The Maximum Domain for an Analytical Approximate Solution to a Nonlinear Differential Equation in the Neighborhood of a Moving Singular Point

Abstract

This paper presents the final stage in the study of the analytical approximate solution to a class of nonlinear differential equations unsolvable in quadrature in the general case in the neighborhood of a perturbed value of a moving singular point. An a priori error estimation is proven. The scope of application of the analytical approximate solution is extended; the formula for calculating this scope is obtained. The proof of the theorem is based on the application of elements of differential calculus. Theoretical results are supported by numerical calculations, which validate their reliability. The authors report a numerical comparison between the results, obtained in the paper, and the findings that were published earlier.

1. Introduction

In the process of solving engineering problems, researchers come across various linear and nonlinear differential equations. Linear equations do not constitute a major problem, the classical theory works well in this case. Nonlinear differential equations may have the following areas of application: they are used to study wave processes in elastic beams [1,2] and to research cantilever structures [3,4]. However, the case of nonlinear differential equations may involve difficulties due to their nonlinearity, as well as the presence of moving singular points. This category of equations belongs to the class that is not solvable in quadrature in the general case. At present, there are two alternative solutions to such equations. Solution One deals with special solutions to such equations using special substitution of variables [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Solution One, dealing with nonlinear differential equations, does not solve the problem as a whole. Solution Two is the development of an analytical approximate method for solving nonlinear differential equations with moving singular points. Therefore, studies are underway to develop new analytical approximate methods of solving these classes of equations. Let us focus on recent publications [24,25,26]. The results, presented in this article are the final stage of a research project [27,28,29,30] that solved the following problems: the theorem of existence and uniqueness of the solution was proved; the analytical approximate solution was obtained, and a priori error estimations were proved; the effect of perturbation of a moving singular point on the structure of the analytical approximate solution was investigated; and the exact criteria for the existence of a moving singular point were obtained for this class of equations both in complex and real domains. In the course of solving the above problems, a considerable reduction in the scope of application of the analytical approximate solution was observed. This paper presents a solution to the problem of extending the scope of an analytical approximate solution in the neighborhood of a moving singular point. It can be solved by switching from the classical method of obtaining estimations to a solution using elements of differential calculus. The results are in good agreement with the classical version, and allow for a substantial extension of the scope of application of the analytical approximate solution in the neighborhood of a moving singular point, as described in [28]. This statement is practically confirmed by the calculations presented in the article.

2. Research Technology

For the initial Cauchy problem

$$y^{\prime \prime \prime }\left(x\right)=y^2\left(x\right)+r\left(x\right),$$

(1)

$$\left\{\begin{array}{c}y\left(x_0\right)=y_0,\hfill \\ y^{\prime }\left(x_0\right)=y_1,\hfill \\ y^{\prime \prime }\left(x_0\right)=y_2.\hfill \end{array}\right.$$

(2)

Study [27] has an analytical approximate solution for the neighborhood of a moving singular point:

$$y\left(x\right)={\left(x^{*}-x\right)}^{-3}·\sum _{i=0}^{\infty }{C}_i{\left(x^{*}-x\right)}^i.$$

(3)

Since currently available methods allow for calculating the value of a moving singular point approximately, we obtain a new structure for the solution instead of (3):

$${\tilde{y}}_{\Im }\left(x\right)={\left({\tilde{x}}^{*}-x\right)}^{-3}·\sum _{i=0}^{\Im }{\tilde{C}}_i{\left({\tilde{x}}^{*}-x\right)}^i.$$

(4)

In this respect, ${\tilde{C}}_i$ are the perturbed values of the coefficients and ${\tilde{x}}^{*}$ is the perturbed value of a moving singular point.

When solving the first problem of the study [27], i.e., proving the existence and uniqueness theorem, we obtained the scope of application of the analytical approximate solution

$$\rho <\frac{1}{\sqrt[6]{M+1}}.$$

Further, when solving the problem of the effect of perturbation of a moving singular point on the analytical approximate solution [28], we demonstrated the narrowing scope of application of the approximate solution

$$\rho <\frac{1}{2\sqrt[6]{M+1}}.$$

In this paper, we can substantially expand the scope of application of the approximate solution, obtained in [28], by using the elements of differential calculus to estimate the solution error (4).

