**3. Fundamental matrices**

### **3.1 The Green's matrix of second-order system of hyperbolic equations**

Let us construct fundamental solutions of Eq. (1) on *D*<sup>0</sup> *<sup>M</sup> RN*þ<sup>1</sup> � �. **Definition 2.** *Ujk*ð Þ *x*, *t* is the Green's matrix of Eq. (1) if it satisfies to equations

$$(L\_{ij}(\partial\_{\mathbf{x}}, \partial\_{\mathbf{t}})U\_{jk}(\mathbf{x}, \mathbf{t}) + \delta\_{ik}\delta(\mathbf{x})\delta(\mathbf{t}) = \mathbf{0}, \quad i, j, k = \overline{\mathbf{1}, M} \tag{11}$$

and next conditions:

$$U\_{jk}(\mathbf{x}, t) = \mathbf{0} \quad for \quad t < \mathbf{0}, \forall \mathbf{x}, \tag{12}$$

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations… DOI: http://dx.doi.org/10.5772/intechopen.92449*

$$U\_{jk}(\mathbf{x}, \mathbf{0}) = \mathbf{0} \quad \text{for} \quad \mathbf{x} \neq \mathbf{0} \tag{13}$$

Here, by definition,

½ � *ui*ð Þ *x*, *t <sup>F</sup>* ¼ 0 (8)

*<sup>F</sup>* ¼ 0 (9)

;

, and has

*<sup>M</sup> RN*þ<sup>1</sup> � �.

vanishes only two last composed right parts of Eq. (7). Hence, it is necessary that

These conditions on the appropriate mobile wave front *Ft* we can write down with the account Eq. (4). By virtue of continuity of function *u*(*x*,*t*) for (*x*,*t*)∈*Ft*, we have

ð Þ *f x*ð þ *εν; t* þ *ενt*Þ � *f x*ð Þ � *εν; t* � *εν<sup>t</sup>*

ð*f x*ð Þ� þ *εn; t f x*ð Þ � *εn; t* Þ ¼ ½ � *f x*ð Þ *; t Ft*

� � ¼ �*νt<sup>=</sup>* ffiffiffiffiffiffiffi

*nlui*, *<sup>t</sup>* þ *cui*, *<sup>l</sup>* ½ �*Ft* ¼ 0, *i* ¼ 1, *M*, *l* ¼ 1, *N* (10)

By virtue of it, the condition (9) will be transformed to the kind (6), where *c*, for

The proof follows from the condition of continuity (5). The expression (10) is

In the physical problems of solid and media, the corresponding condition (6) is a condition for conservation of an impulse at fronts. This condition connects a jump of velocity at a wave fronts with stresses jump. By this cause, such surfaces are

**Definition 1.** The solution of Eq. (1), *u*(*x*,*t*), is named as classical one if it is

limited number of piecewise smooth wave fronts on which conditions jumps (5)

**3.1 The Green's matrix of second-order system of hyperbolic equations**

**Definition 2.** *Ujk*ð Þ *x*, *t* is the Green's matrix of Eq. (1) if it satisfies to equations

*Lij*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> Ujk*ð Þþ *<sup>x</sup>*, *<sup>t</sup> <sup>δ</sup>ikδ*ð Þ *<sup>x</sup> <sup>δ</sup>*ðÞ¼ *<sup>t</sup>* 0, *<sup>i</sup>*, *<sup>j</sup>*, *<sup>k</sup>* <sup>¼</sup> 1, *<sup>M</sup>* (11)

*Ujk*ð Þ¼ *x*, *t* 0 *for t*<0, ∀*x*, (12)

Let us construct fundamental solutions of Eq. (1) on *D*<sup>0</sup>

, twice differentiable almost everywhere on *R<sup>N</sup>*þ<sup>1</sup>

� � <sup>þ</sup> *<sup>F</sup>*, *<sup>t</sup>* � �*Δ<sup>t</sup>* <sup>¼</sup> <sup>0</sup>

> *νiν<sup>i</sup>* p

*<sup>i</sup> ν<sup>m</sup>* � *νtui*, *<sup>t</sup>* � �

*σm*

*Mathematical Theorems - Boundary Value Problems and Approximations*

½ � *f x*ð Þ *; <sup>t</sup> <sup>F</sup>* <sup>¼</sup> lim*<sup>ε</sup>*!þ<sup>0</sup>

From here, we have

named as *shock wave fronts*.

continuous on *RN*þ<sup>1</sup>

and (6) are carried out.

