**3.2 The Green's tensor of elastic medium**

For isotropic elastic medium constants, the matrix is equal to

$$\mathcal{C}\_{ij}^{ml} = \rho \left\{ \lambda \delta\_l^m \delta\_i^j + \mu \left( \delta\_i^m \delta\_j^l + \delta\_j^m \delta\_i^l \right) \right\}.$$

**Theorem 3.4.** *For fixed k and n, the vector <sup>T</sup>*^ *<sup>k</sup>*

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

*<sup>W</sup>*^ *<sup>k</sup> j*

*<sup>∂</sup>tV*^ *<sup>k</sup>*

*V*^ *k*

*<sup>W</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* <sup>¼</sup> *<sup>U</sup>*^ *<sup>k</sup>*

*Lij*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>V</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* and *<sup>W</sup>*^ *<sup>k</sup>*

*Lij*ð Þ *<sup>∂</sup>x*, *<sup>∂</sup><sup>t</sup> <sup>W</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>V</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup> <sup>W</sup>*^ *<sup>k</sup>*

The Green's matrix of the static equations for *<sup>U</sup>*^ *k s*ð Þ

*Lij*ð Þ *<sup>∂</sup>x*, 0 *<sup>U</sup>*^ *k s*ð Þ

*<sup>U</sup>*^ *k s*ð Þ

*<sup>T</sup>*^ *k s*ð Þ

*<sup>i</sup>* ð Þ¼� *<sup>x</sup>*, *<sup>n</sup> <sup>T</sup>*^ *k s*ð Þ

*Lij*ð Þ *<sup>∂</sup>x*, 0 *<sup>T</sup>k s*ð Þ

*<sup>i</sup>* ð Þ¼� *<sup>x</sup>*, *<sup>n</sup> <sup>C</sup>ml*

By analogy with (22), we define the matrix

Obviously, we have the symmetry relations

*<sup>T</sup>*^ *k s*ð Þ

Theorem 3.4 implies the following result.

It is easy to see that this is an elliptic system. The following theorem have been proved [17].

*Gi* <sup>¼</sup> *<sup>C</sup>ml*

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup> <sup>T</sup>*^ *<sup>k</sup>*

which is the primitive of the corresponding matrices with respect to *t*:

*<sup>i</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* , *<sup>∂</sup>tW*^ *<sup>k</sup>*

*<sup>j</sup>* <sup>þ</sup> *<sup>δ</sup><sup>k</sup>*

*<sup>j</sup>* <sup>þ</sup> *nmCml*

*<sup>i</sup>* ð Þ �*x*, *<sup>t</sup>* , *<sup>V</sup>*^ *<sup>k</sup>*

Relation (23) implies the following symmetry properties of the above matrices:

*<sup>j</sup>* ð Þþ *<sup>x</sup> <sup>δ</sup><sup>k</sup>*

*<sup>i</sup>* ð Þ¼� �*x*, *<sup>t</sup>*, *<sup>n</sup> <sup>W</sup>*^ *<sup>k</sup>*

*ik nmδ*, *<sup>l</sup>*ð Þ *x δ*ð Þ*t :*

The matrix *T*^ is called a *multipole matrix*, since it describes the fundamental solutions of system (1) generated by concentrated multipole sources (see [18]). *Primitives of the matrix.* The primitive of the multipole matrix is introduced as

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

*j*

ð Þ *x*, *t*, *n* ∗ *tH t*ð Þ,

*<sup>i</sup>* <sup>¼</sup> *<sup>T</sup>*^ *<sup>k</sup>*

*ki δ*, *<sup>l</sup>*ð Þ *x H t*ðÞ¼ 0*:*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>V</sup>*^ *<sup>i</sup>*

*<sup>i</sup>* ð Þ *x*, *t*, *n :*

*<sup>i</sup>* are fundamental solutions to system (1) of the form

*<sup>i</sup> δ*ð Þ *x H t*ðÞ¼ 0, (25)

*<sup>k</sup>*ð Þ *x*, *t* ,

*<sup>i</sup>* ð Þ! *x* 0, ∥*x*∥ ! ∞*:* (28)

*kj nm∂lU*^ *i s*ð Þ *j :*

*ki δ*, *<sup>l</sup>*ð Þ¼ *x* 0*:*

*<sup>i</sup>* ð Þ¼� �*x*, *<sup>n</sup> <sup>T</sup>*^ *k s*ð Þ

*<sup>i</sup>* is a *fundamental solution of the static equations*:

*<sup>j</sup>* � *nmCml*

*<sup>i</sup>* ð Þ *x*, *t*, �*n :* (26)

*<sup>i</sup>* ð Þ *x* (when the *t*-derivatives in

*<sup>i</sup>* ð Þ *x*, �*n :* (29)

*<sup>i</sup> δ*ð Þ¼ *x* 0, (27)

*system (1) corresponding to*

convolution over time:

It is easy to see that *<sup>V</sup>*^ *<sup>k</sup>*

(1) are zero) is defined by

**Corollary.** *<sup>T</sup>*^ *k s*ð Þ

**65**

*<sup>i</sup>* ð Þ *x*, *t*, *n is the fundamental solution of*

The coefficients of Eq. (1) depend only on two sound velocities

$$c\_1 = \sqrt{(\lambda + 2\mu)/\rho}, \ c\_2 = \sqrt{\mu/\rho}.$$

where *ρ* is the density of medium, and *λ* and *μ* are elastic Lame parameters. These two speeds are velocities of propagation of dilatational and shearing waves. Wave fronts for Green's tensor are two spheres expanding with these velocities.

In the case of plane deformation N = M = 2, an appropriate Green's tensor was constructed in [5, 6]. For the space deformation N = M = 3, the expression of a Green's tensor was represented in [6].

For anisotropic medium in a plane case (N = M = 2), the Green's tensor was constructed in [12, 13]. For such medium, the wave propagation velocities depend on direction *n* and the form of wave fronts essentially depends on coefficients of Eq. (1). Anisotropic mediums with weak and strong anisotropy of elastic properties in the case of plane deformation were considered in [12–15]. In the first case, the topological type of wave fronts is similar to extending spheres. In the second case, the complex wave fronts and *lacunas* appear [16]. *Lacunas* are the mobile unperturbed areas limited by wave fronts and extended with current of time. Such medium has sharply waveguide properties in the direction of vector of maximal speeds. The wave fronts and the components of Green's tensor for weak and strong anisotropy are presented in [15]. The calculations are carried out for crystals of aragonite, topaz and calli pentaborat.

#### **3.3 The fundamental matrices** *<sup>V</sup>*^ **,** *<sup>T</sup>*^ **,***W*^ **,** *<sup>U</sup>*^ ð Þ*<sup>s</sup>* **,** *<sup>T</sup>*^ ð Þ*<sup>s</sup>*

For solution of BVP using Green's matrix *U*^ , we introduce the fundamental matrices ^ *S* and *T*^ with elements given by

$$
\hat{S}\_{ik}^{m}(\mathbf{x},t) = \mathbf{C}\_{\vec{\eta}}^{ml} \, \partial\_l \hat{U}\_j^k, \qquad \Gamma\_i^k(\mathbf{x},t,n) = \hat{S}\_{ik}^m n\_m,\tag{21}
$$

$$\hat{T}\_k^{\hat{i}}(\mathbf{x}, t, n) = -\Gamma\_i^k(\mathbf{x}, t, n) = -\mathbf{C}\_{\hat{\imath}\hat{\jmath}}^{ml} n\_m \partial\_l \hat{U}\_j^k,\tag{22}$$

$$i, j, k = \overline{1, M}, \qquad m, l = \overline{1, N}.$$

Then, the equation for *U*^ can be written as

^ *S l ik*, *<sup>l</sup>* � *<sup>U</sup>*^ *<sup>k</sup> <sup>i</sup>* , *tt* <sup>þ</sup> *<sup>δ</sup><sup>k</sup> <sup>i</sup> δ*ð Þ *x δ*ð Þ¼ *x* 0*:*

From the invariance of the equations for *U*^ under the symmetry transformations *y* ¼ �*x*, some symmetry properties of introduced matrices follows:

$$
\hat{\boldsymbol{U}}\_{i}^{k}(\mathbf{x},t) = \hat{\boldsymbol{U}}\_{i}^{k}(-\mathbf{x},t), \quad \hat{\boldsymbol{U}}\_{i}^{k}(\mathbf{x},t) = \hat{\boldsymbol{U}}\_{k}^{i}(\mathbf{x},t), \quad \hat{\boldsymbol{S}}\_{ik}^{m}(\mathbf{x},t) = -\hat{\boldsymbol{S}}\_{ik}^{m}(-\mathbf{x},t), \tag{23}
$$

$$
\hat{T}\_i^k(\mathbf{x}, t, n) = -\hat{T}\_i^k(-\mathbf{x}, t, n) = -\hat{T}\_i^k(\mathbf{x}, t, -n). \tag{24}
$$

Is easy to prove [17].

