**5. The generalized solution of the Cauchy problem for the KG-equation for** *N* **= 2**

Let us consider the Cauchy problem for the KG equation of the below form:

$$
\Box\_{\iota} \hat{u} \pm m^2 \hat{u} = \hat{f}(\varkappa, t), \varkappa \in \mathbb{R}^N, \quad t > 0,\tag{28}
$$

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

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

*S*� *<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 function of the form [9]:

$$\hat{U} = \frac{H(ct - \left\|\mathbf{x}\right\|)}{2\pi} \frac{ch\left(m\sqrt{c^2t^2 - \left\|\mathbf{x}\right\|^2}\right)}{\sqrt{c^2t^2 - \left\|\mathbf{x}\right\|^2}}\tag{29}$$

ð32Þ

ð33Þ

ð34Þ

is *Bessel function. Because* [10],

*DOI: http://dx.doi.org/10.5772/intechopen.91693*

*the form:*

**47**

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

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

*Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon…*

**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

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

analytic continuation by replacing m with im. It has the form:

with a weak singularity at the front :

$$
\hat{U} \approx \frac{1}{2\pi\sqrt{c^2t^2 - \|\mathbf{x}\|^2}} \qquad \text{by } \|\mathbf{x}\| \to ct - 0 \tag{30}
$$

Its carrier is a light cone: .

**Theorem 4.** *If , , then the solution of the Cauchy problem has the form:*

$$2\pi c^{2}u(\mathbf{x},t)H(t) = \int\_{S\_{r}^{-}(\mathbf{x})} \frac{ch\left(m\sqrt{c^{2}t^{2}-r^{2}}\right)}{\sqrt{c^{2}r^{2}-r^{2}}} \dot{u}\_{0}(\mathbf{y})dV(\mathbf{y}) + $$

$$+\partial\_{t} \int \frac{ch\left(m\sqrt{c^{2}t^{2}-r^{2}}\right)}{\sqrt{c^{2}t^{2}-r^{2}}} u\_{0}(\mathbf{y})dV(\mathbf{y}) - \int\_{0}^{t} d\tau \int \frac{ch\left(m\sqrt{c^{2}\tau^{2}-r^{2}}\right)}{\sqrt{c^{2}\tau^{2}-r^{2}}} f\left(\mathbf{y},t-\tau\right)dV(\mathbf{y})$$

**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 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 formulas (23) and (29).

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