Theorem 1.

We require that the following conditions are satisfied:

1.

$r\left(x\right)\in C^{\infty }$ within $\left|{\tilde{x}}^{*}-x\right|<{\rho }_1$, where $0<{\rho }_1=const;$

2.

$\exists M_i:\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}\le M_i,M_i=const;$

3.

${\tilde{x}}^{*}\le x^{*};$

4.

There is an error estimation for ${\tilde{x}}^{*}:\left|{\tilde{x}}^{*}-x^{*}\right|\le \Delta {\tilde{x}}^{*};$

5.

$\Delta {\tilde{x}}^{*}<\frac{1}{\sqrt[6]{M+1}}.$

In this case, an error estimation is valid for the approximate solution (4) to the problem (1) and (2):

$$\Delta {\tilde{y}}_{\Im }\le {\Delta }_0+{\Delta }_1+{\Delta }_2+{\Delta }_3,$$

in

$$F=F_1\cap F_2\cap F_3,$$

where

$$F_1=\left\{x:x<{{\tilde{x}}_1}^{*}\right\},F_2=\left\{x:\left|{\tilde{x}}^{*}-x\right|<\frac{1}{\sqrt[6]{M+1}}\right\},F_3=\left\{x:\left|{{\tilde{x}}_2}^{*}-x\right|<\frac{1}{\sqrt[6]{M+\Delta M+1}}\right\},$$

$${{\tilde{x}}_1}^{*}={\tilde{x}}^{*}-\Delta {\tilde{x}}^{*},{{\tilde{x}}_2}^{*}={\tilde{x}}^{*}+\Delta {\tilde{x}}^{*},\Im \in {\mathbb{Z}}_{+},$$

$$M=max\left\{\underset{i}{sup}\left\{\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}\right\},\left|y_0\right|,\left|y_1\right|,\left|y_2\right|\right\},$$

$$\Delta M=max\left\{\underset{i}{sup}\left\{\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}\right\},\left|y_0\right|,\left|y_1\right|,\left|y_2\right|\right\}·\Delta {\tilde{x}}^{*},i=0,1,2,\dots ,$$

herewith

$${\Delta }_0=60\frac{\Delta {\tilde{x}}^{*}\left(3{\left|{{\tilde{x}}_1}^{*}-x\right|}^2+\left|{{\tilde{x}}_1}^{*}-x\right|\Delta {\tilde{x}}^{*}+{\left(\Delta {\tilde{x}}^{*}\right)}^2\right)}{{\left|{{\tilde{x}}_1}^{*}-x\right|}^6},$$

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im +1}{6}}·{\left|{{\tilde{x}}_1}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{{\tilde{x}}_1}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +1)(\Im +2)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +1)(\Im +3)(\Im +2)+120}\right)$$

in the case of $\Im +1=6i$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im }{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{\Im (\Im +1)(\Im -1)+120}+$$

$$\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{\Im (\Im +2)(\Im +1)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +1)(\Im +2)(\Im +3)+120}\right)$$

in the case of $\Im +1=6i+1$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -1}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-(M+1){\left|{\tilde{x}}^{*}-x\right|}^6}.\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{(\Im -1)\Im (\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im -1)(\Im +1)\Im +120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+2$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -2}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -2)(\Im -1)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{(\Im -2)\Im (\Im -1)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+3$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -3}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -3)(\Im -2)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -3)(\Im -1)(\Im -2)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+4$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -4}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -4)(\Im -2)(\Im -3)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+5$, which is valid in the domain $F_2$;

$${\Delta }_2\le \frac{\Delta M}{1-\left(M+\Delta M+1\right){\left|{{\tilde{x}}_2}^{*}-x\right|}^6}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{{\tilde{x}}_2}^{*}-x\right|}^k,$$