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

**Theorem 3.4.** *For fixed k and n, the vector <sup>T</sup>*^ *<sup>k</sup> <sup>i</sup>* ð Þ *x*, *t*, *n is the fundamental solution of system (1) corresponding to*

$$G\_i = \mathcal{C}\_{ik}^{ml} n\_m \delta, \iota(\boldsymbol{x}) \delta(t).$$

The matrix *T*^ is called a *multipole matrix*, since it describes the fundamental solutions of system (1) generated by concentrated multipole sources (see [18]).

*Primitives of the matrix.* The primitive of the multipole matrix is introduced as convolution over time:

$$
\hat{W}\_j^k(\mathbf{x}, t, n) = \hat{T}\_j^k(\mathbf{x}, t, n) \ast \iota H(t),
$$

which is the primitive of the corresponding matrices with respect to *t*:

$$
\partial\_t \hat{V}\_i^k = \hat{U}\_i^k(\mathbf{x}, t), \quad \partial\_t \hat{W}\_i^k = \hat{T}\_i^k(\mathbf{x}, t, n).
$$

It is easy to see that *<sup>V</sup>*^ *<sup>k</sup> <sup>i</sup>* and *<sup>W</sup>*^ *<sup>k</sup> <sup>i</sup>* are fundamental solutions to system (1) of the form

$$L\_{\hat{\boldsymbol{\eta}}}(\boldsymbol{\partial}\_{\boldsymbol{\mathcal{X}}},\boldsymbol{\partial}\_{\boldsymbol{t}})\hat{\boldsymbol{\mathcal{V}}}\_{\boldsymbol{j}}^{\boldsymbol{k}}+\delta\_{\boldsymbol{t}}^{\boldsymbol{k}}\delta(\boldsymbol{\varkappa})H(\boldsymbol{t})=\mathbf{0},\tag{25}$$

$$L\_{\hat{\boldsymbol{\eta}}}(\boldsymbol{\partial}\_{\boldsymbol{\mathcal{X}}},\boldsymbol{\partial}\_{\boldsymbol{t}})\hat{\boldsymbol{\mathcal{W}}}\_{\boldsymbol{j}}^{\boldsymbol{k}}+n\_{m}C\_{\boldsymbol{k}i}^{m\boldsymbol{l}}\delta\_{\boldsymbol{\mathcal{I}}}(\boldsymbol{\varkappa})H(\boldsymbol{t})=\mathbf{0}.$$

Relation (23) implies the following symmetry properties of the above matrices:

$$
\hat{\boldsymbol{V}}\_{i}^{k}(\boldsymbol{\varkappa},t) = \hat{\boldsymbol{V}}\_{i}^{k}(-\boldsymbol{\varkappa},t), \quad \hat{\boldsymbol{V}}\_{i}^{k}(\boldsymbol{\varkappa},t) = \hat{\boldsymbol{V}}\_{k}^{i}(\boldsymbol{\varkappa},t),
$$

$$
\hat{\boldsymbol{W}}\_{i}^{k}(\boldsymbol{\varkappa},t,n) = -\hat{\boldsymbol{W}}\_{i}^{k}(-\boldsymbol{\varkappa},t,n) = -\hat{\boldsymbol{W}}\_{i}^{k}(\boldsymbol{\varkappa},t,-n). \tag{26}
$$

The Green's matrix of the static equations for *<sup>U</sup>*^ *k s*ð Þ *<sup>i</sup>* ð Þ *x* (when the *t*-derivatives in (1) are zero) is defined by

$$L\_{\hat{\boldsymbol{\eta}}}(\boldsymbol{\partial\_{\mathbf{x}}},\mathbf{0})\hat{\boldsymbol{U}}\_{\boldsymbol{j}}^{k(\boldsymbol{\epsilon})}(\boldsymbol{\varkappa}) + \delta\_{\boldsymbol{i}}^{k}\delta(\boldsymbol{\varkappa}) = \mathbf{0},\tag{27}$$

$$\hat{U}\_i^{k(\boldsymbol{\epsilon})}(\boldsymbol{x}) \to \mathbf{0}, \quad \|\boldsymbol{x}\| \to \infty. \tag{28}$$

