Solution of an Initial Boundary Value Problem for a Multidimensional Fourth-Order Equation Containing the Bessel Operator

Abstract

In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of a fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If th formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, particularly when they involve operators of the Bessel type. Applying this operator simplifies the problem at hand to a comparable problem, even in the absence of the Bessel operator. An exact solution to the original problem is constructed and analyzed based on the solution to the supplementary problem.

1. Introduction—Formulation of the Problem

Let $x=(x_1,x_2,\dots ,x_n)$ be a point of the n-dimensional Euclidean space $R^n$, $R_{+}^n=\left\{x\in R^n:x_k>0,k=\overline{1,n}\right\}$, $\mathrm{\Omega }=\{(x,t):x\in R_{+}^n,t\in R_{+}^1\}.$

Let us consider the problem of finding a solution to $u\left(x,t\right)\in C^4\left(\mathrm{\Omega }\right)$ in the domain $\mathrm{\Omega }$, using following equation

$$u_{tt}+{\Delta }_B^{2}u=0,\left(x,t\right)\in \mathrm{\Omega },$$

(1)

satisfying the initial

$$u\left(x,0\right)=\phi \left(x\right),u_t\left(x,0\right)=0,x\in R_{+}^n,$$

(2)

and the following boundary conditions

$${\left.{\displaystyle \frac{\partial u}{\partial x_k}}\right|}_{x_k=0}=0,{\left.{\displaystyle \frac{{\partial }^{3}u}{\partial x_k^3}}\right|}_{x_k=0}=0,t>0,k=\overline{1,n},$$

(3)

where ${\Delta }_B^2={\Delta }_B{\Delta }_B$, ${\Delta }_B\equiv {\displaystyle \sum _{k=1}^n}{B}_{{\gamma }_k}^{\left(x_k\right)}$, $B_{{\gamma }_k}^{\left(x_k\right)}={\displaystyle \frac{{\partial }^2}{\partial x_k^2}}+{\displaystyle \frac{2{\gamma }_k+1}{x_k}}{\displaystyle \frac{\partial }{\partial x_k}}$, ${\gamma }_k={\alpha }_k-1/2$, ${\alpha }_k>0$, $k=\overline{1,n},$ $B_{{\gamma }_k}^{\left(x_k\right)}$ is the Bessel operator and $\phi \left(x\right)$ is a given function.

For Equation (1), when ${\gamma }_k=-1/2,k=\overline{1,n}$, transforms into the equation of multidimensional free transverse vibration of a thin elastic plate $u_{yy}+{\Delta }^{2}u=0$, where ${\Delta }^2=\Delta \Delta$ is the biharmonic operator and $\Delta$ is the multidimensional Laplace operator.

In addition, Equation (1) belongs to the class of equations that degenerate in spatial variables at the boundary of the domain. Degenerate equations model processes occurring near the boundaries of different domains. The influence of boundaries leads to the fact that near the boundary, the type or order of the equation describing the process changes. In this case, the equation is said to degenerate.

Let us note that many problems about the vibrations of membranes and plates have important applied significance in structural mechanics, aircraft construction, mechanical engineering, shipbuilding, etc. [1,2].

Many works have been devoted to the study of the initial boundary value problems for Equation (1) for ${\gamma }_k=-1/2,k=\overline{1,n},$—a review of which can be found in [3,4,5]. However, problems for Equation (1) with ${\gamma }_k\ne -1/2,k=\overline{1,n}$ have not been studied to date.

Among the degenerate equations, a special place is occupied by the initial and boundary value problems for partial differential equations with singularities in the coefficients, typical representatives of which are equations with Bessel operators.

The theory of axisymmetric potential [6], Euler–Poisson–Darboux equations [7], Radon transform and tomography [8], gas dynamics and acoustics [9], jet theory in hydrodynamics [10], linearized Maxwell–Einstein equations [11], mechanics, theory of elasticity and plasticity [12], and many other areas also determine the significance of equations from these classes.

The terms B-elliptic, B-hyperbolic, and B-parabolic equations were introduced by I.A. Kipriyanov [13] for equations of the elliptic, hyperbolic, and parabolic forms that have a Bessel operator for one or more variables.

We should note that the complete scope of questions related to equations involving Bessel operators has been thoroughly explored by I.A. Kipriyanov and his students. For more detailed information on this topic, we refer to the monographs by V.V. Katrakhov and S.M. Sitnik [14], as well as S.M. Sitnik and E.L. Shishkina [15].

This work is a continuation of the research of the authors [16,17], devoted to the construction of the exact analytical solutions of the initial-boundary value problems for one-dimensional and multidimensional equations of a beam and a plate using the method of transmutation operators.

Definition 1

([14,18]). For the given two operators $(A,B)$, if there exists a non-null operator T that satisfies the following relation

$$TA=BT,$$

(4)

then operator T is called a transmutation operator (TO).

The spaces or sets of functions that operators A and B, and hence T act on must be identified in order for (4) to be a strict definition. The theory of TO and its applications are presented in the monographs [14,15,18,19,20].

The purpose of this work is to construct an exact solution of the initial boundary value problem (1)–(3) for the equation of multidimensional free transverse vibration of a thin elastic plate (1) with the Bessel operator using the method of transmutation operators.