$${\Delta }_3\le \frac{6\Delta {\tilde{x}}^{*}\Delta M\left(M+\Delta M+1\right){\left|{{\tilde{x}}_2}^{*}-x\right|}^5}{{\left(1-\left(M+\Delta M+1\right){\left|{{\tilde{x}}_2}^{*}-x\right|}^6\right)}^2}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{{\tilde{x}}_2}^{*}-x\right|}^k+$$

$$\frac{\Delta M\Delta {\tilde{x}}^{*}}{1-\left(M+\Delta M+1\right){\left|{{\tilde{x}}_2}^{*}-x\right|}^6}\sum _{k=0}^{5}k{\left(M+\Delta M+1\right)}^k{\left|{{\tilde{x}}_2}^{*}-x\right|}^{k-1}.$$

Proof. 

Stemming from the classical approach, we have:

$$\Delta {\tilde{y}}_{\Im }\left(x\right)=\left|y\left(x\right)-{\tilde{y}}_{\Im }\left(x\right)\right|\le \left|y\left(x\right)-\tilde{y}\left(x\right)\right|+\left|\tilde{y}\left(x\right)-{\tilde{y}}_{\Im }\left(x\right)\right|.$$

Let us evaluate the expression

$$\left|y\left(x\right)-\tilde{y}\left(x\right)\right|$$

using elements of differential calculus [31]:

$$\left|y\left(x\right)-\tilde{y}\left(x\right)\right|\le \underset{U}{sup}\left|\frac{\partial \tilde{y}\left(x\right)}{\partial {\tilde{x}}^{*}}\right|\Delta {\tilde{x}}^{*}+\sum _{i=0}^{\infty }\underset{U}{sup}\left|\frac{\partial \tilde{y}\left(x\right)}{\partial {\tilde{C}}_i}\right|\Delta {\tilde{C}}_i,$$

where

$$U=\left\{x:\left|{\tilde{x}}^{*}-x\right|<\Delta {\tilde{x}}^{*}\right\}.$$

Further, we analyze the expression:

$$\underset{U}{sup}\left|\frac{\partial \tilde{y}\left(x\right)}{\partial {\tilde{x}}^{*}}\right|=\underset{U}{sup}\left|\sum _{i=0}^{\infty }{\tilde{C}}_i\left(i-3\right){\left({\tilde{x}}^{*}-x\right)}^{i-4}\right|\le \sum _{i=0}^{\infty }\underset{U}{sup}\left|{\tilde{C}}_i\right|\left(i-3\right)\underset{U}{sup}{\left|{\tilde{x}}^{*}-x\right|}^{i-4}.$$

Given the decomposition of the function $r\left(x\right)$ into a regular series, using the condition of the theorem,

$$r\left(x\right)=\sum _{i=0}^{\infty }{\tilde{A}}_i{\left({\tilde{x}}^{*}-x\right)}^i,$$

we can write the general case ${\tilde{C}}_i$ in the form of

$${\tilde{C}}_i={\tilde{C}}_i\left({\tilde{A}}_0,{\tilde{A}}_1,...,{\tilde{A}}_m\right).$$

Let us revise the estimations for ${\tilde{C}}_i$ [27]:

$$\left|{\tilde{C}}_{6i}\right|\le \frac{{(M+1)}^i}{(6i-3)(6i-4)(6i-5)+120},$$

$$\left|{\tilde{C}}_{6i+1}\right|\le \frac{{(M+1)}^i}{(6i-2)(6i-3)(6i-4)+120},$$

$$\left|{\tilde{C}}_{6i+2}\right|\le \frac{{(M+1)}^i}{(6i-1)(6i-2)(6i-3)+120},$$

(5)

$$\left|{\tilde{C}}_{6i+3}\right|\le \frac{{(M+1)}^i}{6i(6i-1)(6i-2)+120},$$

$$\left|{\tilde{C}}_{6i+4}\right|\le \frac{{(M+1)}^i}{(6i+1)(6i-1)6i+120},$$

$$\left|{\tilde{C}}_{6i+5}\right|\le \frac{{(M+1)}^i}{(6i+2)(6i+1)6i+120},$$

as well as the estimations for $\Delta {\tilde{C}}_i$ [28]:

$$\Delta C_{6i}\le \frac{\Delta M{(M+\Delta M+1)}^i}{(6i-3)(6i-4)(6i-5)+120},$$

$$\Delta C_{6i+1}\le \frac{\Delta M{(M+\Delta M+1)}^i}{(6i-2)(6i-3)(6i-4)+120},$$