By analogy with (22), we define the matrix

$$
\hat{T}\_i^{k(s)}(\varkappa, n) = -C\_{kj}^{ml} n\_m \partial\_l \hat{U}\_j^{i(s)}.
$$

Obviously, we have the symmetry relations

$$
\hat{T}\_i^{k(\boldsymbol{s})}(\boldsymbol{\varkappa}, \boldsymbol{n}) = -\hat{T}\_i^{k(\boldsymbol{s})}(-\boldsymbol{\varkappa}, \boldsymbol{n}) = -\hat{T}\_i^{k(\boldsymbol{s})}(\boldsymbol{\varkappa}, -\boldsymbol{n}).\tag{29}
$$

Theorem 3.4 implies the following result.

**Corollary.** *<sup>T</sup>*^ *k s*ð Þ *<sup>i</sup>* is a *fundamental solution of the static equations*:

$$L\_{ij}(\partial\_{\mathbf{x}}, \mathbf{0})T\_j^{k(s)} - n\_m \mathbf{C}\_{ki}^{ml} \delta, l(\mathbf{x}) = \mathbf{0}.$$

It is easy to see that this is an elliptic system. The following theorem have been proved [17].

**3.2 The Green's tensor of elastic medium**

Green's tensor was represented in [6].

**3.3 The fundamental matrices** *<sup>V</sup>*^ **,** *<sup>T</sup>*^ **,***W*^ **,** *<sup>U</sup>*^ ð Þ*<sup>s</sup>*

^ *S m*

*T*^*i*

Then, the equation for *U*^ can be written as

*<sup>i</sup>* ð Þ �*x*, *<sup>t</sup>* , *<sup>U</sup>*^ *<sup>k</sup>*

*T*^ *k*

^ *S l ik*, *<sup>l</sup>* � *<sup>U</sup>*^ *<sup>k</sup>*

*S* and *T*^ with elements given by

*ik*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>C</sup>ml*

*<sup>k</sup>*ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup>* <sup>Γ</sup>*<sup>k</sup>*

matrices ^

*U*^ *k*

**64**

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup>*^ *<sup>k</sup>*

Is easy to prove [17].

*Cml*

For isotropic elastic medium constants, the matrix is equal to

*l δ j*

The coefficients of Eq. (1) depend only on two sound velocities

*<sup>c</sup>*<sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>i</sup>* <sup>þ</sup> *μ δ<sup>m</sup> i δl <sup>j</sup>* <sup>þ</sup> *<sup>δ</sup><sup>m</sup> j δl i*

ð Þ *<sup>λ</sup>* <sup>þ</sup> <sup>2</sup>*<sup>μ</sup> <sup>=</sup><sup>ρ</sup>* <sup>p</sup> , *<sup>c</sup>*<sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffi

where *ρ* is the density of medium, and *λ* and *μ* are elastic Lame parameters. These two speeds are velocities of propagation of dilatational and shearing waves. Wave fronts for Green's tensor are two spheres expanding with these velocities. In the case of plane deformation N = M = 2, an appropriate Green's tensor was constructed in [5, 6]. For the space deformation N = M = 3, the expression of a

For anisotropic medium in a plane case (N = M = 2), the Green's tensor was constructed in [12, 13]. For such medium, the wave propagation velocities depend on direction *n* and the form of wave fronts essentially depends on coefficients of Eq. (1). Anisotropic mediums with weak and strong anisotropy of elastic properties in the case of plane deformation were considered in [12–15]. In the first case, the topological type of wave fronts is similar to extending spheres. In the second case, the complex wave fronts and *lacunas* appear [16]. *Lacunas* are the mobile unperturbed areas limited by wave fronts and extended with current of time. Such medium has sharply waveguide properties in the direction of vector of maximal speeds. The wave fronts and the components of Green's tensor for weak and strong anisotropy are presented in [15]. The calculations are carried out for crystals of aragonite, topaz and calli pentaborat.

