Method for Investigation of Convergence of Formal Series Involved in Asymptotics of Solutions of Second-Order Differential Equations in the Neighborhood of Irregular Singular Points

Abstract

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irregular singular points, including the point at infinity, which is generally irregular. The article provides a number of applications of the method for studying the asymptotics of solutions to both ordinary differential equations and partial differential equations.

1. Introduction

This study is devoted to investigation of asymptotics of solutions to the second-order differential equations having irregular singular points. It continues study [1], in which the problem of constructing the asymptotics of solutions to ordinary second-order differential equations in the neighborhoods of their irregular singular points is solved. In addition, this problem was solved for a wide class of partial differential equations. It should be noted that the problem of constructing the asymptotics of solutions in the neighborhoods of regular singular points is solved for many types of partial differential equations and ordinary differential equations by Kondrat’ev [2]. It is shown in [2] that for any linear ordinary differential equation with a regular singular point, the asymptotics of its solution are conormal.

It is shown in [1] that the asymptotics of solutions to the second-order ordinary differential equations in the neighborhood of irregular singularities are representable in the form of expressions containing formal series. Numerous publications are devoted to studying the formal series involved in the asymptotics of solutions to differential equations in the neighborhoods of singular points.

Thomé’s paper [3] is one of the first publications in which it was noted that asymptotics of solutions to differential equations in the neighborhood of irregular singular points contain formal series, which are, generally speaking, divergent. In Poincare’s papers [4,5], it is proven that the divergent series obtained are asymptotic series, and the idea is formulated according to which an integral transform can be used to summarize the asymptotic series obtained; in a particular case, it can be the Laplace transform. An attempt is made to construct uniform asymtotics of this problem with the help of this integral transform. However, the integral Laplace transform is applicable only in some particular cases. Note that the problem formulated by Poincare on constructing asymptotics of solutions in the neighborhood of arbitrary irregular singularities for differential equations with holomorphic coefficients is not solved in its general formulation until now; however, it is solved in [6,7] for ordinary differential equations of arbitrary order with holomorphic coefficients in the neighborhood of an infinitely distant point, which is, generally speaking, an irregular singular point. In [1], it is shown that the asymptotics of solutions to any ordinary second-order differential equations in the neighborhood of irregular singularities can be represented as a sum of the terms of the form $\exp \left({\displaystyle \frac{c}{r^{n+1}}}+{\sum }_{i=1}^n{\displaystyle \frac{{\alpha }_i}{r^{n+1-i}}}\right)r^{\sigma }{\sum }_{i=0}^{\infty }{\beta }_i{r}^i$ or $\exp \left({\displaystyle \frac{c}{r^{n+\frac{3}{2}}}}+{\sum }_{i=1}^{2n+2}{\displaystyle \frac{{\alpha }_i}{r^{n+\frac{3}{2}-\frac{i}{2}}}}\right)r^{\sigma }{\sum }_{j=1}^{\infty }{\beta }_j{r}^{{\displaystyle \frac{j}{2}}}$, where $c,{\alpha }_i$, ${\beta }_j$, $\sigma$, $n\in N$ are constants, ${\sum }_{j=1}^{\infty }{\beta }_j{r}^j$ are formal, generally speaking, divergent series.

In the present paper, a class of equations for which these series converge in some neighborhood of an irregular singular point is identified. Namely, it is shown for equations that can be transformed to the form using exponential and power substitutions

$${\left(-r^{{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+a\left(r\right)v\left(r\right)=0,$$

(1)

where $a\left(r\right)$ is a holomorphic function in a neighborhood of zero such that $a\left(0\right)\ne 0$, $k\in N$. The formal series ${\sum }_{j=1}^{\infty }{\beta }_j{x}^j$ included in the asymptotics of their solutions are convergent in some neighborhood of the singular point $r=0$. By power substitutions we mean substitutions of the form $v=r^{\sigma }{v}_1$, and an exponential substitution is a substitution of the form $v=\exp {\displaystyle \frac{a}{r^{\frac{i}{2}}}}{v}_1$. More details about these substitutions can be found below in Section 3, as well as in work [8].

It is also shown in this study that by using corresponding substitutions, a broad class of the second-order differential equations with regular and irregular singular points can be reduced to the form (1).

So, in this paper, the method is obtained for investigation of the asymptotics of solutions to the second-order differential equations with irregular singularities. The essence of this method consists in the following: if a second-order differential equation can be reduced to (1) with the help of the transformations described in this paper, then the power series involved in the asymptotics of solutions of this equation are convergent in some neighborhood of the irregular singular point.

In Section 4, some examples of application of the method obtained to certain problems of mathematical physics are presented; for instance, for the Heun equation, an explicit form of asymptotic expansions of solutions is found and the divergence of the corresponding power series is investigated. It is shown that the asymptotics of solutions to the Heun equation in the neighborhood of an inifinitely distant point can be represented in the form of the products of exponents and divergent series.

It should be noted that earlier, in [9], the spectrum of the zeros of the solutions represenable by asymptotic power series for the biconfluent Heun differential equation was studied. In [9], the Newton spectral sums (i.e., the sums of the rth powers of zeros) are given by by strict and recurrent ways. In addition, the density of zeros (i.e., the number of zeros per interval unit) is studied. The density is calculated explicitly in terms of the so-called semiclassical or Wentzel–Kramers–Brillouin (WKB) approximation with the help of the general theorem, which is applicable to a second-order linear differential equation under certain conditions. In our study, for this equation, we obtain explicit formulas for the corresponding coefficients and demonstrate that the corresponding asymptotic series are convergent.

Below, it is shown how the created method can be applied to investigate the convergence of the corresponding formal series for elliptic and hyperbolic equations with irregular singular points. It should be noted that earlier, in [10,11], the wave equation with meromorphic coefficients was studied. In those papers, the asymptotics of the solution to the boundary value problem for this equation was constructed and, under a supplementary condition on the meromorphic coefficient, the convergence of the asymptotic series involved in the asymptotic expansions of solutions was proven. In our current study, with the help of the method we invented, this problem is solved without any supplementary conditions on the meromorphic coefficient.

Note that the method created in this paper is applied to a wide class of second-order equations and allows us to construct asymptotics of solutions in the vicinity of their irregular singular points, including an infinity distant singular point which distinguishes it from the methods with consistent asymptotic expansions in WKB or many other methods that allow for construction of asymptotics of solutions of differential equations with respect to a small parameter. In addition, the asymptotics of solutions in the vicinity of an irregular singularity have a more general appearance than the asymptotics of the WKB, since in general the exponents included in the asymptotics of solutions contain polynomials of fractional powers of the variable [1].

2. Materials and Methods

It was demonstrated in [12] that any linear homogeneous differential equation with holomorphic coefficients can be reduced to the form

$$\left({\left(-r^k{\displaystyle \frac{d}{dr}}\right)}^n+\sum _{i=0}^{n-1}{a}_i\left(r\right){\left(-r^k{\displaystyle \frac{d}{dr}}\right)}^i\right)u=0,$$

where k is an integer nonnegative number and $a_i\left(r\right)$ are holomorphic functions. In addition, in this study, the minimum value of k is found. If the condition $k\ge 2$ is fulfilled, than the point $r=0$ is an irregular singular point of the equation; if $k=1$, than the singular point is regular. We consider the case of irregular singularity.

One calls the symbol of the differential operator

$$\widehat{H}={\left(-{\displaystyle \frac{1}{k-1}}{r}^k{\displaystyle \frac{d}{dr}}\right)}^n+\sum _{i=0}^{n-1}{a}_i\left(r\right){\left(-{\displaystyle \frac{1}{k-1}}{r}^k{\displaystyle \frac{d}{dr}}\right)}^i$$

(2)

at $k\ge 2$, where $a_i^0\left(r\right)$ are holomorphic functions, the function $H\left(r,p\right)=p^n+{\sum }_{i=0}^{n-1}{a}_i^0\left(r\right)p^i$.

Definition 1.

One calls the principal symbol of Differential Operator (2) the function

$$H_0\left(p\right)=H\left(0,p\right)=p^n+\sum _{i=0}^{n-1}{a}_i^0\left(0\right)p^i.$$

There are some studies which deal with the case when the roots of the polynomial $H_0\left(p\right)$ are simple. Such equations were considered, for example, in [13,14,15]. In these studies, it was proven that the asymptotics of the solution at $k=2$ can be represented as a product of the corresponding exponents and asymptotic power series, namely

$$u=\sum _{i=1}^n{e}^{{\alpha }_i/r}{r}^{{\sigma }_i}\sum _{k=0}^{\infty }{a}_i^k{r}^k,$$

(3)

where ${\alpha }_i,i=1,\dots ,n$ are the roots of the polynomial $H_0\left(p\right)$; ${\sigma }_j$ and $a_i^k$ are some complex numbers. However, the question concerning interpretation of the divergent series obtained was left open; in other words, there was no method for summation of these divergent series.

By the end of the 1980s, a mathematical apparatus was obtained for summation of such series. It was based on the Laplace–Borel transform acting in the space of the functions of exponential growth and on the concept of a resurgent function first introduced by J. Ecalle [16].

Let us denote by $S_{R,\epsilon }$ the sector $S_{R,\epsilon }=\left\{r\left|-\epsilon <\arg r<\epsilon ,\left|r\right|\right.<R\right\}$.

Definition 2.