(6)

$$\Delta C_{6i+2}\le \frac{\Delta M{(M+\Delta M+1)}^i}{(6i-1)(6i-2)(6i-3)+120},$$

$$\Delta C_{6i+3}\le \frac{\Delta M{(M+\Delta M+1)}^i}{6i(6i-1)(6i-2)+120},$$

$$\Delta C_{6i+4}\le \frac{\Delta M{(M+\Delta M+1)}^i}{(6i+1)(6i-1)6i+120},$$

$$\Delta C_{6i+5}\le \frac{\Delta M{(M+\Delta M+1)}^i}{(6i+2)(6i+1)6i+120}.$$

Now, let us estimate the expression $\underset{U}{sup}\left|{\tilde{C}}_i\right|$, taking into account the recurrence relations provided in [27], and, having performed a number of transformations, we obtain the following:

$$\underset{U}{sup}\left|{\tilde{C}}_i\right|\le {\tilde{C}}_i\left(\left|{\tilde{A}}_0+\Delta {\tilde{A}}_0\right|,\left|{\tilde{A}}_1+\Delta {\tilde{A}}_1\right|,...,\left|{\tilde{A}}_m+\Delta {\tilde{A}}_m\right|\right)\le \frac{{\left(M+\Delta M+1\right)}^{\left[\frac{i}{6}\right]}}{(i-3)(i-4)(i-5)+120}={\mathrm{T}}_i,$$

where

$$\Delta {\tilde{A}}_i=\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}·\Delta {\tilde{x}}^{*},$$

$$\Delta M=max\left\{\underset{i}{sup}\left\{\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}\right\},\left|y_0\right|,\left|y_1\right|,\left|y_2\right|\right\}·\Delta {\tilde{x}}^{*},$$

$$M=max\left\{\underset{i}{sup}\left\{\frac{\left|r^{\left(i\right)}\left({\tilde{x}}^{*}\right)\right|}{i!}\right\},\left|y_0\right|,\left|y_1\right|,\left|y_2\right|\right\},i=0,1,2,\dots ,.$$

Given that

$$\underset{U}{sup}{\left|{\tilde{x}}^{*}-x\right|}^{i-4}\le \left\{\begin{array}{c}{\left|{\tilde{x}}_1^{*}-x\right|}^{i-4},i=0,1,2,3\hfill \\ {\left|{\tilde{x}}_2^{*}-x\right|}^{i-4},i>3\hfill \end{array}\right.\left(where{\tilde{x}}_1^{*}={\tilde{x}}^{*}-\Delta {\tilde{x}}^{*},{\tilde{x}}_2^{*}={\tilde{x}}^{*}+\Delta {\tilde{x}}^{*}\right),$$

and

$$\underset{U}{sup}\left|\frac{\partial \tilde{y}\left(x\right)}{\partial {\tilde{C}}_i}\right|=\underset{U}{sup}{\left|{\tilde{x}}^{*}-x\right|}^{i-3}\le \left\{\begin{array}{c}{\left|{\tilde{x}}_1^{*}-x\right|}^{i-3},i=0,1,2\hfill \\ {\left|{\tilde{x}}_2^{*}-x\right|}^{i-3},i>2\hfill \end{array}\right.,$$

as well as considering the values of the first coefficients ${\tilde{C}}_i$:

$${\tilde{C}}_0=-60,{\tilde{C}}_1=0,{\tilde{C}}_2=0,{\tilde{C}}_3=0,{\tilde{C}}_4=0,{\tilde{C}}_5=0,$$

$${\tilde{C}}_6=\frac{{\tilde{A}}_0}{126},{\tilde{C}}_7=\frac{{\tilde{A}}_1}{144},{\tilde{C}}_8=\frac{{\tilde{A}}_2}{180},{\tilde{C}}_9=\frac{{\tilde{A}}_3}{240},{\tilde{C}}_{10}=\frac{{\tilde{A}}_4}{330},{\tilde{C}}_{11}=\frac{{\tilde{A}}_5}{456},...,$$