**,** *<sup>T</sup>*^ ð Þ*<sup>s</sup>*

*<sup>i</sup>* ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup> <sup>C</sup>ml*

*i*, *j*, *k* ¼ 1, *M*, *m*, *l* ¼ 1, *N:*

*<sup>i</sup>* , *tt* <sup>þ</sup> *<sup>δ</sup><sup>k</sup>*

*y* ¼ �*x*, some symmetry properties of introduced matrices follows:

*<sup>i</sup>* ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup> <sup>T</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup>*^ *<sup>i</sup>*

From the invariance of the equations for *U*^ under the symmetry transformations

*<sup>k</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* , ^ *S m*

*<sup>i</sup>* ð Þ¼� �*x*, *<sup>t</sup>*, *<sup>n</sup> <sup>T</sup>*^ *<sup>k</sup>*

*<sup>i</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>*, *<sup>n</sup>* ^

*<sup>i</sup> δ*ð Þ *x δ*ð Þ¼ *x* 0*:*

*S m*

*ij nm∂lU*^ *<sup>k</sup> j*

*ik*ð Þ¼� *<sup>x</sup>*, *<sup>t</sup>* ^

*S m*

*<sup>i</sup>* ð Þ *x*, *t*, �*n :* (24)

*iknm*, (21)

, (22)

*ik*ð Þ �*x*, *t* , (23)

For solution of BVP using Green's matrix *U*^ , we introduce the fundamental

*ij <sup>∂</sup>lU*^ *<sup>k</sup> j* , Γ*<sup>k</sup>*

n o � �

*:*

*μ=ρ* p ,

*ij* <sup>¼</sup> *ρ λδ<sup>m</sup>*

*Mathematical Theorems - Boundary Value Problems and Approximations*

**Theorem 3.5.** *The following representations take place*

$$
\hat{\boldsymbol{V}}\_i^k(\mathbf{x}, t) = \boldsymbol{U}\_i^{k(\boldsymbol{s})}(\mathbf{x}) \boldsymbol{H}(t) + \boldsymbol{V}\_i^{k(d)}(\mathbf{x}, t), \tag{30}
$$

*ui*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> uS*

boundary conditions (33)–(35) and front conditions (5)–(7).

**Problem 2.** Construct resolving boundary integral equations for the solution of

*Initial-boundary value problem* I. Find a solution of system (1) that satisfies

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

*Initial-boundary value problem* II. Find a solution of system (1) that satisfies

**Remark**. Wavefronts arise if the initial and boundary data do not obey the

In physical problems, they describe shock waves, which are typical when the external actions (forces) have a shock nature and are described by discontinuous or

*E u*ð Þ¼ *K u*ð Þþ *W u*ð Þ, *L u*ð Þ¼ *K u*ð Þ� *W u*ð Þ,

which are called the *densities of internal, kinetic, and total energy* of the system,

ð

*D*� *t*

*Here and below, dV x*ð Þ¼ *dx*<sup>1</sup> … *dxN*, *dV x*ð Þ¼ , *t dV x*ð Þ*dt*; *dS x*ð Þ, *and*, *dS x*ð Þ , *t*

**Proof**. Multiplying (1) by *ui* and summing the result over *i*, after simple algebra,

� �, *<sup>l</sup>* � *uiui*, ð Þ*<sup>t</sup>* , *<sup>t</sup>* <sup>þ</sup> *Giui:*

This equality is integrated over *Dt* taking into account the front discontinuities and using the Gauss-Ostrogradsky theorem and initial conditions (33) and (34) to

ð

*S*�

*ij u <sup>j</sup>*, *mui*

**Theorem 5.1.** *If u is a classical solution of the Dirichlet (Neumann) boundary value*

*<sup>i</sup>* , *<sup>t</sup>*ð Þ¼ *<sup>x</sup>*, 0 *<sup>u</sup>*<sup>1</sup>

*ij ui*, *mu <sup>j</sup>*, *<sup>l</sup>*, *K u*ð Þ¼ 0, 5∥*u*, *<sup>t</sup>*∥<sup>2</sup>

*Gi*ð Þ *x*, *t ui*ð Þ *x*, *t dV x*ð Þþ , *t*

*<sup>i</sup>*ð Þ *<sup>x</sup>* � �*dV x*ð Þ

*<sup>i</sup>* ð Þ *<sup>x</sup> <sup>u</sup>*<sup>1</sup>

*ui*ð Þ *<sup>x</sup>*, *<sup>t</sup> ui*, *<sup>t</sup>*ð Þ� *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>0</sup>

boundary conditions (33), (34), and (36) and front conditions (5)–(7).

