Exact Criteria for the Existence of a Moving Singular Point in a Complex Domain for a Nonlinear Differential Third-Degree Equation with a Polynomial Seventh-Degree Right-Hand Side

Abstract

Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity of transition to phase spaces. Necessary as well as necessary and sufficient conditions for the existence of moving singular points are obtained. The results of these studies are the basis of the algorithm for obtaining moving singular points of nonlinear differential equations in the complex domain. The results of this paper allow us to expand the classes of the studied nonlinear differential equations with moving peculiarities.

1. Introduction

Recent publications testify to the attention to nonlinear differential equations in the study of building structures [1] when the presence of moving singular points is associated with the destruction of the structure. Third-order nonlinear differential equations are closely related to wave processes in beams [2,3], but depending on the nonlinearity, the authors try to reduce the problem to a linear version. This approach is associated with the absence of moving singular points in the resulting equation, and such a replacement essentially changes the mathematical model of the considered physical process. Therefore, if we have the original nonlinear differential equation, in order to solve it, it is necessary to develop mathematical methods where classical methods can’t be applied.

It should be noted that at the moment, not a single mathematical package existing in the world allows calculations for nonlinear differential equations, both for ordinary and partial and fractional derivatives. Since nonlinear differential equations generally belong to the class of non-solvable quadratures, two options are currently possible to solve such equations. The first option is associated with the possibility of solving a nonlinear equation in quadratures, which can be done only in particular cases (e.g., [4,5,6,7]). The second option is related to the author’s developed analytical approximate solution method, in which the finding of a moving singular point is based on the exact criteria for the existence of these points. The works [8,9,10,11,12,13] develop the theory and study some classes of nonlinear differential equations with moving singularities in both real and complex domains. The results obtained are practically implemented in the developed software packages for solving these categories of equations; we present one of the variants of such a complex for classes of equations of the first and second orders [14]. Therefore, based on the previously obtained results of the studied class of nonlinear equations in the complex domain [15], software development requires exact criteria for the existence of a moving singular point for the considered class of third-degree nonlinear differential equations with a seventh-degree polynomial right-hand side.

On the basis of the previously obtained results for the analyzed equation in the complex domain, exact criteria for the existence of a moving singular point are needed for the considered class of nonlinear differential equations of the third degree with a polynomial right-hand side of the seventh degree. These criteria are necessary, as well as necessary and sufficient conditions. They fall into two categories: point and iteration ones. Point criteria are used only to confirm the existence of a moving singular point. Interval criteria are the basis of the algorithm for obtaining the moving singular points themselves.

2. Research Results

We consider a nonlinear differential equation of the following form:

$$y^{‴}\left(z\right)=\sum _{n=0}^7{a}_n\left(z\right)y^n\left(z\right),$$

(1)

which, with the help of the replacement considered in [15], can be reduced to the normal form

$$y^{‴}\left(z\right)=y^7\left(z\right)+r\left(z\right).$$

(2)

We supplement the equation with initial conditions and consider the Cauchy problem:

$$y^{‴}\left(z\right)=y^7\left(z\right)+r\left(z\right),$$

(3)

$$\left\{\begin{array}{c}y\left(z_0\right)=y_0,\hfill \\ y^{\prime }\left(z_0\right)=y_1,\hfill \\ y^{″}\left(z_0\right)=y_2.\hfill \end{array}\right.$$

(4)

In [15], the existence of a moving singular point of the Cauchy problem (3) and (4) was proved; as a result, we pass, by changing the variable:

$$y^2\left(z\right)=\frac{1}{w\left(z\right)},$$

(5)

to the inverse problem for the function $w\left(z\right)$:

$$4w^{‴}{w}^2=18w{w}^{\prime }{w}^{″}+15{w^{\prime }}^3-8w^{7/2}r\left(z\right)-8$$

(6)

$$\left\{\begin{array}{c}w\left(z_0\right)=w_0,\hfill \\ w^{\prime }\left(z_0\right)=w_1,\hfill \\ w^{″}\left(z_0\right)=w_2.\hfill \end{array}\right.$$

(7)

Since the equation is considered in the complex plane, let the function $w\left(z\right)$ be represented as follows: $w\left(z\right)=u\left(x,y\right)+i·v\left(x,y\right)$. This can be characterized by two phase spaces: ${\mathsf{\Phi }}_1\left(x,y,u\left(x,y\right)\right)$ and ${\mathsf{\Phi }}_2\left(x,y,v\left(x,y\right)\right)$.

Let us use the following terminology [10]:

Definition 1.