**3. Fundamental matrices**

and next conditions:

**60**

**Corollary.** On the wave fronts

<sup>¼</sup> lim*<sup>ε</sup>*!þ<sup>0</sup>

therefore the condition (5) is equivalent to (8).

If ð Þ *x*, *t* ∈*Ft*, then ð Þ *x* þ *cn*Δ*t*, *t* þ Δ*t* ∈*Ft*þΔ*<sup>t</sup>*. Therefore,

*F x*ð þ *cnΔt*, *t* þ *Δt*Þ � *F x*ð Þ¼ , *t c F*, *<sup>j</sup>*, *n <sup>j</sup>*

*c* ¼ �*F*, *<sup>t</sup>= F*, *<sup>j</sup>*, *n <sup>j</sup>*

each front, coincides with one of *ck*. The theorem has been proved.

the condition of the continuity of tangent derivative on the wave front.

$$(\delta\_{ik}\delta(\varkappa,t),\varrho\_i(\varkappa,t)) = \varrho\_k(\mathbf{0},\mathbf{0}) \quad \forall \varrho \in D\_M'(\mathbb{R}^{N+1})$$

For construction of Green's matrix, it is comfortable to use Fourier transformation, which brings Eq. (11) to the system of linear algebraic equations of the kind

$$L\_{jk}(-i\xi, -i\alpha)\overline{U}\_{kl}(\xi, \alpha) + \delta\_{jl} = 0, \quad j, k, l = \overline{1, M}$$

Here, ð Þ¼ *<sup>ξ</sup>*, *<sup>ω</sup> <sup>ξ</sup>*1, … ,*ξN*,*<sup>ω</sup>* � � is the Fourier variables appropriate to ð Þ *<sup>x</sup>*, *<sup>t</sup>* .

By permitting the system, we receive transformation of Green's matrix which by virtue of differential polynomials uniformity looks like:

$$\overline{U}\_{jk}(\xi, a) = Q\_{jk}(\xi, a) Q^{-1}(\xi, a) \tag{14}$$

where *Qjk* are the cofactors of the element with index (*k*, *j*) of the matrix f g *L*ð Þ �*iξ*, �*iω* ; and *Q* is the symbol of operator *L*:

$$Q(\xi, o) = \det\{L\_{kj}(-i\xi, -ioo)\}$$

There are the following relations of symmetry and homogeneous:

$$Q\_{jk}(\xi,\boldsymbol{\omega}) = Q\_{jk}(-\xi,\boldsymbol{\omega}) = Q\_{jk}(\xi,-\boldsymbol{\omega}),\\Q(\xi,\boldsymbol{\omega}) = Q(-\xi,\boldsymbol{\omega}) = Q(\xi,-\boldsymbol{\omega})\tag{15}$$

$$Q\_{jk}(\lambda\xi,\lambda\alpha) = \lambda^{2M-2} Q\_{jk}(\xi,\alpha), Q(\lambda\xi,\lambda\alpha) = \lambda^{2M} Q(\xi,\alpha) \tag{16}$$

By virtue of strong hyperbolicity characteristic equation,

$$Q(\xi, a) = 0$$

has 2*M* roots. It is a singular matrix. There is not a classic inverse Fourier transformation of it. It defines the Fourier transformation of the full class of fundamental matrices which are defined with accuracy of solutions of homogeneous system (1). Components of this matrix are not a generalized function. To calculate the inverse transformation, it is necessary to construct regularisation of this matrix in virtue of properties (12) and (13) of Green tensor. The following theorems has been proved [10]:

**Theorem 3.1.** *If cq <sup>q</sup>* <sup>¼</sup> 1, *<sup>M</sup>* � � *are unitary roots of Eq. (4), then the Green's matrix of system (1) has form*

$$\begin{aligned} \mathcal{U}\_{jk}(\mathbf{x},t) &= \sigma\_N \mathcal{H}(t) \sum\_{q=1}^M \int\_{\|\epsilon\|=1} A\_{jk} \left( \mathbf{e}, \mathbf{c}\_q \right) \\ & \quad \times \left\{ \left( (\mathbf{e}, \mathbf{x}) + \mathbf{c}\_q(\mathbf{e})t - i\mathbf{0} \right)^{1-N} - \left( (\mathbf{e}, \mathbf{x}) - \mathbf{c}\_q(\mathbf{e})t - i\mathbf{0} \right)^{1-N} \right\} dS(\epsilon) \end{aligned}$$

*where <sup>σ</sup><sup>N</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*π<sup>i</sup>* �*<sup>N</sup>*ð Þ *<sup>N</sup>* � <sup>2</sup> !, *Ajk <sup>e</sup>*,*cq* � � <sup>¼</sup> *Qjk <sup>e</sup>*,*cq* � �*=*2 *cqQmm e*,*cq* � � � � , and *H t*ð Þ *is Heaviside's function.*

**Theorem 3.2.** *If cq <sup>q</sup>* <sup>¼</sup> 1, *<sup>M</sup>* � � *are roots of Eq. (4) with multiplicity mq*, *then the Green's matrix of system (1) has form*

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\begin{split} U\_{jk}(\boldsymbol{\kappa},t) &= \sigma\_N H(t) \sum\_{q} m\_q \int\_{\boldsymbol{R}^N} Q\_{jk,\boldsymbol{\alpha}}^{\left(m\_q-1\right)} \left(\boldsymbol{e}, \boldsymbol{c}\_q\right) \left(Q\_{\boldsymbol{\omega}} \prescript{\left(m\_q\right)}{\boldsymbol{\alpha}} \left(\boldsymbol{e}, \boldsymbol{c}\_q\right)\right)^{-1} \\ & \times \left\{ \left(\left(\boldsymbol{e}, \boldsymbol{\omega}\right) + \boldsymbol{c}\_q\left(\boldsymbol{e}\right)t - i\boldsymbol{0}\right)^{1-N} - \left(\left(\boldsymbol{e}, \boldsymbol{\omega}\right) - \boldsymbol{c}\_q\left(\boldsymbol{e}\right)t - i\boldsymbol{0}\right)^{1-N} \right\} d\boldsymbol{S}(\boldsymbol{e}). \end{split}$$

*Here, the top index in brackets designate the order of derivative on ω*.

So, the construction of a Green's matrix is reduced to the calculation of integrals on unit sphere. For odd *N*, these theorems allow to build the Green's matrix *ε*approach only. For even *N* and for *ε*�approach, it is required to integrate multidimensional surface integral over unit sphere. However, in a number of cases, this procedure can be simplified.

We notice that if the original of *Q*�<sup>1</sup> is known, i.e.

$$J(\mathfrak{x}, t) = F^{-1} \left[ Q^{-1} (\xi, o) \right],$$

which is built in view of conditions (12), then it is easy to restore the Green's matrix

$$U\_{jk}(\mathbf{x},t) = Q\_{jk}(i\partial\_{\mathbf{x}}, i\partial\_t) f(\mathbf{x},t) \tag{17}$$

For example, *U*<sup>3</sup> is the simple layer on a cone [10] and it is the singular generalized

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations…*

Here, the convolution over *t* undertakes (2 *M* � 2) time, which exists, by virtue of, on semi-infinite at the left of supports of functions [11]. It is easy to check up that the boundary conditions (12) and (13) are carried out as *UN*ð Þ *x*, *t*,*c* which

**Theorem 3.3.** *If the symbol of the operator L is presented in form* (18) *and ck are simple roots of Eq.* (4), *then Ujk*ð Þ *x*, *t is defined by the formula* (17), *where J x*ð Þ , *t looks*

If *ck* have multiplicity *mk* in decomposition as (20), degrees k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>*=c*<sup>2</sup>