*<sup>i</sup>* ð Þ *<sup>x</sup>* , *<sup>u</sup><sup>S</sup>*

ð Þ *x*, *t nl*ð Þ¼ *x gi*

*and the Neumann-type conditions*

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

the following boundary value problems.

These solutions are called *classical*.

**5. Uniqueness of solutions of BVP**

respectively, and *L* is *the Lagrangian*.

ð

*D*� *t*

*wi*ð Þ¼ *<sup>x</sup>*, 0 *<sup>u</sup>*<sup>0</sup>

*W u*ð Þ¼ 0, 5*Cml*

*Lux* ð Þ ð Þ , *t dV x*ð Þ¼ , *t*

*<sup>i</sup>* ð Þ *x*, *t dS x*ð Þ� , *t*

*are the differentials of the area of S and D, respectively*.

*<sup>L</sup>* <sup>¼</sup> *<sup>C</sup>ml*

compatibility conditions

singular functions.

*problem, then*

þ ð

*Dt gi* ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>u</sup><sup>S</sup>*

we obtain the expression

obtain

**67**

Define the functions

*σl i* *<sup>i</sup>* ð Þ *x*, *t* , *x*∈*S*, *t*≥ 0; (35)

ð Þ *x*, *t* , *x*∈*S*, *t*≥ 0, *i* ¼ 1, *N:* (36)

*<sup>i</sup>*ð Þ *x* , *x*∈*S:*

,

$$
\hat{\boldsymbol{W}}\_i^k(\boldsymbol{\kappa}, t) = \boldsymbol{T}\_i^{k(s)}(\boldsymbol{\kappa})\boldsymbol{H}(t) + \boldsymbol{W}\_i^{k(d)}(\boldsymbol{\kappa}, t), \tag{31}
$$

where *Uk s*ð Þ *<sup>i</sup>* ð Þ *<sup>x</sup> H t*ð Þ and *<sup>T</sup>k s*ð Þ *<sup>i</sup>* ð Þ *x H t*ð Þ are regular functions for *x* 6¼ 0. As ∥*x*∥ ! 0,

$$U\_i^{\mathbb{k}(\boldsymbol{\varepsilon})}(\boldsymbol{\infty}) \sim \ln \|\boldsymbol{\infty}\| A\_{ik}^N(\boldsymbol{e}\_{\boldsymbol{\varepsilon}}), \quad T\_i^{\mathbb{k}(\boldsymbol{\varepsilon})}(\boldsymbol{\infty}) \sim \|\boldsymbol{\infty}\|^{-1} B\_{ik}^N(\boldsymbol{e}\_{\boldsymbol{\varepsilon}}), \quad N = 2,$$

$$U\_i^{\mathbb{k}(\boldsymbol{\varepsilon})}(\boldsymbol{\infty}) \sim \|\boldsymbol{\infty}\|^{-N+2} A\_{ik}^N(\boldsymbol{e}\_{\boldsymbol{\varepsilon}}), \quad T\_i^{\mathbb{k}(\boldsymbol{\varepsilon})}(\boldsymbol{\infty}) \sim \|\boldsymbol{\infty}\|^{-N+1} B\_{ik}^N(\boldsymbol{e}\_{\boldsymbol{\varepsilon}}), \quad N > 2. \tag{32}$$

*Here, ex* <sup>¼</sup> *<sup>x</sup>=*∥*x*∥, *AN ik*ð Þ*<sup>e</sup>* , *and B<sup>N</sup> ik*ð Þ*e are continuous and bounded functions on the sphere* <sup>∥</sup>*e*<sup>∥</sup> <sup>¼</sup> <sup>1</sup>*, and Vk d*ð Þ *<sup>i</sup> andWk d*ð Þ *<sup>i</sup> are regular functions that are continuous at x* ¼ 0*and t*> 0*. For any N,*

$$\mathcal{W}\_i^{k(d)}(\mathbf{x}, t) = \mathbf{0} \quad \mathcal{W}\_i^{k(d)}(\mathbf{x}, t) = \mathbf{0} \quad for \; \parallel \mathbf{x} \parallel > \max\_{k=1, \overline{M}} \max\_{\|\epsilon\|=1} c\_k(e)t,$$

and for odd *N*, these relations hold for ∥*x*∥< min *k*¼1, *M* min ∥*e*∥¼1 *ck*ð Þ*e t:*.