In contrast with the cited sources, to solve the problem, we will use a different approach. Namely, taking into account the specifics of equations with singular coefficients, we will use the multidimensional generalized Erdélyi—Kober operator [21,22]. The use of this operator allows us to reduce equations with a singular Bessel operator, which acts on several variables, to non-singular equations. This approach to multidimensional iterated equations was used in [23,24]. In addition, it is very effective and allows us to construct an explicit formula for solving the formulated problem. This formula expresses the solution of the initial-boundary value problem in a compact form through the initial data. It allows us to directly see the nature of the dependence of the solution on the initial functions; in particular, to establish the conditions for the smoothness of the classical solution. This approach was used to solve initial and boundary value problems for hyperbolic equations with the Bessel operator in [25,26,27].

It should be noted that in problems of the general theory of partial differential equations containing the Bessel operator in one or more variables, the main research apparatus is the corresponding integral Fourier or Fourier–Bessel transform. Unlike traditional methods, to achieve the goal of this work, we need to study two problems, which consist of studying the properties of the generalized Erdélyi—Kober operator and their application to the study of the initial-boundary value problem (1)–(3). Therefore, we first study the properties of this operator.

2. Multidimensional Erdélyi–Kober Transmutation Operator

In both theory and practice, classical fractional-order integration and differentiation operators are frequently modified and generalized. Notably among these improvements are the Erdélyi–Kober operators [21].

The multidimensional generalization of the Erdélyi–Kober operator was introduced in [22], in the following form

$$J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n}\left(\begin{array}{c}{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\hfill \\ {\eta }_1,{\eta }_2,\dots ,{\eta }_n\hfill \end{array}\right)f\left(x\right)$$

$$=J_{{\lambda }_1}^{x_1}({\eta }_1,{\alpha }_1)J_{{\lambda }_2}^{x_2}({\eta }_2,{\alpha }_2)\cdots J_{{\lambda }_n}^{x_n}({\eta }_n,{\alpha }_n)f\left(x\right)$$

$$=\left[\prod _{k=1}^n{\displaystyle \frac{2x_k^{-2({\alpha }_k+{\eta }_k)}}{\mathrm{\Gamma }\left({\alpha }_k\right)}}\right]\underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[t_k^{2{\eta }_k+1}{\left(x_k^2-t_k^2\right)}^{{\alpha }_k-1}\right.$$

$$\times \left.{\overline{J}}_{{\alpha }_k-1}\left(\lambda \sqrt{x_k^2-t_k^2}\right)\right]f(t_1,t_2,\dots ,t_n)d{t}_{1}d{t}_2\dots d{t}_n,$$

(5)

where $\lambda ,\alpha ,\eta \in R^n,$ ${\alpha }_k>0,{\eta }_k\ge -1/2,$ $k=\overline{1,n};$ $\mathrm{\Gamma }\left(\alpha \right)$ is Euler’s gamma function; ${\overline{J}}_{\nu }\left(z\right)$ is the Bessel–Clifford function, which is expressed by the Bessel functions $J_{\nu }\left(z\right),$ by the formulas ${\overline{J}}_{\nu }\left(z\right)=\mathrm{\Gamma }(\nu +1){(z/2)}^{-\nu }{J}_{\nu }\left(z\right)$; and $J_{{\lambda }_k}^{x_k}({\eta }_k,{\alpha }_k)$ is a partial integral of the Erdélyi–Kober of the ${\alpha }_k$-order of the k-th variable

$$J_{{\lambda }_k}^{x_k}({\eta }_k,{\alpha }_k)f\left(x\right)={\displaystyle \frac{2x_k^{-2({\alpha }_k+{\eta }_k)}}{\mathrm{\Gamma }\left({\alpha }_k\right)}}\underset{0}{\overset{x_k}{\int }}{(x_k^2-t^2)}^{{\alpha }_k-1}{\overline{J}}_{{\alpha }_k-1}\left({\lambda }_k\sqrt{x_k^2-t^2}\right)$$

$$\times t^{2{\eta }_k+1}f(x_1,x_2,\dots ,x_{k-1},t,x_{k+1},\dots ,x_n)dt.$$

In this work, we studied the main properties of the operator (5) and we showed that the inverse operator had the following form

$$J_{\lambda }^{-1}\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{i\lambda }\left(\begin{array}{c}-\alpha \hfill \\ \eta +\alpha \hfill \end{array}\right)f\left(x\right)$$

$$=2^{n-\left|m\right|}\left[\prod _{k=1}^n{\displaystyle \frac{x_k^{-2{\eta }_k}}{\mathrm{\Gamma }(m_k-{\alpha }_k)}}{\left({\displaystyle \frac{1}{x_k}}{\displaystyle \frac{\partial }{\partial x_k}}\right)}^{m_k}\right]\underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[t_k^{2({\eta }_k+{\alpha }_k)+1}\right.$$

$$\times \left.{\left(x_k^2-t_k^2\right)}^{m_k-1-{\alpha }_k}{\overline{I}}_{m_k-1-{\alpha }_k}\left(\lambda \sqrt{x_k^2-t_k^2}\right)\right]f(t_1,t_2,\dots ,t_n)d{t}_{1}d{t}_2\dots d{t}_n,$$

(6)

where ${\alpha }_k>0,m_k=\left[{\alpha }_k\right]+1,$ ${\eta }_k\ge -1/2,$ $k=\overline{1,n},$ ${\overline{I}}_{\nu }\left(z\right)=\mathrm{\Gamma }(\nu +1){(z/2)}^{-\nu }{I}_{\nu }\left(z\right),$ $I_{\nu }\left(z\right)$ is the modified Bessel function. $m=(m_1,m_2,\dots ,m_n)$ is a multi-index, and $\left|m\right|=m_1+m_2+\cdots +m_n$ its length.