we obtain the following estimation for $\left|y\left(x\right)-\tilde{y}\left(x\right)\right|$:

$$\left|y\left(x\right)-\tilde{y}\left(x\right)\right|\le 60\frac{\Delta {\tilde{x}}^{*}\left(3{\left|{\widetilde{x_1}}^{*}-x\right|}^2+\left|{\widetilde{x_1}}^{*}-x\right|\Delta {\tilde{x}}^{*}+{\left(\Delta {\tilde{x}}^{*}\right)}^2\right)}{{\left|{\widetilde{x_1}}^{*}-x\right|}^6}+$$

$$\Delta {\tilde{x}}^{*}\sum _{i=6}^{\infty }\left(i-3\right){\mathrm{T}}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-4}+\sum _{i=6}^{\infty }\Delta {\tilde{C}}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-3}.$$

Hence,

$$\left|y\left(x\right)-{\tilde{y}}_i\left(x\right)\right|\le 60\frac{\Delta {\tilde{x}}^{*}\left(3{\left|{\widetilde{x_1}}^{*}-x\right|}^2+\left|{\widetilde{x_1}}^{*}-x\right|\Delta {\tilde{x}}^{*}+{\left(\Delta {\tilde{x}}^{*}\right)}^2\right)}{{\left|{\widetilde{x_1}}^{*}-x\right|}^6}+$$

$$\sum _{i=i+1}^{\infty }\left|{\tilde{C}}_i\right|{\left|{\tilde{x}}^{*}-x\right|}^{\frac{i-1}{2}}+\sum _{i=6}^{\infty }\Delta {\tilde{C}}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-3}+$$

$$\Delta {\tilde{x}}^{*}\sum _{i=7}^{\infty }\left(i-3\right){\mathrm{T}}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-4}={\Delta }_0+{\Delta }_1+{\Delta }_2+{\Delta }_3.$$

Therefore, we obtain

$${\Delta }_0=60\frac{\Delta {\tilde{x}}^{*}\left(3{\left|{\widetilde{x_1}}^{*}-x\right|}^2+\left|{\widetilde{x_1}}^{*}-x\right|\Delta {\tilde{x}}^{*}+{\left(\Delta {\tilde{x}}^{*}\right)}^2\right)}{{\left|{\widetilde{x_1}}^{*}-x\right|}^6}.$$

To estimate ${\Delta }_1$, we use theorem 2 from [27] is used, and therefore:

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im +1}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-(M+1){\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +1)(\Im +2)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +1)(\Im +3)(\Im +2)+120}\right)$$

in the case of $\Im +1=6i$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im }{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-{(M+1)}^{}{\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{\Im (\Im +1)(\Im -1)+120}+$$

$$\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{\Im (\Im +2)(\Im +1)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +1)(\Im +2)(\Im +3)+120}\right)$$

in the case of $\Im +1=6i+1$,

$${\Delta }_1\le \frac{{\left(M+1\right)}^{\frac{\Im -1}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-\left(M+1\right){\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{(\Im -1)\Im (\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im -1)(\Im +1)\Im +120}$$

$$+\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+2$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -2}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-\left(M+1\right){\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -2)(\Im -3)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -2)(\Im -1)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{(\Im -2)\Im (\Im -1)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+3$,

$${\Delta }_1\le \frac{{\left(M+1\right)}^{\frac{\Im -3}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-\left(M+1\right){\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -3)(\Im -2)(\Im -4)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -3)(\Im -1)(\Im -2)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+4$,

$${\Delta }_1\le \frac{{(M+1)}^{\frac{\Im -4}{6}}·{\left|{\tilde{x}}^{*}-x\right|}^{\Im +1}}{1-(M+1){\left|{\tilde{x}}^{*}-x\right|}^6}\times$$