Let us say that the function f analytical on $S_{R,\epsilon }$ is a function of k-exponential growth at zero, if there are nonnegative constants C and α such that in the sector $S_{R,\epsilon }$, the inequality

$$\left|f\right|<C{e}^{\alpha {\displaystyle \frac{1}{{\left|r\right|}^k}}}$$

is fulfilled.

Let us denote by $E_k\left(S_{R,\epsilon }\right)$ the space of functions of k-exponential growth that are holomorphic in the sector $S_{R,\epsilon }$ and by $E\left(C\right)$ the space of integer functions of exponential growth.

Definition 3.

The k-Laplace–Borel transform of the function $f\left(r\right)\in E_k\left(S_{R,\epsilon }\right)$ is the mapping $B_k:E_k\left(S_{R,\epsilon }\right)\to \mathrm{E}({\tilde{\Omega }}_{R,\epsilon })/\mathrm{E}(\mathrm{C})$,

$$B_{k}f=\underset{0}{\overset{r_0}{\int }}{e}^{-p/r^k}f\left(r\right){\displaystyle \frac{dr}{r^{k+1}}}.$$

(4)

Here, $r_0$ is an arbitrary point belonging to the sector $S_{R,\epsilon }$; $E\left({\tilde{\Omega }}_{R,\epsilon }\right)$ is the space of the functions of exponential growth holomorphic in the domain ${\tilde{\Omega }}_{R,\epsilon }=\left\{p\left|-{\displaystyle \frac{\pi }{2}}-\epsilon <\arg p<{\displaystyle \frac{\pi }{2}}+\epsilon ,\left|p\right|>R\right.\right\}$.

The inverse k-Laplace–Borel transform is defined by the formula

$${B_k}^{-1}\tilde{f}={\displaystyle \frac{1}{2\pi i}}\underset{\tilde{\gamma }}{\int }{e}^{p/r^k}\tilde{f}\left(p\right)dp.$$

(5)

The contour $\tilde{\gamma }$ is shown in [1].

Let us formulate the properties of the k-Laplace–Borel transform.

It was proven in [17] that for the k-Laplace–Borel Transform (4), the following formulas hold:

$$\begin{array}{cc}& B_k\circ \left(-{\displaystyle \frac{1}{k}}{r}^{k+1}{\displaystyle \frac{d}{dr}}\right)f\left(r\right)=p{B}_{k}f,\hfill \\ \hfill & {\displaystyle \frac{d}{dp}}\circ B_{k}f=-B_k\left({\displaystyle \frac{1}{r^k}}f\left(r\right)\right).\hfill \end{array}$$

(6)

In [8], the equality

$$B_k{r}^{\sigma }{B_k}^{-1}\left[\tilde{f}\right]={\displaystyle \frac{e^{i\pi \frac{\sigma }{k}}}{k\Gamma \left(\frac{\sigma }{k}\right)}}\underset{\infty }{\overset{p}{\int }}{\left(p-p^{\prime }\right)}^{{\displaystyle \frac{\sigma }{k}}-1}\tilde{f}\left(p^{\prime }\right)d{p}^{\prime },\sigma >0$$

(7)

was obtained.

It should be noted that if $\sigma =k$, then (7) takes the form

$$B_k{r}^{\sigma }{B_k}^{-1}\left[\tilde{f}\right]=-{\displaystyle \frac{1}{k}}\underset{p_0}{\overset{p}{\int }}\tilde{f}\left(p^{\prime }\right)d{p}^{\prime }.$$

Let us introduce the denotation

$${\widehat{r}}^{\sigma }\tilde{f}=B_k{r}^{\sigma }{B_k}^{-1}\left[\tilde{f}\right]$$

(8)

In [8], the following property was proven:

Property 1.

The images of the k-Laplace—Borel transform of the functions $r^{\sigma },\sigma \in R$ have the form

$$B_k{r}^{\sigma }=\left\{\begin{array}{c}{\displaystyle \frac{\pi }{k\Gamma \left(\frac{\sigma }{k}\right)\sin \pi \frac{\sigma }{k}}}{p}^{{\displaystyle \frac{\sigma }{k}}-1},\forall {\displaystyle \frac{\sigma }{k}}\notin Z\hfill \\ {\displaystyle \frac{\Gamma \left(1-\frac{\sigma }{k}\right)}{k}}{p}^{{\displaystyle \frac{\sigma }{k}}-1},{\displaystyle \frac{\sigma }{k}}=0,-1,\dots \hfill \\ {\left(-1\right)}^{{\displaystyle \frac{\sigma }{k}}}{\displaystyle \frac{p^{\frac{\sigma }{k}-1}\ln p}{k\left(\frac{\sigma }{k}-1\right)!}},{\displaystyle \frac{\sigma }{k}}\in N\hfill \end{array}\right.$$

(9)

Later, this apparatus was actively used by B.-W. Schulze, B. Yu. Sternin and V. E. Shatalov to investigate degenerate equations that arise when considering elliptical equations on the manifolds with the cusp-type singularities and constructing the asymptotics of solutions to equations with a small parameter. In some cases, they succeeded in constructing asymptotics of solutions in the weighted Sobolev spaces for the equations with a cuspidal singularity [18,19,20].

The mathematical apparatus for interpretation and construction of the asymptotic expansions of the form (3) based on the Laplace–Borel transform is called resurgent analysis. The main idea of resurgent analysis consists in the fact that the formal Laplace–Borel transforms are power series with respect to a dual variable p that converges in the neighborhood of the points $p={\lambda }_j$, where ${\lambda }_j$ are the roots of the principal symbol of the corresponding differential operator. At that, the inverse Borel transform provides a regular method for summation of the power series. With the help of resurgent analysis methods, the question about constructing uniform asymtotics of solutions to the equations whose principal symbol has simple roots was solved in [8]. In these studies, the following theorem was proven:

Theorem 1.

Let f be a resurgent function; then, the solution of equation $H\left(r,-{\displaystyle \frac{1}{k}}{r}^{k+1}{\displaystyle \frac{d}{dr}}\right)u=f$ is a resurgent function. If the polynomial $H_0\left(p\right)$ has simple roots at points $p_1\dots ,p_m$, then, at $k\in N$, the asymptotics of the solution to the uniform equation

$$H\left(r,-{\displaystyle \frac{1}{k}}{r}^{k+1}{\displaystyle \frac{d}{dr}}\right)u=0$$

has the form

$$u\left(r\right)\approx \sum _{j=1}^m\exp \left({\displaystyle \frac{p_j}{r^k}}+\sum _{i=1}^{k-1}{\displaystyle \frac{{{\alpha }^1}_{k-i}}{r^{k-i}}}\right)r^{{\sigma }_j}\sum _{i=0}^{\infty }{b}_i^j{r}^i.$$

If $k+1={\displaystyle \frac{b}{n}}$, $b\in N,k\in N,b>n$, then the asymptotics of solution have the form

$$u\approx \sum _j\exp \left({\displaystyle \frac{p_j}{r^{\frac{b}{n}-1}}}+\sum _{i=1}^{m-k-1}{\displaystyle \frac{{{\alpha }^1}_{m-k-i}}{r^{\frac{b-i}{n}-1}}}\right)r^{{\sigma }_j}\sum _{i=0}^{\infty }{b}_i^j{r}^{{\displaystyle \frac{i}{n}}}.$$

(10)

Here, the summation is performed over the entire set of the roots of the polynomial $H_0\left(p\right)$.

For the second-order equations in the case of simple and multiple roots, asymptotics of solutions were constructed in [1]. Our goal is to create a method for investigation of the convergence of the asymptotic series involved in these asymptotics and prove their convergence for a wide class of second-order differential equations.

3. Results

Let us study the asymptotics of solutions to the equation

$${\left(-r^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+a\left(r\right)\lambda v\left(r\right)=0,$$

(11)

where $a\left(r\right)={\sum }_{j=0}^{\infty }{a}_j{r}^j$; $a\left(0\right)\ne 0$ is a holomorphic function; $k\in Z$ in the space $E_{{\displaystyle \frac{k}{2}}+1}\left(S_{R,\epsilon }\right)$ in the neighborhood of $r=0$. Let us introduce denotation $n=\left[{\displaystyle \frac{k}{2}}\right]$.

Theorem 2.

The asymptotics of solutions to Equation (11) at $\lambda \ne 0$ in the neighborhood of zero have the following form:

I. If $k\ge -1$, then

at even k,

$$u\left(r\right)\approx \exp \left({\displaystyle \frac{c_1}{r^{n+1}}}+\sum _{i=1}^n{\displaystyle \frac{{{\alpha }^1}_i}{r^{n+1-i}}}\right)r^{{\sigma }_1}\sum _{i=0}^{\infty }{\beta }_i^1{r}^i+\exp \left({\displaystyle \frac{c_2}{r^{n+1}}}+\sum _{i=1}^n{\displaystyle \frac{{{\alpha }^2}_i}{r^{n+1-i}}}\right)r^{{\sigma }_2}\sum _{i=0}^{\infty }{\beta }_i^2{r}^i;$$

(12)

at odd k,

$$u\left(r\right)\approx \exp \left({\displaystyle \frac{c_1}{r^{n+\frac{3}{2}}}}+\sum _{i=1}^{2n+2}{\displaystyle \frac{{{\alpha }^1}_i}{r^{n+\frac{3}{2}-\frac{i}{2}}}}\right)r^{{\sigma }_1}\sum _{j=1}^{\infty }{\beta }_j^1{r}^{{\displaystyle \frac{j}{2}}}+\exp \left({\displaystyle \frac{c_2}{r^{n+\frac{3}{2}}}}+\sum _{i=1}^{2n+2}{\displaystyle \frac{{{\alpha }^2}_i}{r^{n+\frac{3}{2}-\frac{i}{2}}}}\right)r^{{\sigma }_2}\sum _{j=1}^{\infty }{\beta }_j^2{r}^{{\displaystyle \frac{j}{2}}}.$$

(13)

Here $c_1$, $c_2$ are the roots of the polynomial $H_0\left(p\right)={\left({\displaystyle \frac{1}{1+k}}\right)}^{2}a\left(0\right)\lambda +p^2$; the power series ${\sum }_{i=0}^{\infty }{\beta }_i^1{r}^i$ and ${\sum }_{i=0}^{\infty }{\beta }_i^2{r}^i$ converge in some neighborhood of zero.