� � can appear. Using the property of convolution transformation, we

Then, the procedure of construction of a Green's matrix is similar to the

We notice that as follows from (20) in a case of N = 1, 2, the convolution

0

0

But already for N = 3 and more, the construction of convolutions is non-trivial, and for their determination, its definition in a class of generalized functions should

*<sup>u</sup>*^*<sup>i</sup>* <sup>¼</sup> *Uik* <sup>∗</sup> *<sup>G</sup>*^ *<sup>k</sup>:*

For regular functions, it has integral representation in form of retarded potential:

If Eqs. (1) are invariant, concerning the group of orthogonal transformations, then *ck* do not depend on *n*. In physical problems, the isotropy of medium is reduced

*dV y*ð Þ <sup>ð</sup>*<sup>t</sup>*

receive their original in kind of complete convolution over (*x*,*t*):

operation is reduced to calculate regular integrals of simple kind:

*UN*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *tH t*ðÞ¼ <sup>ð</sup>*<sup>t</sup>*

ð

*RN*

*=c* 2 *k*

h i � ��*<sup>m</sup>*

*Ak*ð*H t*ð Þ ∗ *<sup>t</sup>* … ð Þ *H t*ð Þ ∗ *tUN*ð Þ *x*, *t*,*ck*Þ *:* (20)

¼ ð*UN*ð Þ *x*, *t*,*c* ∗ … *<sup>m</sup>* ∗ *UN*ð Þ *x*, *t*,*c*Þ

*UN*ð Þ *x*, *t* � *τ dτ*

*UN*ð Þ *x* � *y*, *t* � *τ UN*ð Þ *y*, *τ dτ*

*<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � : sup*<sup>t</sup> <sup>G</sup>*^ <sup>∈</sup>ð Þ 0, <sup>∞</sup> , the appropriate

*Uik*ð Þ *x* � *y*, *τ Gk*ð Þ *y*, *t* � *τ dV y* ð Þ

*k*

� ��*<sup>m</sup>*

function. In this case, *J x*ð Þ , *t* is convolution over *t* Green's function with *H t*ð Þ:

*J x*ð Þ¼ , *<sup>t</sup>* <sup>X</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92449*

satisfies them. We formulate this result as:

*<sup>F</sup>*�<sup>1</sup> k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

*UN*ð Þ *x*, *t* ∗ *tUN*ð Þ¼ *x*, *t*

For any regular function *G*^ ∈ *D*<sup>0</sup>

solution of Eq. (1) looks like the convolution

*u*^*i*ð Þ¼ *x*, *t H t*ð Þ

to the specified property.

∞ð

0 *dτ* ð

*RN*

*like* (20).

*m* ¼ 1, *mk*

described one.

be used.

**63**

*M*

*k*¼1

In the case of invariance of Eq. (1) relative to group of orthogonal transformations, a symbol of the operator *Lij* is a function of only two variables k k*ξ* ,*ω* and can be presented in the form:

$$Q(\xi, o) = (i o)^{2M} q\left(||\xi|| o^{-1}\right). \tag{18}$$

It essentially simplifies the construction of the original using the Green's functions of classical wave equations. For this purpose, it is necessary to spread out *Q*�<sup>1</sup> ð Þ *ξ*, *ω* on simple fractions. In the case of simple roots,

$$\mathcal{Q}(\xi,\boldsymbol{\omega}) = \prod\_{k=1}^{M} \left( \left\| \xi \right\|^2 - \boldsymbol{\omega}^2 / c\_k^2 \right)$$

$$\mathcal{Q}^{-1}(\xi,\boldsymbol{\omega}) = (-i\boldsymbol{\alpha})^{-2M+2} \sum\_{k=1}^{M} A\_k \left( \left\| \xi \right\|^2 - \boldsymbol{\alpha}^2 / c\_k^2 \right)^{-1} \tag{19}$$

where *Ak* is the decomposition constant. It is easy to see that summand in round brackets under summation sign is the symbol of the classical wave operator

$$D\_k = c\_k^{-2} \partial\_t^2 - \Delta\_N.$$

Here, *Δ<sup>N</sup>* is the Laplacian for which the Green's function *UN*ð Þ *x*, *t* has been investigated well [11].