$$\left(\frac{1}{(\Im -4)(\Im -2)(\Im -3)+120}+\frac{\left|{\tilde{x}}^{*}-x\right|}{(\Im -1)(\Im -2)(\Im -3)+120}+\right.$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^2}{\Im (\Im -1)(\Im -2)+120}+\frac{{\left|{\tilde{x}}^{*}-x\right|}^3}{(\Im +1)\Im (\Im -1)+120}+$$

$$\frac{{\left|{\tilde{x}}^{*}-x\right|}^4}{(\Im +2)(\Im +1)\Im +120}+\left.\frac{{\left|{\tilde{x}}^{*}-x\right|}^5}{(\Im +3)(\Im +2)(\Im +1)+120}\right)$$

in the case of $\Im +1=6i+5$.

These estimations are valid in the region of

$$F_2=\left\{x:\left|{\tilde{x}}^{*}-x\right|<\frac{1}{\sqrt[6]{M+1}}\right\}.$$

Then, we proceed to the estimations for ${\Delta }_2$:

$${\Delta }_2=\sum _{i=7}^{\infty }\Delta {\tilde{C}}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-3}.$$

Given the regularity of the estimations for $\Delta {\tilde{C}}_i$ (6) and the estimations obtained for

$$\underset{U}{sup}{\left|{\tilde{x}}^{*}-x\right|}^{i-3},$$

we will have:

$${\Delta }_2=\sum _{i=7}^{\infty }\left|\Delta {\tilde{C}}_i\right|{\left|{\tilde{x}}_2^{*}-x\right|}^{i-3}\le \sum _{i=1}^{\infty }\sum _{k=0}^5\frac{\Delta M{\left(M+\Delta M+1\right)}^{\left[\frac{i}{6}\right]+k}}{(i-3+k)(i-4+k)(i-5+k)+120}{\left|{\tilde{x}}_2^{*}-x\right|}^{6n+k}\le$$

$$\sum _{k=0}^5\frac{\Delta M{\left(M+\Delta M+1\right)}^k}{1-\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^6}{\left|{\tilde{x}}_2^{*}-x\right|}^k=$$

$$\frac{\Delta M}{1-\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^6}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{\tilde{x}}_2^{*}-x\right|}^k.$$

Thus, the following expression is obtained:

$${\Delta }_2<\frac{\Delta M}{1-\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^6}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{\tilde{x}}_2^{*}-x\right|}^k.$$

This estimation is valid in the region of

$$F_2=\left\{x:\left|{\tilde{x}}_2^{*}-x\right|<\frac{1}{\sqrt[6]{M+1}}\right\}.$$

Let us demonstrate the validity of the estimation of ${\Delta }_3$:

$${\Delta }_3\le \Delta {\tilde{x}}^{*}{\left(\sum _{i=6}^{\infty }{T}_i{\left|{\tilde{x}}_2^{*}-x\right|}^{i-3}\right)}_{{\tilde{x}}_2}^{\prime }.$$

The series, having the sign of a derivative, is expanded, taking into account estimations (5) for coefficients ${\tilde{C}}_i$, and the corresponding number of series is obtained. Then, the series thus obtained are written down in the concise form with a new summation index

$$\Delta {\tilde{x}}^{*}{{\left(\frac{\Delta M}{1-\left(M+\Delta M+1\right){\left|{{\tilde{x}}_2}^{*}-x\right|}^6}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{{\tilde{x}}_2}^{*}-x\right|}^k\right)}_{{{\tilde{x}}_2}^{*}}}^{\prime }.$$

Differentiation is carried out in respect of the above expression. As a result, the estimation for ${\Delta }_3$ is obtained:

$${\Delta }_3\le \frac{6\Delta {\tilde{x}}^{*}\Delta M\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^5}{{\left(1-\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^6\right)}^2}\sum _{k=0}^5{\left(M+\Delta M+1\right)}^k{\left|{\tilde{x}}_2^{*}-x\right|}^k+$$

$$\frac{\Delta M\Delta {\tilde{x}}^{*}}{1-\left(M+\Delta M+1\right){\left|{\tilde{x}}_2^{*}-x\right|}^6}\sum _{k=0}^{5}k{\left(M+\Delta M+1\right)}^k{\left|{\tilde{x}}_2^{*}-x\right|}^{k-1}.$$

Estimation ${\Delta }_3$ works in the region of

$$F_3=\left\{x:\left|{\tilde{x}}_2^{*}-x\right|<\frac{1}{\sqrt[6]{M+\Delta M+1}}\right\}.$$

We finally obtain scope of application

$$F=F_1\cap F_2\cap F_3,$$

in which the statements of the presented theorem are valid. □

3. Discussion of the Results

Example 1.