II. If $k<-1$ or $\lambda =0$, then the asymptotics of the solution is conormal or the solution is regular.

Proof. 

1.

Let us assume that $\lambda \ne 0$ and $k\ge -1$.

Equation (11) can be rewritten in the form

$${\left(-{\displaystyle \frac{1}{1+\frac{k}{2}}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+{\left({\displaystyle \frac{1}{1+\frac{k}{2}}}\right)}^{2}a\left(r\right)\lambda v\left(r\right)=0.$$

(14)

The principal symbol of the differential operator in Equation (14) is $H_0\left(p\right)={\left({\displaystyle \frac{1}{1+\frac{k}{2}}}\right)}^{2}a\left(0\right)\lambda +p^2$. Since $a\left(0\right)\ne 0$, then the roots are simple. We denote the roots of this polynomial by $c_1,c_2$. They are simple; therefore, we can apply Theorem 1 to Equation (14). This theorem implies that if k is even, then the asymptotics of the solution to Equation (14) is constructed according to Formula (12); if k is odd, then the asymptotics is constructed according to Formula (13). Without loss of generality, to be more specific, we assume that k is even. The question arises about finding the numbers ${{\alpha }^j}_i,{\sigma }_j,j=1,2$ involved in Asymptotics (12) and (13). The algorithm for their calculation is as follows. To find the numbers ${\sigma }_1,{{\alpha }^1}_i,i=1,...,n$, we shift the root $c_1$ to zero by using the exponential substitution $v=e^{{\displaystyle \frac{c_1}{r^{n+1}}}}{v}_1$. Let us rewrite Equation (14) with respect to the function $v_1\left(r\right)$:

$${\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)}^2{v}_1+2c_1\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)v_1+{\left({\displaystyle \frac{1}{1+n}}\right)}^2(a\left(r\right)-a\left(0\right))\lambda v_1\left(r\right)=0.$$

(15)

Since $a\left(r\right)-a\left(0\right)={\sum }_{j=1}^{\infty }{a}_j{r}^j={\sum }_{j=1}^{n+1}{a}_j{r}^j+{\sum }_{j=n+2}^{\infty }{a}_j{r}^j$, then Equation (15) can be rewritten as follows:

$$\begin{array}{cc}& {\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)}^2{v}_1\left(r\right)+2c_1\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)v_1\left(r\right)+\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2\sum _{j=1}^n{a}_j{r}^j{v}_1\left(r\right)+\hfill \\ \hfill & +a_{n+1}+\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2{a}_{n+1}{r}^{n+1}{v}_1\left(r\right)+\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2{r}^{n+2}\sum _{j=0}^{\infty }{a}_{j+n+2}{r}^j{v}_1\left(r\right)=0.\hfill \end{array}$$

(16)

The principal symbol of the operator in Equation (16) has the form $2c_{1}p+p^2$; it has two simple roots: $p=0,p=-2c_1$. Theorem 1 implies that the asymptotics of solution to Equation (16) is representable as a sum of two asymptotic terms

$$u\left(r\right)\approx \exp \left(\sum _{i=1}^n{\displaystyle \frac{{{\alpha }^1}_i}{r^{n+1-i}}}\right)r^{{\sigma }_1}\sum _{i=0}^{\infty }{\beta }_i^1{r}^i+\exp \left({\displaystyle \frac{-2c_1}{r^{n+1}}}+\sum _{i=1}^n{\displaystyle \frac{{{\alpha }^2}_i}{r^{n+1-i}}}\right)r^{{\sigma }_2}\sum _{i=0}^{\infty }{\beta }_i^2{r}^i.$$

(17)

The first term of Asymptotics (17) corresponds to the root $p=0$; the second term to the root $p=-2c_1$. Our aim is to investigate the first term of the asymptotics, namely to prove the convergence of series ${\sum }_{i=0}^{\infty }{\beta }_i^1{r}^i$ in some neighborhood of zero. To find numbers ${{\alpha }^1}_i,i=1,\dots ,n$, let us use exponential substitution $v_n=\exp \left({\sum }_{i=1}^n{\displaystyle \frac{{{\alpha }^1}_i}{r^{n+1-i}}}\right)v_1$ and choose numbers ${{\alpha }^1}_i,i=1,\dots ,n$ so that the terms of the form $\lambda a_j{r}^{j}v\left(r\right)$, where $j\le n$, are zero.

For example, to zero term $\lambda a_{1}rv\left(r\right)$, it is necessary to use exponential substitution $v_2=\exp {\displaystyle \frac{{\alpha }_n^1}{r^n}}{v}_1$, etc. With substitutions $v_{i-1}=\exp {\displaystyle \frac{{\alpha }_i^1}{r^i}}{v}_i,i=1,\dots ,n$, all the terms in the sum $\lambda {\sum }_{j=2}^n{a}_j^{\prime }{r}^j{v}_n\left(r\right)$ are zero. To find ${\sigma }_1$, we have to use the last power substitution $v_n=r^{{\sigma }_1}{v}_{n+1}$ and choose the corresponding ${\sigma }_1$ so as to zero the term with the (n + 1)th power of r. All other terms are minor and do not influence on the asymptotics’ form. One can read more about this method in [8].

After performing the above substitutions, we obtain the following equation with respect to function $v_{n+1}\left(r\right)$:

$$\begin{array}{cc}& {\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)}^2{v}_{n+1}\left(r\right)+2c_1\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)v_{n+1}\left(r\right)+\hfill \\ \hfill & +\sum _{i=1}^{n+1}{a}_i^1{r}^i\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+n}{\displaystyle \frac{d}{dr}}\right)v_{n+1}\left(r\right)+\lambda r^{n+2}{\left({\displaystyle \frac{1}{1+n}}\right)}^2\sum _{j=0}^{\infty }{a}_{j+n+2}^{\prime }{r}^j{v}_{n+1}\left(r\right)=0\hfill \end{array}$$

(18)

Here, $a_i^1,i=1,\dots ,n+1$, $a_{j+n+2}^{\prime },j=0,\dots$ denote the corresponding coefficients, which were obtained after performing the above substitutions. It is obvious that since series ${\sum }_{j=0}^{\infty }{a}_j{r}^j$ converges in some neighborhood of zero, then for all i, there is the constant D such that $\left|a_i\right|<D^i$. This implies that for the coefficients of series ${\sum }_{j=0}^{\infty }{a}_{j+n+2}^{\prime }{r}^j$, there is $D_1$ such that conditions $\left|a_i^{\prime }\right|<{D_1}^i$ are fulfilled for all i. Applying the Laplace–Borel transform to Equation (18), we obtain

$$p^2{\widehat{v}}_{n+1}\left(p\right)+2c_{1}p{\widehat{v}}_{n+1}\left(p\right)+\sum _{i=1}^{n+1}{\alpha }_i^1{\widehat{r}}^{i}p{\widehat{v}}_{n+1}\left(p\right)+\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2{\widehat{r}}^{n+2}\sum _{j=0}^{\infty }{a}_{j+n+2}^{\prime }{\widehat{r}}^j{\widehat{v}}_{n+1}\left(p\right)=f\left(p\right).$$

(19)

Here, ${\widehat{v}}_{n+1}\left(p\right)=B_k{v}_{n+1}\left(r\right)$; $f\left(p\right)$ is an arbitrary holomorphic function. It follows from Equation (19) that

$${\widehat{v}}_{n+1}\left(p\right)=-{\displaystyle \frac{{\displaystyle \sum _{i=1}^{n+1}}{a}_i^1{\widehat{r}}^{i}p{\widehat{v}}_{n+1}\left(p\right)}{p\left(p+2c_1\right)}}-{\displaystyle \frac{\lambda {\widehat{r}}^{n+2}{\left(\frac{1}{1+n}\right)}^2{\displaystyle \sum _{j=0}^{\infty }}{a}_{j+n+2}^{\prime }{\widehat{r}}^j{\widehat{v}}_{n+1}\left(p\right)}{p\left(p+2c_1\right)}}+{\displaystyle \frac{f}{p\left(p+2c_1\right)}}$$

(20)

The free term in the right-hand side of (20) can be written in the form

$${\displaystyle \frac{f}{p\left(p-2c_1\right)}}={\displaystyle \frac{M}{p\left(p-2c_1\right)}}+G\left(p\right).$$

(21)

