**6. Generalized solution to the Cauchy problem for the KG equation for** *N* **= 3**

For *N* = 3, the Green function (28) (for) is a singular generalized function of the form [9]:

$$4\pi \hat{U} = cH\left(t\right)\delta(c^2t^2 - r^2) - mcf\_0(r, t) \tag{31}$$

where is a simple layer on a light cone [9]. The function *is defined by the expression:*

*Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon… DOI: http://dx.doi.org/10.5772/intechopen.91693*

$$f\_0(r,t) = \frac{H\left(ct - r\right)J\_1\left(m\sqrt{c^2t^2 - r^2}\right)}{\sqrt{c^2t^2 - r^2}}\tag{32}$$

is *Bessel function. Because* [10],

where ^*f x*ð Þ , *<sup>t</sup>* is a generalized function.

with a weak singularity at the front :

Its carrier is a light cone: .

*S*�

function of the form [9]:

*problem has the form:*

formulas (23) and (29).

**for** *N* **= 3**

form [9]:

**46**

the solution in Theorem 4 replacing *m* with im*.*

The function *is defined by the expression:*

Let us introduce designations *r* ¼ k k *y* � *x* , *St*ð Þ¼ *x* f g *y*∈*S*, *r*<*ct*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*<sup>t</sup>* ð Þ¼ *x y*∈ *S*� f g , *r*<*ct* and *St*ð Þ¼ *x* f g *y*∈*S*, *r*< *ct* , which we will use further. In the flat case (*N* = 2), the Green's function of Eq. (28) is a regular generalized

**Theorem 4.** *If , , then the solution of the Cauchy*

**Proof.** The integral notation of formula (23) leads to the formula of the theorem. All integrals are proper due to the regularity of integrands. The carrier of the kernel

Similarly, we construct a solution to the Cauchy problem in the case. The solution of the problem in this case allows analytic continuation. It can be obtained from

For *N* = 3, the Green function (28) (for) is a singular generalized function of the

**6. Generalized solution to the Cauchy problem for the KG equation**

where is a simple layer on a light cone [9].

of integrals is a circle expanding over time with the center at the *point x*. Note that if the initial conditions and the right-hand side of equation (1) (source) belong to the class of singular functions admitting convolution with the Green's function of the equation, to construct a solution to the Cauchy problem, use

ð29Þ

ð30Þ

ð31Þ

$$J\_1(z) \sim \mathbf{0}, \mathfrak{F}z \qquad \text{when} \quad z \to \mathbf{0} \tag{33}$$

at the front , the second term has a finite jump:

$$\left[f\_0(r, r/c)\right] = -\frac{m}{2} \tag{34}$$

**Theorem 5.** *The solution of the Cauchy problem for the KG-Eq.* (28) *for N = 3 has the form:*

**Proof.** It follows from the representation of a generalized solution for the Cauchy problem taking into account the form of the fundamental solution (30). The solution of the Cauchy problem for Eq. (28) in the case also allows analytic continuation by replacing m with im. It has the form:

$$\begin{split} 4\pi c u(\mathbf{x},t) &= \frac{1}{2t} \int\_{r=ct} \dot{u}\_0(\mathbf{y}) dS(\mathbf{y}) - m \int\_{S\_t^-(\mathbf{x})} \frac{I\_1\left(m\sqrt{c^2t^2 - r^2}\right)}{\sqrt{c^2t^2 - r^2}} \dot{u}\_0(\mathbf{y}) \,dV(\mathbf{y}) + \\ &+ \frac{1}{2ct^2} \int\_{r=ct} u\_0(\mathbf{y}) dS(\mathbf{y}) + \frac{1}{2ct} \hat{\mathcal{O}}\_t \int\_{r=ct} u\_0(\mathbf{y}) dS(\mathbf{y}) - \\ &- m \hat{\mathcal{O}}\_t \left[ \int\_{S\_t^-(\mathbf{x})} \frac{I\_1\left(m\sqrt{c^2t^2 - r^2}\right)}{\sqrt{c^2t^2 - r^2}} u\_0(\mathbf{y}) \,dV(\mathbf{y}) \right] + \\ &+ \frac{c}{2} \int\_{S\_t^-(\mathbf{x})} r^{-1} f(\mathbf{y}, t - \frac{r}{c}) \,dV(\mathbf{y}) - mc^2 \Big] d\tau \int\_{S\_t^-(\mathbf{x})} \frac{f(\mathbf{y}, t - \tau) I\_1\left(m\sqrt{c^2\tau^2 - r^2}\right)}{\sqrt{c^2\tau^2 - r^2}} \,dW(\mathbf{y}). \end{split}$$

If the initial functions and the right-hand side of Eq. (1) belong to the class of singular functions admitting convolution with the Green function of Eq. (28), to

construct the solution, one should use the formula in ultraprecise form (23). We construct solutions to initial-boundary value problems.