Taking into account ${\overline{J}}_{\nu }\left(0\right)=1$, in the limit for ${\lambda }_k\to 0,k=\overline{1,n},$ we obtain

$$J_0\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{0,0,\dots ,0}\left(\begin{array}{c}{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\hfill \\ {\eta }_1,{\eta }_2,\dots ,{\eta }_n\hfill \end{array}\right)f\left(x\right)$$

$$=\prod _{k=1}^n\left[{\displaystyle \frac{2x_k^{-2({\alpha }_k+{\eta }_k)}}{\mathrm{\Gamma }\left({\alpha }_k\right)}}\right]\underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[t_k^{2{\eta }_k+1}{\left(x_k^2-t_k^2\right)}^{{\alpha }_k-1}\right]f\left(t\right)d{t}_1\dots d{t}_n.$$

(7)

This operator can be considered a multidimensional analog of the ordinary Erdélyi–Kober operator. In this case, the inverse operator of (6) will have the following form

$$J_0^{-1}\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=2^{n-\left|m\right|}\left[\prod _{k=1}^n{\displaystyle \frac{x_k^{-2{\eta }_k}}{\mathrm{\Gamma }(m_k-{\alpha }_k)}}{\left({\displaystyle \frac{1}{x_k}}{\displaystyle \frac{\partial }{\partial x_k}}\right)}^{m_k}\right]$$

$$\underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[t_k^{2({\eta }_k+{\alpha }_k)+1}\right.\left.{\left(x_k^2-t_k^2\right)}^{m_k-1-{\alpha }_k}\right]f(t_1,t_2,\dots ,t_n)d{t}_{1}d{t}_2\dots d{t}_n.$$

(8)

In addition, the following theorems were proved in [22,23,24]:

Let ${\left[B_{{\eta }_k}^{x_k}\right]}^0=E,$ where E is the unit operator, ${\left[B_{{\eta }_k}^{x_k}\right]}^{m_k}=$ ${\left[B_{{\eta }_k}^{x_k}\right]}^{m_k-1}\left[B_{{\eta }_k}^{x_k}\right]$ is the $m_k$ th power of the operator $B_{{\eta }_k}^{x_k},k=\overline{0,n}.$

Theorem 1

([23,24]). Let ${\alpha }_k>0,{\eta }_k\ge -1/2;$ $f\left(x\right)\in C^{2m_0}\left({\mathrm{\Omega }}^n\right);$ $x_k^{2{\eta }_k+1}{\left[B_{{\eta }_k}^{x_k}\right]}^{p_k+1}f\left(x\right)$ functions are integrable in a neighborhood of the origin and $\underset{x_k\to 0}{lim}{x}_k^{2{\eta }_k+1}(\partial /\partial x_k){\left[B_{{\eta }_k}^{x_k}\right]}^{p_k}f\left(x\right)=0,$ $p_k=\overline{0,m_k-1},$ $k=\overline{1,n}.$ Then,

$${\left[B_{{\eta }_k+{\alpha }_k}^{x_k}+{\lambda }_k^2\right]}^{m_k}{J}_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right){\left[B_{{\eta }_k}^{x_k}\right]}^{m_k}f\left(x\right),k=\overline{1,n},$$

where $m_0=max\{m_1,m_2,\dots ,m_n\}.$

We should note that Theorem 1 also holds when any or all of the ${\lambda }_k=0,k=\overline{1,n}.$

Corollary 1.

Assume that the conditions of Theorem 1 are fulfilled. Then,

$$\sum _{k=1}^n{\left[B_{{\eta }_k+{\alpha }_k}^{x_k}+{\lambda }_k^2\right]}^{m_k}{J}_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)\sum _{k=1}^n{\left[B_{{\eta }_k}^{x_k}\right]}^{m_k}f\left(x\right),$$

in addition, if $f\left(x\right)\in C^{2\left|m\right|}\left({\mathrm{\Omega }}^n\right),$ then

$$\prod _{k=1}^n{\left[B_{{\eta }_k+{\alpha }_k}^{x_k}+{\lambda }_k^2\right]}^{m_k}{J}_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)\prod _{k=1}^n{\left[B_{{\eta }_k}^{x_k}\right]}^{m_k}f\left(x\right).$$

Theorem 2

([23,24]). Let ${\alpha }_k>0,{\eta }_k\ge -1/2,$ $q\in N;$ $f\left(x\right)\in C^{2q}\left({\mathrm{\Omega }}^n\right);$ the functions $x_k^{2{\eta }_k+1}{\left[B_{{\eta }_k}^{x_k}\right]}^{l+1}f\left(x\right)$ are integrable in a neighborhood of the origin and

$$\underset{x_k\to 0}{lim}{x}_k^{2{\eta }_k+1}(\partial /\partial x_k){\left[B_{{\eta }_k}^{x_k}\right]}^{l}f\left(x\right)=0,l=\overline{0,q-1},k=\overline{1,n}.$$

Then,

$${\left[\sum _{k=1}^n\left(B_{{\eta }_k+{\alpha }_k}^{x_k}+{\lambda }_k^2\right)\right]}^q{J}_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right)f\left(x\right)=J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ \eta \hfill \end{array}\right){\left[\sum _{k=1}^n{B}_{{\eta }_k}^{x_k}\right]}^{q}f\left(x\right).$$

Corollary 2.