Here, M is a constant, $G\left(p\right)$ is a function holomorphic in some neighborhood of zero. Let us introduce the denotations

$$\begin{array}{cc}& {\widehat{A}}_1=-{\displaystyle \frac{1}{2c_{1}p}}\sum _{i=1}^{n+1}{\alpha }_i^1{\widehat{r}}^{i}p=\sum _{i=1}^{n+1}{\widehat{\mathrm{A}}}_1^{\mathrm{i}}\hfill \\ & {\widehat{A}}_2=-{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{1}{2c_{1}p}}\lambda \sum _{j=0}^{\infty }{a}_{j+n+2}^{\prime }{\widehat{r}}^{j+n+2}=\sum _{i=1}^{\infty }{\widehat{\mathrm{A}}}_2^{\mathrm{i}}\hfill \end{array}$$

where

$$\begin{array}{cc}& {\widehat{\mathrm{A}}}_1^{\mathrm{i}}=-{\displaystyle \frac{1}{2c_{1}p}}{a}_i^1{\widehat{r}}^{i}p\hfill \\ & {\widehat{\mathrm{A}}}_2^{\mathrm{i}}=-{\displaystyle \frac{1}{2c_{1}p}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2\lambda a_{i+n+2}^{\prime }{\widehat{r}}^{\mathrm{i}+\mathrm{n}+2}\hfill \end{array}$$

Let us use denotation $\widehat{A}={\widehat{A}}_1+{\widehat{A}}_2$. With these denotations, the operator $\widehat{A}$ can be represented in the form $\widehat{A}={\widehat{A}}_1+{\widehat{A}}_2=-{\displaystyle \frac{1}{2c_1}}\left({\displaystyle \frac{1}{p}}{\displaystyle \sum _{i=1}^{n+1}}{a}_i^1{\widehat{r}}^{i}p+\lambda {\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \sum _{j=0}^{\infty }}{a}_{j+n+2}^{\prime }{\widehat{r}}^{j+n+2}\right)$.

To construct the asymptotics of the solution to Equation (20), we apply the method of successive approximations. At the first step of this method, we construct asymptotics $\widehat{A}{\displaystyle \frac{1}{p}}=\left({\widehat{A}}_1+{\widehat{A}}_2\right){\displaystyle \frac{1}{p}}$ in the neighborhood of zero. With this purpose, we proceed as follows.

• We find the asymptotic expansion for ${\widehat{A}}_1{\displaystyle \frac{1}{p}}$.

  $${\widehat{A}}_1{\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{2c_1}}{\displaystyle \frac{1}{p}}\sum _{i=1}^{n+1}{\alpha }_i^1{B}_{n+1}{r}^i{B}_{n+1}^{-1}1=0$$
• We find the asymptotic expansion for ${\widehat{A}}_2{\displaystyle \frac{1}{p}}$. To construct this asymptotics, we consider two cases.

In the first case, we assume that ${\displaystyle \frac{n+2+j}{n+1}}\notin N$. Let us introduce the following denotation: C = $B_{n+1}^{-1}{\displaystyle \frac{1}{p}}={\displaystyle \frac{n+1}{\pi }}$. Then,

$$\begin{array}{cc}\hfill {\widehat{A}}_2^j{\displaystyle \frac{1}{p}}& =-{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{1}{p}}{a}_{j+n+2}^{\prime }{\widehat{r}}^{j+n+2}{B}_{n+1}^{-1}{\displaystyle \frac{1}{p}}=\hfill \\ \hfill & =-C{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{1}{p}}{a}_{j+n+2}^{\prime }{B}_{n+1}{r}^{j+n+2}=\hfill \\ \hfill & =-C{\displaystyle \frac{\lambda }{2c_1}}{a}_{j+n+2}^{\prime }{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{\pi }{\left(n+1\right)\Gamma \left(\frac{n+2+j}{n+1}\right)\sin \pi \frac{n+2+j}{n+1}}}{p}^{{\displaystyle \frac{j+n+2}{n+1}}-2}=\hfill \\ & =-C{\displaystyle \frac{\lambda }{2c_1}}{a}_{j+n+2}^{\prime }{\left({\displaystyle \frac{1}{1+n}}\right)}^3{\displaystyle \frac{\pi }{\Gamma \left(1+\frac{j+1}{n+1}\right)\sin \pi \left(1+\frac{j+1}{n+1}\right)}}{p}^{{\displaystyle \frac{j+1}{n+1}}-1}\hfill \end{array}$$

(22)

In the second case, ${\displaystyle \frac{n+2+j}{n+1}}\in \mathrm{N}$; then,

$$\begin{array}{cc}\hfill {\widehat{A}}_2^j{\displaystyle \frac{1}{p}}& =-{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{1}{p}}{a}_{j+n+2}^{\prime }{\widehat{r}}^{j+n+2}{B}_{n+1}^{-1}{\displaystyle \frac{1}{p}}=\hfill \\ \hfill & =-C{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{a}_{j+n+2}^{\prime }{\displaystyle \frac{1}{p}}{B}_{n+1}{r}^{j+n+2}=\hfill \\ \hfill & ={\left(-1\right)}^{{\displaystyle \frac{n+2+j}{n+1}}}C{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^2{a}_{j+n+2}^{\prime }{\displaystyle \frac{p^{\frac{n+2+j}{n+1}-2}\ln p}{\left(n+1\right)\left(\frac{n+2+j}{n+1}-1\right)!}}=\hfill \\ & ={\left(-1\right)}^{{\displaystyle \frac{n+2+j}{n+1}}}C{\displaystyle \frac{\lambda }{2c_1}}{\left({\displaystyle \frac{1}{1+n}}\right)}^3{a}_{j+n+2}^{\prime }{\displaystyle \frac{p^{\frac{1+j}{n+1}-1}\ln p}{\left(\frac{1+j}{n+1}\right)!}}\hfill \end{array}$$

(23)

It follows from (22) and (23) that

$$\begin{array}{cc}\hfill \widehat{A}{\displaystyle \frac{1}{\mathrm{p}}}& =-{\displaystyle \frac{1}{2c_1}}\left({\widehat{A}}_1+{\widehat{A}}_2\right){\displaystyle \frac{1}{\mathrm{p}}}=\hfill \\ & =-{\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^3(\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{a}_{j+n+1}^{\prime }{\displaystyle \frac{\pi }{\Gamma \left(1+\frac{j}{n+1}\right)\sin \pi \left(1+\frac{j}{n+1}\right)}}{p}^{{\displaystyle \frac{j}{n+1}}-1}+\hfill \\ & +\sum _{j=1}^{\infty }{\left(-1\right)}^{{\displaystyle \frac{n+1+j\left(n+1\right)}{n+1}}}{a}_{j\left(n+1\right)+n+1}^{\prime }{\displaystyle \frac{p^{\frac{j\left(n+1\right)}{n+1}-1}\ln p}{\left(\frac{j\left(n+1\right)}{n+1}\right)!}})=\hfill \\ & =\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=1}{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\hfill \end{array}$$

Here, the following denotations are introduced:

$$\begin{array}{c}{b}_1^j={\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^3{a}_{j+n+1}^{\prime }{\displaystyle \frac{\pi }{\Gamma \left(1+\frac{j}{n+1}\right)\sin \pi \left(1+\frac{j}{n+1}\right)}},{\displaystyle \frac{j}{n+‘1}}\notin N\hfill \\ b_1^j={\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^3{a}_{j\left(n+1\right)+n+1}^{\prime }{\left(-1\right)}^{j+1}{\displaystyle \frac{1}{j!}},{\displaystyle \frac{j}{n+‘1}}\in N\hfill \end{array}$$

It is obvious that if ${\displaystyle \frac{j}{n+1}}\notin N$, then

$$\begin{array}{cc}& \left|B_{n+1}^{-1}{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}\right|=\hfill \\ & =\left|{\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^3{a}_{j+n+1}^{\prime }{\displaystyle \frac{\pi }{\Gamma \left(1+\frac{j}{n+1}\right)\sin \pi \left(1+\frac{j}{n+1}\right)}}{\displaystyle \frac{\left(n+1\right)\Gamma \left(\frac{j}{n+1}\right)\sin \pi \frac{j}{n+1}}{\pi }}{r}^j\right|=\hfill \\ \hfill & =\left|{\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2{a}_{j+n+1}^{\prime }{\displaystyle \frac{1}{j}}{r}^i\right|\hfill \end{array}$$

(24)

And if ${\displaystyle \frac{j}{n+1}}\in N$, then we obtain

$$\begin{array}{cc}\hfill & \left|B_{n+1}^{-1}\ln p{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\right|=\left|{\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^3{a}_{j\left(n+1\right)+n+1}^{\prime }{\displaystyle \frac{1}{j!}}\left(n+1\right)\left(j-1\right)!r^{j\left(n+1\right)}\right|=\hfill \\ \hfill & =\left|{\displaystyle \frac{C}{2c_1}}\lambda {\left({\displaystyle \frac{1}{1+n}}\right)}^2{a}_{j\left(n+1\right)+n+1}^{\prime }{\displaystyle \frac{1}{j}}{r}^{j\left(n+1\right)}\right|.\hfill \end{array}$$

(25)

Note that (24) and (25) imply that the power series ${\sum }_{j=1}^{\infty }{a}_j^1{r}^j$, where

$$\sum _{j=1}^{\infty }{a}_j^1{r}^j=B_{n+1}^{-1}\left(\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=1}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\right),$$

(26)

converges in some neighborhood of zero.