A line, as a set of points defined by a continuous function in an explicit, implicit, and parametric way in a Cartesian coordinate system on a plane, is called regular in the direction of the $Ox$ axis if there is no straight line parallel to the $Ox$ axis intersecting this line at more than one point.

Definition 2.

A line, as a set of points defined by a continuous function in an explicit, implicit, and parametric way in a Cartesian coordinate system on a plane, is called regular in the direction of the $Oy$ axis if there is no straight line parallel to the $Oy$ axis intersecting this line at more than one point.

Definition 3.

A line, as a set of points defined by a continuous function in an explicit, implicit, and parametric way in a Cartesian coordinate system on a plane, is called regular if it is regular in the direction of the Ox axis and the $Oy$ axis simultaneously.

Definition 4.

A line, as a set of points defined by a continuous function in an explicit, implicit, and parametric way in a Cartesian coordinate system on a plane is called irregular in the direction of the $Ox$ axis if there is no straight line parallel to the $Ox$ axis intersecting this line in fewer than two points.

Definition 5.

A line, as a set of points defined by a continuous function in an explicit, implicit, and parametric way in a Cartesian coordinate system on a plane, is called irregular in the direction of the $Oy$ axis if there is no straight line parallel to the $Oy$ axis and intersecting this line in fewer than two points.

Theorem 1 (the case for a regular line).

$z^{\ast }$ is a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4) if and only if the imaginary and real parts of the function $w\left(z\right)$, the solution to the inverse Cauchy problem (6) and (7), in some neighborhood of the domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$, satisfy the following conditions:

(1)

They are continuous functions with respect to their arguments;

(2)

They change signs when passing through the point $z^{\ast }\left(x^{\ast },y^{\ast }\right)$, moving sequentially along the correct line l in the direction of the axes Ox and Oy, according to $l:\left\{z^{\ast }\in l\subset G,l\in \left({\mathsf{\Phi }}_1\cup {\mathsf{\Phi }}_2\right)\right\}.$

Proof. Necessity. 

Let $z^{\ast }$ be a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4). Let us prove that the imaginary and real parts $w\left(z\right)$ of the function in some neighborhood of the domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ satisfy conditions 1 and 2 of theorem 1.

Based on the previously obtained results [6], the function $y\left(z\right)$ can be represented as:

$$y\left(z\right)={(z^{\ast }-z)}^{-\frac{1}{2}}\sum _{n=0}^{\infty }{C}_n{(z^{\ast }-z)}^{\frac{n}{2}}$$

(8)

where $C_0=\sqrt[6]{\frac{15}{8}},C_1=C_2=\cdots =C_6=0.$ From this, we conclude that the main part of the function $y\left(z\right)$ will look like $y\left(z\right)=O\left(\sqrt[6]{\frac{15}{8}}/\sqrt{z^{\ast }-z}\right)$. Taking into account the transition to the inverse problem for the function $w\left(z\right)$ in the domain G, we have the correct part as follows: $w\left(z\right)=o\left((z^{\ast }-z)/\sqrt[3]{\frac{15}{8}}\right)$, what is more,

$$sign\left(u\left(x,y\right)\right)=sign\left(x^{\ast }-x\right),sign\left(v\left(x,y\right)\right)=sign\left(y^{\ast }-y\right).$$

(9)

We assume that the moving singular point $z^{\ast }$ is in the first quarter of the phase spaces plane. Part of a circle $\left|z\right|=\left|z^{\ast }\right|$ in a domain G can be considered as a regular line. Moving along this circle in accordance with the line l direction, we note that for points $z\in l:argz<arg{z}^{\ast }\Rightarrow \left\{\begin{array}{c}x>x^{\ast }\hfill \\ y<y^{\ast }\hfill \end{array}\right.$ and for points $z\in l:argz>arg{z}^{\ast }\Rightarrow \left\{\begin{array}{c}x<x^{\ast }\hfill \\ y>y^{\ast }\hfill \end{array}\right.$; therefore, the imaginary and real parts of the function $w\left(z\right)$ are continuous functions with respect to their arguments, which change their sign when passing through the point $z^{\ast }$. The proof is similar if the moving singular point is in other quarters. □

Proof. Sufficiency. 

Let the imaginary and real parts of the function $w\left(z\right)$ in some sufficient neighborhood of the domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ satisfy conditions 1 and 2 of theorem 1. Let us prove that $z^{\ast }$ is a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4).