Assume that the conditions of Theorem 2 are fulfilled. Then, for ${\eta }_k=-1/2,k=\overline{1,n},$

$${\left[\sum _{k=1}^n\left(B_{{\alpha }_k-1/2}^{x_k}+{\lambda }_k^2\right)\right]}^q{J}_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ -1/2\hfill \end{array}\right)f\left(x\right)=J_{\lambda }\left(\begin{array}{c}\alpha \hfill \\ -1/2\hfill \end{array}\right){\Delta }^{q}f\left(x\right),$$

in particular, for ${\lambda }_k=0,$ we have the equality

$${\Delta }_B^q{J}_0\left(\begin{array}{c}\alpha \hfill \\ -1/2\hfill \end{array}\right)f\left(x\right)=J_0\left(\begin{array}{c}\alpha \hfill \\ -1/2\hfill \end{array}\right){\Delta }^{q}f\left(x\right).$$

(9)

3. Solving the Problem

We will seek a solution to problem (1)–(3) in the form

$$u\left(x,t\right)=J_0^{\alpha }\upsilon \left(x,t\right),$$

(10)

where $\upsilon \left(x,t\right)$ is an unknown four times continuously differentiable function, and $J_0^{\alpha }=J_0\left(\begin{array}{c}\alpha \hfill \\ -1/2\hfill \end{array}\right)$ is the multidimensional Erdélyi–Kober operator (7).

Let us substitute (10) into Equation (1), the initial conditions (2) and boundary conditions (3), taking into account Corollary 2 (see (9)) and Formula (8), then we come to the problem of solving the following equation $\upsilon \left(x,t\right)$

$${\upsilon }_{tt}+{\Delta }^2\upsilon =0,\left(x,t\right)\in \mathrm{\Omega },$$

(11)

satisfying the initial

$$\upsilon \left(x,0\right)=\mathrm{\Phi }\left(x\right),{\upsilon }_t\left(x,0\right)=0,x\in R_{+}^n,$$

(12)

and the boundary conditions

$${\left.{\displaystyle \frac{\partial \upsilon }{\partial x_k}}\right|}_{x_k=0}=0,{\left.{\displaystyle \frac{{\partial }^3\upsilon }{\partial x_k^3}}\right|}_{x_k=0}=0,t\in R_{+}^1,$$

(13)

where

$$\mathrm{\Phi }\left(x\right)={\left[J_0^{\alpha }\right]}^{-1}\phi \left(x\right)=\prod _{j=1}^n\left[{\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-{\alpha }_j\right)}}\right]{\displaystyle \frac{{\partial }^n}{\partial x_1\partial x_2\dots \partial x_n}}$$

$$\times \underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[s_k^{2{\alpha }_k}{\left(x_k^2-s_k^2\right)}^{-{\alpha }_k}\right]\phi \left(s_1,s_2,\dots ,s_n\right)d{s}_{1}d{s}_2\dots d{s}_n.$$

(14)

We extend the function $\mathrm{\Phi }\left(x\right)$ given in (12), taking into account the boundary conditions (13), to $x_k⩽0,$ $k=\overline{1,n}$ in an even manner and denote the extended function by ${\mathrm{\Phi }}_1\left(x\right)$.

Then, in the half-space ${\mathrm{\Omega }}^{+}=\left\{\left(x,t\right):x\in R^n,t\in R_{+}^1\right\}$ we obtain the problem of finding a solution to Equation (11) that satisfies the initial conditions

$$\upsilon \left(x,0\right)={\mathrm{\Phi }}_1\left(x\right),{\upsilon }_t\left(x,0\right)=0,x\in R^n.$$

(15)

Let

$${\mathrm{\Phi }}_1\left(x\right)\in C^{n+3}\left(R^n\right),{\left|x\right|}^{n+5}\left|{\mathrm{\Phi }}_1\left(x\right)\right|<M,{\left|x\right|}^{n+1}\left|D^{\beta }{\mathrm{\Phi }}_1\left(x\right)\right|<M,\left|\beta \right|⩽n+3,$$

(16)

where $M=const>0,$ $\beta$ is a multi index and $\left|\beta \right|$ is its length. Then, the solution to problem (11) and (15) has the form [28]:

$$\upsilon \left(x,t\right)={\displaystyle \frac{1}{{\left(2\sqrt{\pi t}\right)}^n}}\underset{R^n}{\int }{\mathrm{\Phi }}_1\left(\xi \right)cos\left({\displaystyle \frac{{\left|x-\xi \right|}^2}{4t}}-{\displaystyle \frac{\pi n}{4}}\right)d\xi .$$