Let us compute the Cauchy problem (1) and (2), having the following conditions

$$r\left(x\right)=x^5;y\left(0\right)=0.2;y^{\prime }\left(0\right)=-0.3;y^{\prime \prime }\left(0\right)=0.95;{\tilde{x}}^{*}=4.2135;\Delta {\tilde{x}}^{*}=0.0001.$$

The problem (1) and (2) is not solvable in quadrature under these conditions. We apply the obtained results to calculate the approximate solution to (4) at $\Im =9$ for the value of argument $x_1$ taking into account the scope

$$F=F_1\cap F_2\cap F_3,$$

to which the theorem [28] is applicable; the scope is determined using the value of $\rho =0.56123$. The results of the calculations are presented in

Table 1. Characteristics of the approximate solution.

${\mathit{x}}_1$	${\tilde{\mathit{y}}}_9\left({\mathit{x}}_1\right)$	${\mathbf{\Delta }}_4$	${\mathbf{\Delta }}_5$
$3.7$	$442.2903$	$0.04$	$0.002$

, where ${\tilde{y}}_9\left(x_1\right)$ is analytical approximate solution (4), ${\Delta }_4$ is the error estimation obtained according to [28], and ${\Delta }_5$ is the error estimation according to this article; mind that a priori estimations from this paper and those obtained in [28] have the same order of magnitude.

Example 2.

Let us consider the Cauchy problem (1) and (2), with conditions

$$r\left(x\right)=x^5;y\left(0\right)=0.2;y^{\prime }\left(0\right)=-0.3;y^{\prime \prime }\left(0\right)=0.95;{\tilde{x}}^{*}=4.2135;\Delta {\tilde{x}}^{*}=0.0001.$$

We consider the value $x_2=3.4$, the scope defined using the formula:

$$F=F_1\cap F_2\cap F_3,$$

for which

$${\rho }_2=0.891.$$

In this case, the value of $x_2=3.4$ is beyond the scope that is defined as relevant in [28].

The values presented in

Table 2. Characteristics of the approximate solution.

${\mathit{x}}_2$	${\tilde{\mathit{y}}}_9\left({\mathit{x}}_2\right)$	${\mathbf{\Delta }}_6$	${\mathbf{\Delta }}_7$
$3.4$	$109.2247$	$0.05$	$0.003$

are as follows: ${\tilde{y}}_9\left(x_2\right)$ is the approximate solution (4), ${\Delta }_6$ is the error, according to the present paper, and ${\Delta }_7$ is the a posteriori error of solution (4). For the option ${\Delta }_7=0.003$, it follows from the theorem, proved in this paper, that $\Im =13.$ The summands from 10 to 13 do not affect the accuracy of calculations $\epsilon =0.003.$ This means that, at $\Im =9$, the obtained value of ${\tilde{y}}_9\left(x_2\right)$ has an accuracy of $\epsilon =0.003$.

The graphical interpretation of the analytical approximate solution in the neighborhood of the perturbed value of the moving singular point is illustrated by Figure 1 using the data provided to the authors. Calculations were conducted using the Maple mathematical package.

4. Conclusions

This paper solves the problem of extending the boundaries of the scope of application of the analytical approximate solution in the neighborhood of the perturbed value of a moving singular point. The studies presented in this work complement the results of [28] for one class of nonlinear differential equations with moving singular points. A new variant of the a priori estimation is obtained. The proof of the a priori estimation is based on the application of elements of differential calculus. Theoretical results are numerically and graphically confirmed by numerical experiments. It is noteworthy that the results presented by the authors agree well with the data of the numerical experiment described in [28].