ

<sup>i</sup> exist by virtue of the

<sup>i</sup> at D. Then if the surface integral exists, its

D Uj i

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

D Tj i

Proof. Let consider Eq. (45) for ð Þ <sup>x</sup>; <sup>z</sup> <sup>∈</sup> <sup>D</sup>�. Let (x<sup>∗</sup>

uið Þ¼ <sup>x</sup>; <sup>z</sup> ui <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ¼ <sup>b</sup>gj

ð

D Tj i

ð Þ x; y; z; τ pj

ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup>

D Uj i

ð Þ x; y; z; τ pj

<sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ

Holder property of u on D and weak singularity U<sup>j</sup>

corresponding to each of these functions if they exist.

ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; n yð Þ ; <sup>τ</sup> ujð Þ� <sup>y</sup>; <sup>τ</sup> ujðx<sup>∗</sup>; <sup>z</sup><sup>∗</sup><sup>Þ</sup> � �dD yð Þþ ; <sup>τ</sup>

ð

D Uj i

In the last relation, we have used the obvious properties: integrals with U<sup>j</sup>

δ j <sup>i</sup> � ð

0 @

�V:P: ð

142 Differential Equations - Theory and Current Research

6.1, we have

lim ð Þ! x;z x∗;;z<sup>∗</sup> ð Þ

� lim ð Þ! x;z x∗;z<sup>∗</sup> ð Þ

<sup>¼</sup> <sup>b</sup>gj ∗ Ub j i

�V:P: ð

<sup>¼</sup> <sup>b</sup>gj ∗ Ub j i

�V:P: ð

�V:P: ð

D Tj i

D Tj i

<sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ<sup>δ</sup>

D Tj i

<sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ lim

ð

D Tj i

<sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ

ð Þ! x;z x∗;z<sup>∗</sup> ð Þ

ð Þ! x;z x∗;z<sup>∗</sup> ð Þ

ð

D Uj i

<sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þþ

j <sup>i</sup> <sup>¼</sup> <sup>b</sup>gj ∗ Ub j i

ð

D Uj i

<sup>þ</sup>uj <sup>x</sup><sup>∗</sup>; <sup>z</sup><sup>∗</sup> ð Þ lim

This theorem gives us resolving system of integral equations for defining unknown values of boundary displacements.

Note also that the subsonic analog of the Somigliana formula was obtained for generalized functions. But since they are regular, from the Dubois-Reymond lemma ([3]: 97), the solution is classical. However, if the acting loads are described by singular generalized functions, which often takes place in physical problems, then one should use a representation of a generalized solution in the convolution form (Eq. (43)) with the evaluation of convolutions by the definition (see [3]: 133).