(17)

Considering

$$Re\left\{exp\left(i\left({\displaystyle \frac{{\left|x-\xi \right|}^2}{4t}}-{\displaystyle \frac{\pi n}{4}}\right)\right)\right\}=cos\left({\displaystyle \frac{{\left|x-\xi \right|}^2}{4t}}-{\displaystyle \frac{\pi n}{4}}\right)$$

we rewrite equality (17) in the form

$$\upsilon \left(x,t\right)=Rew\left(x,t\right),$$

where

$$w\left(x,t\right)={\displaystyle \frac{e^{-i\frac{\pi n}{4}}}{{\left(2\sqrt{\pi t}\right)}^n}}\underset{R^n}{\int }{\mathrm{\Phi }}_1\left(\xi \right)exp\left[i{\displaystyle \frac{{\left|x-\xi \right|}^2}{4t}}\right]d\xi .$$

(18)

We rewrite equality (14) in the form

$$\mathrm{\Phi }\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)={\displaystyle \frac{{\partial }^n{\mathrm{\Phi }}_0\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)}{\partial {\xi }_1\partial {\xi }_2\dots \partial {\xi }_n}},$$

(19)

where $0<{\alpha }_k<1,$ $k=\overline{1,n}$,

$${\mathrm{\Phi }}_0\left(\xi \right)=\prod _{j=1}^n\left[{\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-{\alpha }_j\right)}}\right]\underset{0}{\overset{{\xi }_1}{\int }}\underset{0}{\overset{{\xi }_2}{\int }}\dots \underset{0}{\overset{{\xi }_n}{\int }}\prod _{k=1}^n\left[s_k^{2{\alpha }_k}{\left({\xi }_k^2-s_k^2\right)}^{-{\alpha }_k}\right]\phi \left(s\right)ds.$$

(20)

Changing the variables in the last integral by $s_k={\xi }_k{\mu }_k,$ $k=\overline{1,n},$ we obtain

$${\mathrm{\Phi }}_0\left(\xi \right)=\prod _{j=1}^n\left[{\displaystyle \frac{{\xi }_j}{\mathrm{\Gamma }\left(1-{\alpha }_j\right)}}\right]\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\dots \underset{0}{\overset{1}{\int }}\prod _{k=1}^n\left[{\mu }_k^{2{\alpha }_k}{\left(1-{\mu }_k^2\right)}^{-{\alpha }_k}\right]\phi \left({\xi }_1{\mu }_1,\dots ,{\xi }_n{\mu }_n\right)d{\mu }_1\dots d{\mu }_n.$$

Hence, it follows that ${\left.{\mathrm{\Phi }}_0\left(\xi \right)\right|}_{{\xi }_j=0}=0$, $j=\overline{1,n}$.

Let

$$\phi \left(x\right)\in C^{2n+3}\left(R^n\right),{\left|x\right|}^{2n+5}\left|\phi \left(x\right)\right|<M_1,{\left|x\right|}^{2n+1}\left|D^{\beta }\phi \left(x\right)\right|<M_1,\left|\beta \right|⩽2n+3,$$

(21)

$M_1=const>0$, then, by virtue of (19) and (20), the conditions (16) hold.

Taking into account ${\left|x-\xi \right|}^2={\displaystyle \sum _{k=1}^n}{\left(x_k-{\xi }_k\right)}^2$, we rewrite Formula (18) in the form

$$w\left(x,t\right)={\displaystyle \frac{e^{-i\frac{\pi n}{4}}}{{\left(2\sqrt{\pi t}\right)}^n}}\underset{R^n}{\int }{\mathrm{\Phi }}_1\left(\xi \right)\prod _{j=1}^{n}exp\left[i{\displaystyle \frac{{\left(x_j-{\xi }_j\right)}^2}{4t}}\right]d\xi$$

$$={\displaystyle \frac{e^{-i\frac{\pi n}{4}}}{{\left(2\sqrt{\pi t}\right)}^n}}\prod _{j=1}^n\left\{{\displaystyle \frac{1}{2\sqrt{\pi t}}}\underset{-\infty }{\overset{+\infty }{\int }}exp\left[i{\displaystyle \frac{{\left(x_j-{\xi }_j\right)}^2}{4t}}\right]d{\xi }_j\right\}$$

$$\times \left[{\displaystyle \frac{1}{2\sqrt{\pi t}}}\underset{-\infty }{\overset{+\infty }{\int }}exp\left[i{\displaystyle \frac{{\left(x_n-{\xi }_n\right)}^2}{4t}}\right]{\mathrm{\Phi }}_1\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)d{\xi }_n\right].$$