At the second step of the method of successive approximations, we apply operator $\widehat{A}={\widehat{A}}_1+{\widehat{A}}_2$ to expression

$$\sum _{j=1,i\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=1}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}.$$

(27)

It is clear that ${\widehat{A}}^2{\displaystyle \frac{1}{p}}$ has an asymptotic expansion with the same powers of p that the asymptotics of (27), except for the lower powers at $j=1$, i.e., it has the form

$$\begin{array}{cc}\hfill {\widehat{A}}^2{\displaystyle \frac{1}{p}}& =A\left(\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=1}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\right)=\hfill \\ & =\left({\widehat{A}}_1+{\widehat{A}}_2\right)\left(\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}\right)+\left({\widehat{A}}_1+{\widehat{A}}_2\right)\left(\ln p\sum _{j=1}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\right)\hfill \\ & =\left(\sum _{i=1}^{n+1}{\widehat{\mathrm{A}}}_1^{\mathrm{i}}+\sum _{i=1}^{\infty }{\mathrm{A}}_2^{\mathrm{i}}\right)\left(\sum _{j=1,j\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}\right)+\hfill \\ & +\left(\sum _{i=1}^{n+1}{\widehat{\mathrm{A}}}_1^{\mathrm{i}}+\sum _{i=1}^{\infty }{\mathrm{A}}_2^{\mathrm{i}}\right)\left(\ln p\sum _{j=1}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\right)=\hfill \\ & =\sum _{j=2,j\ne k\left(n+1\right)}^{\infty }{b}_2^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=2,}^{\infty }{b}_2^{j\left(n+1\right)}{p}^{{\displaystyle \frac{j\left(n+1\right)}{n+1}}-1}\hfill \end{array}.$$

Here, $b_2^k$ are the coefficients at the corresponding powers of p. Our next task is to estimate the coefficients of power series

$$\sum _{i=2}^{\infty }{a}_i^2{r}^i=B_{n+1}^{-1}\left(\sum _{j=2,j\ne k\left(n+1\right)}^{\infty }{b}_2^j{p}^{{\displaystyle \frac{j}{n+1}}-1}+\ln p\sum _{j=2}^{\infty }{\left(-1\right)}^{j+1}{b}_2^{j\left(n+1\right)}{p}^{j-1}\right).$$

It is obvious that $B_{n+1}^{-1}{\widehat{A}}_1{\sum }_{j=2,i\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}$ is a power series with respect to the variable r. Namely,

$$B_{n+1}^{-1}{\widehat{A}}_1\sum _{j=2,j\ne m\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}=\sum _{j=2,j\ne m\left(n+1\right)}^{\infty }{a}_2^{1,k}{r}^k.$$

Here, $a_2^{1,k}$ is the coefficient at $r^k$ in power series $B_{n+1}^{-1}{\widehat{A}}_1{\sum }_{j=2,i\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}$; similarly, $a_2^{2,k}$ is the coefficient at the kth power of r in power series $B_{n+1}^{-1}{\widehat{A}}_2{\sum }_{j=2,i\ne k\left(n+1\right)}^{\infty }{b}_1^j{p}^{{\displaystyle \frac{j}{n+1}}-1}$; $a_2^{3,k}$ and $a_2^{4,k}$ denote the corresponding coefficients in power series $B_{n+1}^{-1}{\widehat{A}}_1\ln p{\sum }_{k=2}{\left(-1\right)}^{j+1}{b}_2^j{p}^{j-1}$ and $B_{n+1}^{-1}{\widehat{A}}_2\ln p{\sum }_{k=2}{\left(-1\right)}^{j+1}{b}_2^j{p}^{j-1}$, respectively.

It is obvious that equality

$$a_k^2=a_2^{1,k}+a_2^{2,k}+a_2^{3,k}+a_2^{4,k}$$

(28)

holds.

To be more specific, let us assume that $k\ne m\left(n+1\right),m\in N$. For $k=m\left(n+1\right),m\in N$, the calculations do not differ from those performed below.

1.

Our next task is to estimate coefficient $a_2^{i,k},i=1,2,3,4$.Let us first estimate $a_2^{2,k}$.To solve this problem, we estimate the coefficient at power $r^i$ provided that $i\ne m\left(n+1\right),m\in N$ in expression $B_{n+1}^{-1}\left({\widehat{A}}_2+{\widehat{A}}_1\right){\sum }_{i=2,i\ne m\left(n+1\right)}^{\infty }{b}_2^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$. With this purpose, we apply operator $A_2^j$ to one of the terms $b_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$ of the last sum; after that, we perform the same with operator $A_1^j$, i.e., calculate $A_1^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$.

At first, let us calculate ${\widehat{A}}_2^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$. It follows from (9) that

$$\begin{array}{cc}\hfill {\widehat{A}}_2^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}& =b_1^i{\left({\displaystyle \frac{1}{1+n}}\right)}^2{\displaystyle \frac{\lambda }{2c_1}}{a}_{j+n+2}^{\prime }{\displaystyle \frac{1}{p}}{\widehat{r}}^{j+n+2}{p}^{{\displaystyle \frac{i}{n+1}}-1}=\hfill \\ \hfill & =b_1^i{\displaystyle \frac{\lambda a_{j+n+2}^{\prime }\Gamma \left(\frac{i}{n+1}\right)\sin \pi \frac{i}{n+1}}{2c_1\pi \left(1+n\right)}}{\displaystyle \frac{1}{p}}{B}_{n+1}{r}^{i+j+n+2}.\hfill \end{array}$$

(29)

Since (9) implies equality

$$\begin{array}{c}\hfill B_{n+1}{r}^{n+2+i+j}={\displaystyle \frac{\pi }{\left(n+1\right)\Gamma \left(\frac{n+2+i+j}{n+1}\right)\sin \pi \frac{n+2+i+j}{n+1}}}{p}^{{\displaystyle \frac{n+2+i+j}{n+1}}-1}\end{array},$$

then, substituting this equality into (29), we obtain

$$\begin{array}{cc}\hfill {\widehat{A}}_2^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}=& b_1^i{\displaystyle \frac{\lambda a_{j+n+2}^{\prime }\Gamma \left(\frac{i}{n+1}\right)\sin \pi \frac{i}{n+1}}{2c_1\pi \left(1+n\right)}}{\displaystyle \frac{1}{p}}{B}_{n+1}{r}^{i+j+n+2}=\hfill \\ \hfill =& {\displaystyle \frac{b_1^i\lambda a_{j+n+2}^{\prime }}{{\left(n+1\right)}^2}}{\displaystyle \frac{\Gamma \left(\frac{i}{n+1}\right)\sin \pi \frac{i}{n+1}}{\Gamma \left(\frac{n+2+i+j}{n+1}\right)\sin \pi \frac{n+2+i+j}{n+1}}}{p}^{{\displaystyle \frac{1+i+j}{n+1}}-1}=\hfill \\ \hfill =& -{\displaystyle \frac{{\lambda }^{2}C\pi }{2c_1{\left(n+1\right)}^4\sin \pi \left(\frac{1+i+j}{n+1}+1\right)}}{\displaystyle \frac{a_{i+n+1}^{\prime }{a}_{j+n+2}^{\prime }}{i\Gamma \left(\frac{1+i+j}{n+1}+1\right)}}{p}^{{\displaystyle \frac{1+i+j}{n+1}}-1}\hfill \end{array}$$

Since equality

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\lambda }^{2}C\pi }{2c_1{\left(n+1\right)}^4\sin \pi \left(\frac{k}{n+1}+1\right)}}{\displaystyle \frac{a_{i+n+1}^{\prime }{a}_{k-i+n+1}^{\prime }}{i\Gamma \left(\frac{k}{n+1}+1\right)}}{B}_{n+1}^{-1}{p}^{{\displaystyle \frac{k}{n+1}}-1}\right|=\hfill \\ & =\left|{\displaystyle \frac{{\lambda }^{2}C}{2c_1{\left(n+1\right)}^3}}{\displaystyle \frac{a_{i+n+1}^{\prime }{a}_{k-i+n+1}^{\prime }}{ki}}{r}^k\right|\hfill \end{array}$$

(30)

holds, we obtain, from (30), the estimate

$$\left|a_2^{2,k}\right|=\left|{\displaystyle \frac{{\lambda }^{2}C}{2c_1{\left(n+1\right)}^3}}\sum _{i=1}^k{\displaystyle \frac{a_{i+n+1}^{\prime }{a}_{k-i+n+1}^{\prime }}{ki}}\right|\le {\displaystyle \frac{{\lambda }^{2}C}{2c_1{\left(n+1\right)}^3}}{D_1}^{k+2n+2}{\displaystyle \frac{1}{k}}\sum _{i=1}^k{\displaystyle \frac{1}{i}}$$

(31)

Let us use denotation ${\tilde{C}}_2\left(k\right)={\displaystyle \frac{{\lambda }^{2}C}{2c_1{\left(n+1\right)}^3}}{\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{\displaystyle \frac{1}{i}}$. It is evident that ${\lim }_{k\to \infty }{\tilde{C}}_2\left(k\right)=0$. Therefore, we can choose a constant $A_2>0$ such that condition $\left|{\tilde{C}}_2\left(k\right)\right|<A_2$ is fulfilled. Using this inequality and (31), we obtain the estimate

$$\left|a_2^{2,k}\right|\le A_2{D_1}^{k+2n+2}.$$

2.