From Theorem 3.1 follows the support *UN*ð Þ *x*, *t*,*c* is:

$$K\_c^+ = \{(\mathbf{x}, t) : \|\mathbf{x}\| \le ct, t > 0\}$$

in *R<sup>N</sup>*þ<sup>1</sup> for even *N* and it is sound cone

$$K\_{\mathfrak{c}} = \{ (\mathfrak{x}, t) : ||\mathfrak{x}|| = \mathfrak{ct}, t > \mathbf{0} \}$$

for odd *N:*

*Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations… DOI: http://dx.doi.org/10.5772/intechopen.92449*

For example, *U*<sup>3</sup> is the simple layer on a cone [10] and it is the singular generalized function. In this case, *J x*ð Þ , *t* is convolution over *t* Green's function with *H t*ð Þ:

$$J(\mathbf{x}, t) = \sum\_{k=1}^{M} A\_k(H(t) \ast\_t \dots (H(t) \ast\_t U\_N(\mathbf{x}, t, c\_k))).\tag{20}$$

Here, the convolution over *t* undertakes (2 *M* � 2) time, which exists, by virtue of, on semi-infinite at the left of supports of functions [11]. It is easy to check up that the boundary conditions (12) and (13) are carried out as *UN*ð Þ *x*, *t*,*c* which satisfies them. We formulate this result as:

**Theorem 3.3.** *If the symbol of the operator L is presented in form* (18) *and ck are simple roots of Eq.* (4), *then Ujk*ð Þ *x*, *t is defined by the formula* (17), *where J x*ð Þ , *t looks like* (20).

If *ck* have multiplicity *mk* in decomposition as (20), degrees k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>*=c*<sup>2</sup> *k* � ��*<sup>m</sup> m* ¼ 1, *mk* � � can appear. Using the property of convolution transformation, we receive their original in kind of complete convolution over (*x*,*t*):

$$F^{-1}\left[\left(||\xi||^2 - \alpha^2/c\_k^2\right)^{-m}\right] = \left(U\_N(\mathfrak{x},t,\mathfrak{c}) \ast \dots \ast U\_N(\mathfrak{x},t,\mathfrak{c})\right)$$

Then, the procedure of construction of a Green's matrix is similar to the described one.

We notice that as follows from (20) in a case of N = 1, 2, the convolution operation is reduced to calculate regular integrals of simple kind:

$$U\_N(\mathbf{x}, t) \* \iota H(t) = \int\_0^t U\_N(\mathbf{x}, t - \tau) d\tau$$

$$U\_N(\mathbf{x}, t) \* \iota\_I U\_N(\mathbf{x}, t) = \int\_{\mathbb{R}^N} dV(\mathbf{y}) \begin{cases} U\_N(\mathbf{x} - \mathbf{y}, t - \tau) U\_N(\mathbf{y}, \tau) d\tau \\\\ 0 \end{cases}$$

But already for N = 3 and more, the construction of convolutions is non-trivial, and for their determination, its definition in a class of generalized functions should be used.

For any regular function *G*^ ∈ *D*<sup>0</sup> *<sup>M</sup> <sup>R</sup><sup>N</sup>*þ<sup>1</sup> � � : sup*<sup>t</sup> <sup>G</sup>*^ <sup>∈</sup>ð Þ 0, <sup>∞</sup> , the appropriate solution of Eq. (1) looks like the convolution

$$
\hat{u}\_{\dot{\imath}} = U\_{\dot{\imath}k} \* \hat{G}\_k .
$$

For regular functions, it has integral representation in form of retarded potential:

$$\hat{u}\_i(\varkappa, t) = H(t) \int\_0^\infty d\tau \int\_{\mathcal{R}^N} U\_{ik}(\varkappa - \jmath, \tau) G\_k(\jmath, t - \tau) dV(\jmath)$$

If Eqs. (1) are invariant, concerning the group of orthogonal transformations, then *ck* do not depend on *n*. In physical problems, the isotropy of medium is reduced to the specified property.