Taking into account the parity of the function ${\mathrm{\Phi }}_1\left(\xi \right)$, we obtain

$$w\left(x,t\right)=e^{-i{\displaystyle \frac{\pi n}{4}}}\prod _{j=1}^{n-1}\left\{{\displaystyle \frac{1}{2\sqrt{\pi t}}}\underset{0}{\overset{+\infty }{\int }}\left[exp\left(-{\displaystyle \frac{{\left(x_j-{\xi }_j\right)}^2}{4it}}\right)+exp\left(-{\displaystyle \frac{{\left(x_j+{\xi }_j\right)}^2}{4it}}\right)\right]d{\xi }_j\right\}$$

$$\times \left\{{\displaystyle \frac{1}{2\sqrt{\pi t}}}\underset{0}{\overset{+\infty }{\int }}\left[exp\left(-{\displaystyle \frac{{\left(x_n-{\xi }_n\right)}^2}{4it}}\right)+exp\left(-{\displaystyle \frac{{\left(x_n+{\xi }_n\right)}^2}{4it}}\right)\right]\mathrm{\Phi }\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)d{\xi }_n\right\}.$$

Let

$$G_{\epsilon }\left(x_k,{\xi }_k,t\right)={\displaystyle \frac{1}{2\sqrt{\pi t}}}\left[exp\left(-{\displaystyle \frac{{\left(x_k-{\xi }_k\right)}^2}{4a_{\epsilon }}}\right)+exp\left(-{\displaystyle \frac{{\left(x_k+{\xi }_k\right)}^2}{4a_{\epsilon }}}\right)\right],a_{\epsilon }=\epsilon +it,\epsilon >0.$$

Then. $w\left(x,t\right)=\underset{\epsilon \to +0}{lim}{w}_{\epsilon }\left(x,t\right)$, where

$$w_{\epsilon }\left(x,t\right)=e^{-i{\displaystyle \frac{\pi n}{4}}}\prod _{j=1}^{n-1}\left\{\underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon }\left(x_j,{\xi }_j,t\right)d{\xi }_j\right\}\underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon }\left(x_n,{\xi }_n,t\right){\displaystyle \frac{{\partial }^n{\mathrm{\Phi }}_0\left(\xi \right)}{\partial {\xi }_1\partial {\xi }_2\dots \partial {\xi }_n}}d{\xi }_n.$$

(22)

Applying the integrating by parts rule to the last integral of (22), and taking into account $\underset{{\xi }_j\to \infty }{lim}{\mathrm{\Phi }}_0\left(\xi \right)=0$, we obtain

$$w_{\epsilon }\left(x,t\right)=e^{-i{\displaystyle \frac{\pi n}{4}}}\prod _{j=1}^{n-1}\left\{\underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon {\xi }_j}\left(x_j,{\xi }_j,t\right)d{\xi }_j\right\}\underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon {\xi }_n}\left(x_n,{\xi }_n,t\right){\mathrm{\Phi }}_0\left({\xi }_1,{\xi }_2,\dots ,{\xi }_n\right)d{\xi }_n.$$

(23)

Next, applying the Formula [29]

$$\underset{0}{\overset{+\infty }{\int }}{e}^{-a{\lambda }^2}cos\left(b\lambda \right)d\lambda =\sqrt{{\displaystyle \frac{\pi }{4a}}}{e}^{-{\displaystyle \frac{b^2}{4a}}},Rea>0,$$

at $b=x_j-{\xi }_j,$ $a_{\epsilon }=\epsilon +it,\epsilon >0$, $t>0$, we have

$$G_{\epsilon }\left(x_j,{\xi }_j,t\right)={\displaystyle \frac{2}{\pi }}{\displaystyle \frac{\sqrt{a_{\epsilon }}}{\sqrt{t}}}\underset{0}{\overset{+\infty }{\int }}{e}^{-a_{\epsilon }{\lambda }^2}cos\left(x_j\lambda \right)cos\left({\xi }_j\lambda \right)d\lambda ,$$

Computing the derivative of this function, we obtain

$$G_{\epsilon {\xi }_j}\left(x_j,{\xi }_j,t\right)=-{\displaystyle \frac{2}{\pi }}{\displaystyle \frac{\sqrt{a_{\epsilon }}}{\sqrt{t}}}\underset{0}{\overset{+\infty }{\int }}{e}^{-a_e{\lambda }^2}cos\left(x_j\lambda \right)sin\left({\xi }_j\lambda \right)\lambda d\lambda .$$

(24)

Substituting the (20) expression of the function ${\mathrm{\Phi }}_0\left(\xi \right)$ into (23) and taking (24) into account, we have

$$w_{\epsilon }\left(x,t\right)=e^{-i{\displaystyle \frac{\pi n}{4}}}\prod _{j=1}^n\left[{\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-{\alpha }_j\right)}}\right]\prod _{j=1}^n\left\{\underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon {\xi }_j}\left(x_j,{\xi }_j,t\right)d{\xi }_j\underset{0}{\overset{{\xi }_j}{\int }}{s}_j^{2{\alpha }_j}{\left({\xi }_j^2-s_j^2\right)}^{-{\alpha }_j}d{s}_j\right\}$$

$$\times \underset{0}{\overset{+\infty }{\int }}{G}_{\epsilon {\xi }_n}\left(x_n,{\xi }_n,t\right)d{\xi }_n\underset{0}{\overset{{\xi }_n}{\int }}{s}_n^{2{\alpha }_n}{\left({\xi }_n^2-s_n^2\right)}^{-{\alpha }_n}\phi \left(s\right)d{s}_n.$$