Now, we estimate coefficients $a_2^{1,k}$ at $r^k$ in power series $B_{n+1}^{-1}{\widehat{A}}_1{\sum }_{i=2,i\ne k\left(n+1\right)}^{\infty }{b}_2^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$. At first, we apply operator ${\widehat{\mathrm{A}}}_1^{\mathrm{i}}=-{\displaystyle \frac{1}{2c_{1}p}}{\alpha }_i^1{\widehat{r}}^{i}p$ to one of the terms $b_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$ of power series $B_{n+1}^{-1}{\widehat{A}}_1{\sum }_{i=2,i\ne k\left(n+1\right)}^{\infty }{b}_2^i{p}^{{\displaystyle \frac{i}{n+1}}-1}$, namely

$$\begin{array}{c}\hfill {\widehat{A}}_1^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}=b_1^i{\displaystyle \frac{1}{p}}{\displaystyle \frac{{\alpha }_j^{1}n{\widehat{r}}^{j}p}{2c_1}}{p}^{{\displaystyle \frac{i}{n+1}}-1}=b_1^i{\displaystyle \frac{{\alpha }_j^{1}n}{2c_{1}p}}{\displaystyle \frac{\left(n+1\right)\Gamma \left(\frac{i}{n+1}\right)\sin \pi \frac{i}{n+1}}{\pi }}{B}_{n+1}{r}^{i+j+n+1}\end{array}.$$

(32)

It follows from (9) that

$$B_{n+1}{r}^{n+1+i+j}={\displaystyle \frac{\pi }{\left(n+1\right)\Gamma \left(\frac{n+1+i+j}{n+1}\right)\sin \pi \frac{n+1+i+j}{n+1}}}{p}^{{\displaystyle \frac{n+1+i+j}{n+1}}-1}$$

(33)

Substituting (33) into (32), we obtain

$$\begin{array}{cc}\hfill {\widehat{A}}_1^j{b}_1^i{p}^{{\displaystyle \frac{i}{n+1}}-1}=& b_1^i{\alpha }_j^{1}n{\displaystyle \frac{\Gamma \left(\frac{i}{n+1}\right)\sin \pi \frac{i}{n+1}}{\Gamma \left(\frac{n+1+i+j}{n+1}\right)\sin \pi \frac{n+1+i+j}{n+1}}}{p}^{{\displaystyle \frac{i+j}{n+1}}-1}=\hfill \\ \hfill =& {\alpha }_j^{1}n{\displaystyle \frac{C\lambda }{2c_1\left(n+1\right)}}{\displaystyle \frac{a_{i+n+1}^{\prime }}{i}}{\displaystyle \frac{1}{\Gamma \left(\frac{n+1+i+j}{n+1}\right)\sin \pi \frac{n+1+i+j}{n+1}}}{p}^{{\displaystyle \frac{i+j}{n+1}}-1}.\hfill \end{array}$$

(34)

Since

$$B_{n+1}^{-1}{p}^{{\displaystyle \frac{i+j}{n+1}}-1}={\displaystyle \frac{\left(n+1\right)\Gamma \left(\frac{i+j}{n+1}\right)\sin \pi \frac{i+j}{n+1}}{\pi }}{r}^{i+j},$$

then

$$\begin{array}{cc}& \left|B_{n+1}^{-1}{\widehat{A}}_1^j{b}_1^i{p}^{{\displaystyle \frac{k}{n+1}}-1}\right|=\hfill \\ & =\left|{\alpha }_{k-i}^{1}n{\displaystyle \frac{C\lambda }{2c_1\left(n+1\right)}}{\displaystyle \frac{a_{i+n+1}^{\prime }}{i}}{\displaystyle \frac{1}{\Gamma \left(\frac{n+1+k}{n+1}\right)\sin \pi \frac{n+1+k}{n+1}}}{\displaystyle \frac{\left(n+1\right)\Gamma \left(\frac{k}{n+1}\right)\sin \pi \frac{k}{n+1}}{\pi }}{r}^k\right|=\hfill \\ & =\left|{\alpha }_{k-i}^{1}n{\displaystyle \frac{C\lambda }{2c_1}}{\displaystyle \frac{a_{i+n+1}^{\prime }}{i\pi }}{\displaystyle \frac{\Gamma \left(\frac{k}{n+1}\right)}{\Gamma \left(\frac{k}{n+1}+1\right)}}{r}^k\right|=\left|{\alpha }_{k-i}^{1}n{\displaystyle \frac{C\lambda }{2c_1}}{\displaystyle \frac{a_{i+n+1}^{\prime }}{i\pi k}}{r}^k\right|\hfill \end{array}$$

(35)

Let us estimate the corresponding coefficient at $r^k$. It follows from (35) that

$$\left|a_2^{1,k}\right|=\left|{\displaystyle \frac{\lambda C}{2c_1\pi }}n\sum _{i=1}^k{\displaystyle \frac{{\alpha }_{k-i}^1{a}_{i+n+1}^{\prime }}{ki}}\right|={\displaystyle \frac{\lambda Cn}{2c_1\pi }}{\displaystyle \frac{1}{k}}\sum _{i=1}^k\left|{\displaystyle \frac{{\alpha }_{k-i}^1{a}_{i+n+1}^{\prime }}{i}}\right|.$$

(36)

Let us use denotation ${\tilde{C}}_1\left(k\right)={\displaystyle \frac{\lambda Cn}{2c_1\pi }}{\displaystyle \frac{1}{k}}{\sum }_{i=1}^k{\displaystyle \frac{1}{i}}$. It is obvious that ${\lim }_{k\to \infty }{\tilde{C}}_1\left(k\right)=0$. Hence, we can choose the constant $A_1>0$ such that condition $\left|\tilde{C}\left(k\right)\right|<A_1$ is fulfilled. Using this inequality and (36), we obtain the estimate $\left|a_2^{1,k}\right|<A_1{D_1}^{k+2n+2}$.

The estimates for $\left|a_2^{3,k}\right|, \left|a_2^{4,k}\right|$ can be determined in the same way. So, we obtain

$$\left|a_2^{3,k}\right|\le A_2{D_1}^{k+2n+2},\left|a_2^{4,k}\right|\le A_1{D_1}^{k+2n+2}.$$

Let us introduce the denotation $A=\max \left(A_1,A_2\right)$.

Then,

$$\left|a_k^2\right|<\left|a_2^{1,k}\right|+\left|a_2^{2,k}\right|+\left|a_2^{3,k}\right|+\left|a_2^{4,k}\right|<4A{D_1}^{k+2\left(n+1\right)}.$$

It is evident that at the ith step of this method of successive approximations, we obtain the estimate

$$\left|a_k^i\right|<{\left(4A\right)}^i{D_1}^{k+i\left(n+1\right)}.$$

The estimate

$$\left|{\beta }_k^1\right|\le \sum _{i=1}^k\left|a_k^i\right|<4A{D_1}^{k+n+1}+4A{D_1}^{k+2\left(n+1\right)}+....+{\left(4A\right)}^k{D_1}^{k+k\left(n+1\right)}$$

is valid.

Then, the estimate

$$\left|{\beta }_k^1\right|\le {D_1}^k\sum _{i=1}^k{\left(4A{D_1}^{n+1}\right)}^i={D_1}^k{\displaystyle \frac{4A{D_1}^{n+1}\left({\left(4A{D_1}^{n+1}\right)}^k-1\right)}{4A{D_1}^{n+1}-1}}<$$

$$<{D_1}^k{\left(4A{D_1}^{n+1}\right)}^{k+1}<{\left(4A{D_1}^{n+2}\right)}^{k+1}$$

is also valid.

It is obvious that power series ${\sum }_{i=1}^{\infty }{\left(4A{D_1}^{n+2}\right)}^{i+1}{x}^i$ converges in some neighborhood of zero. The first statement of Theorem 2 is proven.

• We let $\lambda =0$ or $k<-1$.

If $\lambda =0$, then Equation (11) has the form

$${\left(-r^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)=0$$

(37)

or

$$r^2{\left({\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+\left(2+{\displaystyle \frac{k}{2}}\right)r{\displaystyle \frac{d}{dr}}=0.$$

(38)

Since $2+{\displaystyle \frac{k}{2}}\ne 0$, then (38) is a Fuchsian equation with a regular singular point at zero.

If $k\le -2$, Equation (11) can be written in the form

$$r^2{\left({\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+\left(2+{\displaystyle \frac{k}{2}}\right)r{\displaystyle \frac{d}{dr}}+{\displaystyle \frac{1}{r^{k+2}}}a\left(r\right)\lambda v\left(r\right)=0.$$

(39)

It is evident that (39) is a Fuchsian equation with a regular singular point at zero.

Theorem 2 is proven. □

Remark 1.

It follows from (16) that with the help of exponential substitutions, Equation (11) can be reduced to the form

$${\left(-{\displaystyle \frac{1}{1+\frac{k}{2}}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+c\left(-{\displaystyle \frac{1}{1+\frac{k}{2}}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)v\left(r\right)+a\left(r\right)v\left(r\right)=0,$$

(40)

where $a\left(r\right)$ is a function holomorphic in svome neighborhood of zero, such that $a\left(0\right)\ne 0$. Hence, the power series involved in the asymptotics of solutions to Equation (40), which have the form of (12) and (13), converge in some neighborhood of zero.

Remark 2.