*Ujk*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>σ</sup>NH t*ð Þ<sup>X</sup>

this procedure can be simplified.

be presented in the form:

investigated well [11].

for odd *N:*

**62**

*Q*�<sup>1</sup>

*Q*�<sup>1</sup>

*q mq* ð

We notice that if the original of *Q*�<sup>1</sup> is known, i.e.

*RN*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*<sup>Q</sup>*ð Þ *mq*�<sup>1</sup> *jk*,*<sup>ω</sup> e*,*cq*

*Here, the top index in brackets designate the order of derivative on ω*.

on unit sphere. For odd *N*, these theorems allow to build the Green's matrix *ε*approach only. For even *N* and for *ε*�approach, it is required to integrate

*J x*ð Þ¼ , *<sup>t</sup> <sup>F</sup>*�<sup>1</sup> *<sup>Q</sup>*�<sup>1</sup>

*<sup>Q</sup>*ð Þ¼ *<sup>ξ</sup>*,*<sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* <sup>2</sup>*<sup>M</sup>*

ð Þ *ξ*, *ω* on simple fractions. In the case of simple roots,

*<sup>Q</sup>*ð Þ¼ *<sup>ξ</sup>*,*<sup>ω</sup>* <sup>Y</sup>

ð Þ¼ � *<sup>ξ</sup>*, *<sup>ω</sup>* ð Þ *<sup>i</sup><sup>ω</sup>* �2*M*þ<sup>2</sup><sup>X</sup>

� � *Q*,

� ð Þþ *<sup>e</sup>*, *<sup>x</sup> cq*ð Þ*<sup>e</sup> <sup>t</sup>* � *<sup>i</sup>*<sup>0</sup> � �1�*<sup>N</sup>* � ð Þ� *<sup>e</sup>*, *<sup>x</sup> cq*ð Þ*<sup>e</sup> <sup>t</sup>* � *<sup>i</sup>*<sup>0</sup> � �1�*<sup>N</sup>* n o

So, the construction of a Green's matrix is reduced to the calculation of integrals

multidimensional surface integral over unit sphere. However, in a number of cases,

which is built in view of conditions (12), then it is easy to restore the Green's matrix

In the case of invariance of Eq. (1) relative to group of orthogonal transformations, a symbol of the operator *Lij* is a function of only two variables k k*ξ* ,*ω* and can

It essentially simplifies the construction of the original using the Green's functions of classical wave equations. For this purpose, it is necessary to spread out

k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

*M*

*k*¼1

where *Ak* is the decomposition constant. It is easy to see that summand in round

*<sup>t</sup>* � *ΔN:*

� �

*=c* 2 *k*

*Ak* k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

*=c* 2 *k*

(19)

� ��<sup>1</sup>

*M*

*k*¼1

brackets under summation sign is the symbol of the classical wave operator

Here, *Δ<sup>N</sup>* is the Laplacian for which the Green's function *UN*ð Þ *x*, *t* has been

*<sup>c</sup>* ¼ f g ð Þ *x*, *t* : k k*x* ≤ *ct*, *t*>0

*Kc* ¼ f g ð Þ *x*, *t* : k k*x* ¼ *ct*, *t*>0

*Dk* ¼ *c* �2 *<sup>k</sup> <sup>∂</sup>*<sup>2</sup>

From Theorem 3.1 follows the support *UN*ð Þ *x*, *t*,*c* is:

*K*<sup>þ</sup>

in *R<sup>N</sup>*þ<sup>1</sup> for even *N* and it is sound cone

ð Þ *<sup>ξ</sup>*,*<sup>ω</sup>* � �,

*Ujk*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> Qjk*ð Þ *<sup>i</sup>∂x*, *<sup>i</sup>∂<sup>t</sup> J x*ð Þ , *<sup>t</sup>* (17)

*<sup>q</sup>* k k*<sup>ξ</sup> <sup>ω</sup>*�<sup>1</sup> � �*:* (18)

ð Þ *mq <sup>ω</sup> e*,*cq* � � � ��<sup>1</sup>

*dS e*ð Þ