In the last equality, through successively changing the order of integration, we obtain

$$w_{\epsilon }\left(x,t\right)=e^{-i{\displaystyle \frac{\pi n}{4}}}\prod _{j=1}^n\left[{\displaystyle \frac{1}{\mathrm{\Gamma }\left(1-{\alpha }_j\right)}}\right]\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)\prod _{j=1}^n\left[s_j^{2{\alpha }_j}{P}_{\epsilon }\left(x_j,s_j,t\right)\right]ds,$$

(25)

where

$$P_{\epsilon }\left(x_j,s_j,t\right)=\underset{s_j}{\overset{+\infty }{\int }}{G}_{\epsilon {\xi }_j}\left(x_j,{\xi }_j,t\right){\left({\xi }_j^2-s_j^2\right)}^{-{\alpha }_j}d{\xi }_j$$

$$=-{\displaystyle \frac{2}{\pi }}{\displaystyle \frac{\sqrt{a_{\epsilon }}}{\sqrt{t}}}\underset{0}{\overset{+\infty }{\int }}{e}^{-a_{\epsilon }{\lambda }^2}cos\left(x_j\lambda \right)\lambda \left\{\underset{s_j}{\overset{\infty }{\int }}{\left({\xi }_j^2-s_j^2\right)}^{-{\alpha }_j}sin\left({\xi }_j\lambda \right)\lambda d{\xi }_j\right\}d\lambda .$$

Applying the Mehler–Sonin formula to the internal integral [30] (p. 93), we obtain

$$P_{\epsilon }\left(x_j,s_j,t\right)=-{\displaystyle \frac{2^{\frac{1}{2}-{\alpha }_j}}{\sqrt{\pi }}}{\displaystyle \frac{\sqrt{a_{\epsilon }}}{\sqrt{t}}}\mathrm{\Gamma }\left(1-{\alpha }_j\right)s_j^{{\displaystyle \frac{1}{2}}-{\alpha }_j}\underset{0}{\overset{\infty }{\int }}{e}^{-a_{\epsilon }{\lambda }^2}{\lambda }^{{\alpha }_j+{\displaystyle \frac{1}{2}}}cos\left(x_j\lambda \right)J_{{\alpha }_j-{\displaystyle \frac{1}{2}}}\left(s_j\lambda \right)d\lambda ,$$

(26)

where $J_{\nu }\left(z\right)$ is the Bessel function of the first type.

Substituting (26) into (25), we obtain

$$w_{\epsilon }\left(x,t\right)={(-1)}^n{e}^{-i{\displaystyle \frac{\pi n}{4}}}{\displaystyle \frac{2^{\frac{n}{2}-\left|\alpha \right|}}{{\left(\sqrt{\pi }\right)}^n}}{\left({\displaystyle \frac{\sqrt{a_{\epsilon }}}{\sqrt{t}}}\right)}^n\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)\prod _{j=1}^n\left(s_j^{1/2+{\alpha }_j}{P}_{1\epsilon }\left(x_j,s_j,t\right)\right)d{s}_n,$$

(27)

where

$$P_{1\epsilon }\left(x_j,s_j,t\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-a_{\epsilon }{\lambda }^2}{\lambda }^{{\alpha }_j+1/2}{J}_{{\alpha }_j-1/2}\left(s_j\lambda \right)cos\left(x_j\lambda \right)d\lambda .$$

Applying the theorem on the passage to the limit under the improper integral sign and taking into account $\underset{\epsilon \to +0}{lim}\sqrt{a_{\epsilon }}=\sqrt{it}=e^{i{\displaystyle \frac{\pi }{4}}}\sqrt{t},t>0,$ from (27), we obtain

$$\upsilon \left(x,t\right)=Re\underset{\epsilon \to +0}{lim}{w}_{\epsilon }\left(x,t\right)={(-1)}^n{\displaystyle \frac{2^{\frac{3n}{2}-\left|\alpha \right|}}{{\left(\sqrt{\pi }\right)}^n}}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)\prod _{j=1}^n\left(s_j^{1/2+{\alpha }_j}{P}_2\left(x_j,s_j,t\right)\right)ds,$$

(28)

where $P_2\left(x_j,s_j,t\right)=\underset{0}{\overset{\infty }{\int }}cos\left(t{\lambda }^2\right){\lambda }^{{\alpha }_j+1/2}{J}_{{\alpha }_j-1/2}\left(s_j\lambda \right)cos\left(x_j\lambda \right)d\lambda .$

Substituting equality (28) into (10), we obtain

$$u\left(x,t\right)=J_0^{\alpha }\upsilon \left(x,t\right)={(-1)}^n\prod _{j=1}^n\left[{\displaystyle \frac{2x_j^{1-2{\alpha }_j}}{\mathrm{\Gamma }\left({\alpha }_j\right)}}\right]{\displaystyle \frac{2^{\frac{3n}{2}-\left|\alpha \right|}}{{\left(\sqrt{\pi }\right)}^n}}\underset{0}{\overset{x_1}{\int }}\underset{0}{\overset{x_2}{\int }}\dots \underset{0}{\overset{x_n}{\int }}\prod _{k=1}^n\left[{\left(x_k^2-{\xi }_k^2\right)}^{{\alpha }_k-1}\right]$$

$$\times \left\{\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)\prod _{j=1}^n\left[s_j^{1/2-{\alpha }_j}{P}_2\left(x_j,s_j,t\right)\right]ds\right\}d\xi .$$