Let us consider equation

$${\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+c{r}^{1+{\displaystyle \frac{k}{2}}}\left(-{\displaystyle \frac{1}{1+n}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)v\left(r\right)+a\left(r\right)v\left(r\right)=0.$$

(41)

Here, $a\left(r\right)$ is a function holomorphic in some neighborhood of zero. After performing power substitution $v\left(r\right)=r^{\sigma }{v}_1\left(r\right)$, Equation (41) takes the form

$${\left(-r^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^2{v}_1\left(r\right)+\left(-2\sigma +c\right)r^{1+{\displaystyle \frac{k}{2}}}\left(-r^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)v_1\left(r\right)+$$

$$+\sigma \left(\sigma +1+{\displaystyle \frac{k}{2}}-c\right)r^{2\left({\displaystyle \frac{k}{2}}+1\right)}{v}_1\left(r\right)+a\left(r\right)v_1\left(r\right)=0.$$

Let us choose $\sigma ={\displaystyle \frac{c}{2}}$; then, Equation (41) can be rewritten as follows:

$${\left(-r^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^2{v}_1\left(r\right)+\sigma \left(1+{\displaystyle \frac{k}{2}}-\sigma \right)r^{2\left({\displaystyle \frac{k}{2}}+1\right)}{v}_1\left(r\right)+a\left(r\right)v_1\left(r\right)=0.$$

Theorem 2 implies that the asymptotics of solutions to Equation (41) has the form of (12) or (13), where the power series are convergent in some neighborhood of zero.

4. Examples of Application of the Method of Successive Approximations

4.1. Application 1— Coherent Heun Equation

Let us consider the problem of constructing the asymptotics of solutions in the neighborhood of an irregular singular point for one of the equations of quantum mechanics.

Problem formulation.

Let us consider the stationary coherent Heun equation [21],

$$z{\omega }^{″}\left(z\right)+\left(c-z\right){\omega }^{\prime }\left(z\right)-a\omega \left(z\right)=0.$$

(42)

Here, c and a are some constants. This equation has two singular points, namely a regular singular point at zero and an irregular singular point at infinity. We seek the asympotitcs of solutions at $z\to \infty$ in the space of the functions of exponential growth. The asymptotics of solutions in the neighborhood of the regular singular point are conormal. The problem is to construct the asymptotics of solutions to Equation (42) in the neighborhood of the irregular singularity. Let us use the substitution $r={\displaystyle \frac{1}{z}}$ and modify Equation (42).

Since the identity

$${\displaystyle \frac{d}{dz}}\omega \left(z\right)={\displaystyle \frac{d\omega }{dr}}{\displaystyle \frac{dr}{dz}}=-{\displaystyle \frac{1}{z^2}}{\displaystyle \frac{d\omega }{dr}}=r^2{\displaystyle \frac{d\omega }{dr}}$$

holds, then the Heun equation can be rewritten in the form

$${\left(r^2{\displaystyle \frac{d}{dr}}\right)}^2+\left(r^2{\displaystyle \frac{d}{dr}}\right)\omega -cr\left(r^2{\displaystyle \frac{d}{dr}}\right)\omega -ar\omega =0.$$

(43)

Let us use substitution $\omega \left(r\right)=r^{\sigma }{\omega }_1\left(r\right)$ as it was performed in Remark 2. Then, Equation (43) takes the form

$$\begin{array}{cc}\hfill & {\left(-r^2{\displaystyle \frac{d}{dr}}\right)}^2{\omega }_1\left(r\right)+\left(-r^2{\displaystyle \frac{d}{dr}}\right){\omega }_1\left(r\right)+\left(2\sigma -c\right)r\left(-r^2{\displaystyle \frac{d}{dr}}\right){\omega }_1\left(r\right)+\hfill \\ \hfill & +\left(\sigma \left(\sigma +1\right)r^2+\sigma r-c\sigma r^2-ar\right){\omega }_1\left(r\right)=0\hfill \end{array}$$

(44)

Let us choose $\sigma ={\displaystyle \frac{c}{2}}$; then, Equation (44) can be rewritten in the form

$${\left(-r^2{\displaystyle \frac{d}{dr}}\right)}^2{\omega }_1\left(r\right)+\left(-r^2{\displaystyle \frac{d}{dr}}\right){\omega }_1\left(r\right)+\left(\left(\sigma -{\sigma }^2\right)r^2+\left(\sigma -a\right)r\right){\omega }_1\left(r\right)=0.$$

(45)

It is obvious that Equation (45) has the from (40). It follows from Theorem 2 that the asymptotics of solutions to Equation (45) in the neighborhood of zero has the form

$${\omega }_1\left(r\right)\approx \exp \left({\displaystyle \frac{1}{r}}\right)r^{{\sigma }_1}\sum _{i=0}^{\infty }{\beta }_i^1{r}^i+r^{{\sigma }_2}\sum _{i=0}^{\infty }{\beta }_i^2{r}^i,$$

where $g_1\left(r\right)={\sum }_{i=0}^{\infty }{\beta }_i^1{r}^i,g_2\left(r\right)={\sum }_{i=0}^{\infty }{\beta }_i^2{r}^i$ are the functions holomorphic in some neighborhood of zero. This implies the following lemma.

Lemma 1.

The asymptotics of solutions to Equation (42) in the neighborhood of an infinitely distant point have the form

$$\omega \left(z\right)\approx \exp \left(z\right)z^{-{\sigma }_1}{g}_1^{\prime }\left(z\right)+z^{-{\sigma }_2}{g}_2^{\prime }\left(z\right).$$

Here, $g_1\left(z\right)={\sum }_{i=0}^{\infty }{\beta }_i^1{z}^{-i}$, $g_2\left(z\right)={\sum }_{i=0}^{\infty }{\beta }_i^2{z}^{-i}$ are the functions holomorphic in the neighborhood of an infinitely distant point.

Let us present some examples of the problems on constructing asymptotics of solutions to partial differential equations.

4.2. Application 2—Boundary Value Problem for the Wave Operator Equation

Problem formulation.

Let us designate by $\Omega \subset R^n$ the region with a smooth boundary $\partial \Omega$; by $Q_{\infty }=\Omega \times (0\le t<\infty )$ a cylinder in the space $R^{n+1}$; and by $\partial Q_{\infty }=\partial \Omega \times (0\le t<\infty )$ the lateral boundary of the cylinder.

The boundary value problem is as follows:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{d}{dt}}\right)}^{2}u(x,t)-a^0\left(t\right)\Delta u\left(x,t\right)=0\end{array}$$

(46)

$${\left.\left(\alpha u+\beta {\displaystyle \frac{\partial u}{\partial n}}\right)\right|}_{\partial Q_{\infty }}=0.$$

(47)

Here, $\alpha \left(x\right)\ge 0,\beta \left(x\right)\ge 0,a^0\left(t\right)>0$; function $a^0\left(t\right)>0$ is meromorphic at infinity, which means that function $a^0\left(t\right)$ can be expanded into a convergent series:

$$a^0\left(t\right)=t^k\sum _{j=0}^{\infty }{\displaystyle \frac{a_j}{t^j}}.$$

(48)

Here, $k\in Z$. Without loss of generality, we consider $a_0>0$. Our task is to construct the asymptotics of solutions at $t\to \infty$.

We seek the solution of Problems (46) and (47) in the form $u(x,t)=Y\left(x\right)v\left(t\right)$. Then, for the function $Y\left(x\right)$, we obtain the Sturm–Liouville problem

$$\Delta Y\left(x\right)+\lambda Y\left(x\right)=0$$

(49)

$$\alpha Y+{\left.\beta {\displaystyle \frac{\partial Y}{\partial n}}\right|}_{\partial \Omega }=0$$

(50)

Its solution is well known. The Sturm–Liouville operator is a semibounded operator; it has a denumerable number of eigenvalues. For function $v\left(t\right)$, we obtain the equation

$${\left({\displaystyle \frac{d}{dt}}\right)}^{2}v\left(t\right)+a^0\left(t\right)\lambda v\left(t\right)=0.$$

(51)

So, the problem of constructing asymptotics of solutions of the boundary value Problems (46) and (47) is reduced to studying the problem of constructing the asymptotics of solutions to the ordinary differential equation for which infinity is a singular point. To construct the asymptotics of solution to Equation (51) at $t\to \infty$, we use substitution $t={\displaystyle \frac{1}{r}}$ and rewrite Equation (51) in the form

$${\left(-r^2{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+a^1\left(r\right)\lambda v\left(r\right)=0$$

(52)

Here, $a^1\left(r\right)={\displaystyle \frac{1}{r^k}}{\sum }_{j=0}^{\infty }{a}_j{r}^j$. Let us write Equation (52) in the form

$$r^k{\left(-r^2{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+a\left(r\right)\lambda v\left(r\right)=0,$$

(53)

where $a\left(r\right)=r^k{a}^1\left(r\right)={\sum }_{j=0}^{\infty }{a}_j{r}^j$ is a holomorphic function. Since equality

$$r^k{\left(-r^2{\displaystyle \frac{d}{dr}}\right)}^2=\left(r^{{\displaystyle \frac{k}{2}}+2}{\displaystyle \frac{d}{dr}}\right)\left(r^{{\displaystyle \frac{k}{2}}+2}{\displaystyle \frac{d}{dr}}\right)-{\displaystyle \frac{k}{2}}{r}^{k+3}{\displaystyle \frac{d}{dr}}$$

holds, then Equation (53) can be written as follows:

$$\begin{array}{c}\hfill {\left(-{\displaystyle \frac{2}{k+2}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+{\displaystyle \frac{k}{2}}{r}^{1+{\displaystyle \frac{k}{2}}}\left(-{\displaystyle \frac{2}{k+2}}{r}^{2+{\displaystyle \frac{k}{2}}}{\displaystyle \frac{d}{dr}}\right)+{\left({\displaystyle \frac{2}{k+2}}\right)}^{2}a\left(r\right)\lambda v\left(r\right)=0\end{array}.$$

(54)

As above, we introduce denotation $n=\left[{\displaystyle \frac{k}{2}}\right]$. It follows from Equation (54) and Remark 2 to Theorem 2 that the following lemma is valid.