Changing the order of integration, we have

$$u\left(x,t\right)={(-1)}^n{\displaystyle \frac{2^{\frac{3n}{2}-\left|\alpha \right|}}{{\left(\sqrt{\pi }\right)}^n}}\prod _{j=1}^n\left[{\displaystyle \frac{x_j^{1-2{\alpha }_j}}{\mathrm{\Gamma }\left({\alpha }_j\right)}}\right]\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)\prod _{k=1}^n\left[s_k^{1/2+{\alpha }_k}\right]$$

$$\times \left\{\prod _{j=1}^n\left[\underset{0}{\overset{x_k}{\int }}{\left(x_k^2-{\xi }_k^2\right)}^{{\alpha }_k-1}{P}_2\left(x_j,s_j,t\right)\right]d\xi \right\}ds.$$

Now, we will compute the internal integral

$$\underset{0}{\overset{x_k}{\int }}{\left(x_k^2-{\xi }_k^2\right)}^{{\alpha }_k-1}{P}_2\left(x_j,s_j,t\right)d{\xi }_k$$

$$=\underset{0}{\overset{\infty }{\int }}cos\left(t{\lambda }^2\right){\lambda }^{{\alpha }_j+1/2}{J}_{{\alpha }_j-1/2}\left(s_j\lambda \right)d\lambda \underset{0}{\overset{x_k}{\int }}{\left(x_k^2-{\xi }_k^2\right)}^{{\alpha }_k-1}cos\left({\xi }_k\lambda \right)d{\xi }_k.$$

Hence, applying the Poisson Formula [30] (p. 93), we obtain

$$u\left(x,t\right)=\prod _{k=1}^n\left[x_k^{1/2-{\alpha }_k}\right]\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)$$

$$\times \prod _{j=1}^n\left[s_j^{1/2+{\alpha }_j}\underset{0}{\overset{\infty }{\int }}cos\left(t{\lambda }^2\right)\lambda J_{{\alpha }_j-1/2}\left(s_j\lambda \right)J_{{\alpha }_j-1/2}\left(x_j\lambda \right)d\lambda \right]ds.$$

Applying the following Formula [29] (p. 201)

$$\underset{0}{\overset{\infty }{\int }}\lambda cos\left(a{\lambda }^2\right)J_{\nu }\left(b\lambda \right)J_{\nu }\left(c\lambda \right)d\lambda ={\displaystyle \frac{1}{2a}}{J}_{\nu }\left({\displaystyle \frac{bc}{2a}}\right)sin\left[{\displaystyle \frac{b^2+c^2}{4a}}-{\displaystyle \frac{\nu \pi }{2}}\right],\left[Re\nu >-1,a,b,c>0\right],$$

for $a=t>0,b=s_j>0,c=x_j>0,$ $\nu ={\alpha }_j-1/2>-1;$ ${\alpha }_j>-1/2$, we have

$$\underset{0}{\overset{\infty }{\int }}\lambda cos\left(t{\lambda }^2\right)J_{{\alpha }_j-1/2}\left(s_j\lambda \right)J_{{\alpha }_j-1/2}\left(x_j\lambda \right)d\lambda$$

$$={\displaystyle \frac{1}{2t}}{J}_{{\alpha }_j-1/2}\left({\displaystyle \frac{x_j{s}_j}{2t}}\right)sin\left[{\displaystyle \frac{x_j^2+s_j^2}{4t}}-{\displaystyle \frac{\pi }{2}}\left({\alpha }_j-1/2\right)\right].$$

Thus, we have obtained the representation of the solution of the problem as follows

$$u\left(x,t\right)={\displaystyle \frac{1}{{\left(2t\right)}^n}}\prod _{k=1}^n\left[x_k^{{\displaystyle \frac{1}{2}}-{\alpha }_k}\right]\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\dots \underset{0}{\overset{\infty }{\int }}\phi \left(s\right)$$

$$\times \prod _{j=1}^n\left[s_j^{{\displaystyle \frac{1}{2}}+{\alpha }_j}{J}_{{\alpha }_j-{\displaystyle \frac{1}{2}}}\left({\displaystyle \frac{x_j{s}_j}{2t}}\right)sin\left[{\displaystyle \frac{x_j^2+s_j^2}{4t}}-{\displaystyle \frac{\pi }{2}}\left({\alpha }_j-{\displaystyle \frac{1}{2}}\right)\right]\right]ds.$$

(29)

The following theorem is true

Theorem 3.

Let conditions (21) be fulfilled. Then, the function $u(x,t)$ defined by (29) will be a solution to the problem (1)–(3).

We should note that the Formula (29), when $n=1,$ coincides with the formula obtained in [16].

4. Conclusions

By employing the Erdélyi–Kober transmutation operator, an exact solution to the problem is obtained. Although modern computer technology has advanced significantly, constructing exact solutions to boundary value problems for partial differential equations remains a crucial and pressing challenge. These solutions offer deeper insights into the qualitative characteristics of the processes and phenomena being modeled, reveal the properties of mathematical models, and can serve as benchmarks for asymptotic, approximate, and numerical methods.