Lemma 2.

The asymptotics of solutions to Problems (46) and (47) at $t\to \infty$ in the space of functions of k-exponential growth are representable as a linear combination of functions $u_i\left(x,t\right),i=0,1\dots$, where

$$u_i\left(x,t\right)\approx v_i\left(t\right)Y_{{\lambda }_i}\left(x\right)$$

(55)

Here, $Y_{{\lambda }_i}\left(x\right)$ are the solutions of Problems (49) and (50), corresponding to the eigenvalue ${\lambda }_i$; $v_i\left(t\right)$ are the asymptotics of solutions to Equation (51), which have the following form:

1. at ${\lambda }_i\ne 0$ and $k\ge -1$:

at even k

$$\exp \left(c_1{t}^{n+1}+\sum _{i=1}^n{{\alpha }^1}_i{t}^{n+1-i}\right)t^{-{\sigma }_1}{g}_1\left(t\right)+\exp \left(c_2{t}^{n+1}+\sum _{i=1}^n{{\alpha }^2}_i{t}^{n+1-i}\right)t^{-{\sigma }_2}{g}_2\left(t\right);$$

at odd k

$$\exp \left(c_1{t}^{n+{\displaystyle \frac{3}{2}}}+\sum _{i=1}^{2n+2}{{\alpha }^1}_i{t}^{n+{\displaystyle \frac{3}{2}}-{\displaystyle \frac{i}{2}}}\right)t^{-{\sigma }_1}\left(g_1^1\left(t\right)+\sqrt{{\displaystyle \frac{1}{t}}}{g}_1^2\left(t\right)\right)+$$

$$+\exp \left(c_2{t}^{n+{\displaystyle \frac{3}{2}}}+\sum _{i=1}^{2n+2}{{\alpha }^2}_i{t}^{n+{\displaystyle \frac{3}{2}}-{\displaystyle \frac{i}{2}}}\right)t^{{\sigma }_2}\left(g_2^1\left(t\right)+\sqrt{{\displaystyle \frac{1}{t}}}{g}_2^2\right)$$

Here, $g_1^i,g_2^i,g_i,i=1,2$ are the functions holomorphic in the neighborhood of an infinitely distant point, the numbers $c_1,c_2$ are the roots of the polynomial ${\displaystyle \frac{1}{{\left(1+\frac{k}{2}\right)}^2}}a\left(0\right){\lambda }_i+p^2$.

2. at $k\le -2$ or $\lambda =0$, all the asymptotics of solutions are conormal or the solutions are regular.

4.3. Application 3—The Laplace Equation on a Plane

The problem formulation is as follows.

It is required to find the asymptotics of solutions to equation

$$\left({\displaystyle \frac{1}{R}}{\displaystyle \frac{\partial }{\partial R}}\left(R{\displaystyle \frac{\partial }{\partial R}}\right)+{\displaystyle \frac{1}{R^2}}{\displaystyle \frac{{\partial }^2}{\partial {\phi }^2}}\right)u\left(R,\phi \right)+a^0\left(R\right)u\left(R,\phi \right)=0$$

(56)

at $R\to \infty$, where $a^0\left(R\right)=R^k{\sum }_{j=0}^{\infty }{\displaystyle \frac{a_j}{R^j}}$ is a function meromorphic at infinity.

We seek the solution in the form $u\left(R,\phi \right)=G\left(\phi \right)V\left(R\right)$. For function $G\left(\phi \right)$, we obtain the problem

$$-G^{″}\left(\phi \right)=\mu G\left(\phi \right),G\left(\phi \right)=G\left(\phi +2\pi \right).$$

(57)

As is known, the solutions of Problem (57) have the form of function $G_k\left(\phi \right)=C_1\cos k\phi +C_2\sin k\phi$, where $k\in Z$ and its eigenvalues are $\mu =k^2$;

$V\left(R\right)$ are the solutions of equation

$$R{\displaystyle \frac{d}{dR}}\left(R{\displaystyle \frac{d}{dR}}\right)V\left(R\right)+R^2{a}_0\left(R\right)V\left(R\right)-\mu V\left(R\right)=0$$

(58)

corresponding to them.

Let us make the change of variables $r={\displaystyle \frac{1}{R}}$; then, Equation (58) takes the form

$${\left(r{\displaystyle \frac{d}{dr}}\right)}^{2}V\left(r\right)+{\displaystyle \frac{1}{r^2}}a\left(r\right)V\left(R\right)-\mu V\left(R\right)=0,$$

(59)

where $a\left(r\right)=r^{-k}{\sum }_{j=0}^{\infty }{a}_j{r}^j$.

Since

$$r^{2+k}{\left(r{\displaystyle \frac{d}{dr}}\right)}^2={\left(r^{{\displaystyle \frac{4+k}{2}}}{\displaystyle \frac{d}{dr}}\right)}^2-\left({\displaystyle \frac{k+2}{2}}\right)r^{{\displaystyle \frac{4+k}{2}}-1}\left(r^{{\displaystyle \frac{4+k}{2}}}{\displaystyle \frac{d}{dr}}\right),$$

(60)

substituting (60) into (59), we obtain equation

$$\begin{array}{c}\hfill {\left(r^{{\displaystyle \frac{k}{2}}+2}{\displaystyle \frac{d}{dr}}\right)}^2-\left({\displaystyle \frac{k+2}{2}}\right)r^{{\displaystyle \frac{k}{2}}+1}\left(r^{{\displaystyle \frac{k}{2}}+2}{\displaystyle \frac{d}{dr}}\right)+\left(a\left(r\right)-r^{2+k}\mu \right)V\left(r\right)=0\end{array}$$

(61)

We obtained an equation of type (41). Therefore, Remark 2 to Theorem 2 can be applied to Equation (61). Hence, the asymptotics of solutions to Problem (61) can be constructed as in Application 2.

Remark 3.

The boundary value problem of constricting the asymptotics of solutions at $t\to \infty$ for the heat conduction equation with a meromorphic coefficient is solved in [7]. Namely, in [7], the asymptotics of solutions to problem

$$\left({\displaystyle \frac{d}{dt}}\right)u(x,t)-a^0\left(t\right)\Delta u\left(x,t\right)=0,$$

(62)

$${\left(\alpha u+\left.\beta {\displaystyle \frac{\partial u}{\partial n}}\right)\right|}_{\partial Q_{\infty }}=0,$$

(63)

where $\alpha \left(x\right)>0,\beta \left(x\right)\ge 0$, $a^0\left(t\right)\ge 0$ is a meromorphic function in the neighborhood of an infinitely distant point, are found. Here, the differential operator with respect to t has the first order; therefore, Theorem 2 cannot be applied to this problem.

Note also that in the presented examples given for the wave equation and for the Laplace equation, we assume that the corresponding coefficients in these equations are meromorphic functions in the neighborhood of infinity; in other words, the singular point in these examples is infinity. We can consider the same problems under the assumption that these coefficients have a meromorphic singularity at zero or any other finite point. If this singular point is an irregular singularity, by applying Theorem 2, we can construct asymptotics of solutions in the vicinity of this singular point in exactly the same way as was performed earlier in present work.

5. Conclusions

In this study, the method is created to estimate the coefficients of the power series involved in the asymptotics of solutions to some second-order differential equations with irregular singularities. The essence of this method is to reduce a second-order differential equation to form (11) by using the corresponding substitutions and then apply Theorem 2. This method can be applied to a wide class of ordinary differential equations and to partial differential equations. Namely, this method is applicable to the equations that can be reduced to the form

$${\left(-r^m{\displaystyle \frac{d}{dr}}\right)}^{2}v\left(r\right)+a\left(r\right)v\left(r\right)=0,$$

where $a\left(r\right)$ is a holomorphic function such that $a\left(0\right)\ne 0$, and to all the equations that can be reduced to this form with the help of the exponential substitutions described in Section 3 of this paper.

In addition to the examples considered in present paper, our method is applicable, for example, to the study of the asymptotics of the Schrödinger equation with various types of potentials that are meromorphic functions. For example, in [22], the stationary Schrödinger equation with various types of singular potentials, many of which are meromorphic functions, is considered. For each of these problems, Theorem 2 can be applied and can thereby construct the asymptotics of the potentials in the neighborhoods of these singular points. Also, a further direction of research can be the study of inhomogeneous equations of the second equation with irregular singularities in both the non-resonant and resonant cases. The beginning of such studies was laid by [23] for the non-resonant case and [24] for the resonant case. The question of the convergence of the series included in the corresponding asymptotic series still remains open. It is also possible to continue the study of the telegraph equation, namely to refine the result obtained in work [25] and to prove, using the created method, the convergence of the corresponding formal series included in the asymptotics of the solutions of these problems.