Let us consider a circle passing through a point $z^{\ast }$. Without integrity limitation, we take its part lying in the first quadrant as a regular line. From equality $\left|z\right|=\left|z^{\ast }\right|$, we can express one of the variables, y for example, through the other one, namely $y=\sqrt{x^{\ast 2}-x^2+y^{\ast 2}}.$ In this case, the functions $u\left(x,y\right)$ and $v\left(x,y\right)$ join the category of functions that depend on one variable. Since, according to the condition of the theorem, these functions are continuous in the domain under consideration and take different values at the ends of the interval $\left[x_1,x_2\right]\subset G$, according to the Bolzano–Cauchy theorem, there is a point at which these functions are simultaneously equal to zero. Then, taking into account the replacement (5), the function $w\left(z\right)$ should have the structure as follows: $w\left(z\right)=(z^{\ast }-z){\displaystyle \sum _{n=0}^{\infty }}{D}_n{(z^{\ast }-z)}^{\frac{n}{2}}.$

Taking into account the replacement (5), we obtain: $y\left(z\right)=\pm {(z^{\ast }-z)}^{-\frac{1}{2}}{\displaystyle \sum _{n=0}^{\infty }}{C}_n{(z^{\ast }-z)}^{\frac{n}{2}}$. The last equality indicates that $z^{\ast }$ is a movable singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4). □

Theorem 2 (case of irregular line).

$z^{\ast }$ is a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4) if and only if the imaginary and real parts of the function $w\left(z\right)$, as well as the solution to the inverse Cauchy problem (6) and (7), in some domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$, satisfy the following conditions:

(1)

They are continuous functions with respect to their arguments;

(2)

They change signs when passing through the point $z^{\ast }\left(x^{\ast },y^{\ast }\right)$, moving sequentially along the irregular line $l_1,l_2$ in the wrong direction of the axes Ox and Oy, according to $l_1,l_2:\left\{z^{\ast }\in l_1\subset G,z^{\ast }\in l_2\subset G,l_1\in {\mathsf{\Phi }}_1,l_2\in {\mathsf{\Phi }}_2\right\}.$

Proof. Necessity. 

Let $z^{\ast }$ be a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4). Let us prove that the imaginary and real parts of the function $w\left(z\right)$ in some neighborhood of the domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ satisfy conditions 1 and 2 of theorem 2.

Based on the previously obtained results [6], the function $y\left(z\right)$ can be represented in the way of (7), where $C_0=\sqrt[6]{\frac{15}{8}},C_1=C_2=\cdots =C_6=0.$ From this, we conclude that the main part of the function $y\left(z\right)y\left(z\right)=O\left(\sqrt[6]{\frac{15}{8}}/\sqrt{z^{\ast }-z}\right)$. Taking into account the replacement (5), for the inverse function of the Cauchy problem (6) and (7), we have: $w\left(z\right)=o\left((z^{\ast }-z)/\sqrt[3]{\frac{15}{8}}\right)$.

Let the straight line $l_1:y=const$ be an irregular line in the direction of the axis $Ox$. When moving along the indicated line $l_1$, taking into account expression (9) and Theorem 2, the function $u\left(x,const\right)\in C\left(G\right)$ changes sign when passing through the point $z^{\ast }$. Thus, we obtain the first coordinate of the moving singular point $x^{\ast }$.

Then, we take the straight line $l_2:x=x^{\ast }=const$ as an irregular line in the direction of the axis $Oy$. By analogy, we find that the condition $v\left(x^{\ast },y\right)\in C\left(G\right)$ is satisfied for the function $v\left(x^{\ast },y\right)$ and changes sign when passing through the point $z^{\ast }$. Therefore, we also determine the second coordinate of the point $z^{\ast }$, namely $y^{\ast }$. □

Proof. Sufficiency. 

Let the imaginary and real parts of the function $w\left(z\right)$ in some neighborhood of the domain G of phase spaces ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ satisfy conditions 1 and 2 of theorem 2. Let us prove that $z^{\ast }$ is a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4).

On the basis of conditions 1 and 2 of Theorem 2, we have that $z^{\ast }$ is a regular point for solving problem (6) and (7): $w\left(z^{\ast }\right)=0$. This means that the function $w\left(z\right)=u\left(x,y\right)+i·v\left(x,y\right)$ can be represented as:

$$w\left(z\right)=(z^{\ast }-z)\sum _{n=0}^{\infty }{D}_n{(z^{\ast }-z)}^{\frac{n}{2}}.$$

Taking into account the replacement $y^2\left(z\right)=\frac{1}{w\left(z\right)}$, we obtain

$$y\left(z\right)=\pm {(z^{\ast }-z)}^{-\frac{1}{2}}\sum _{n=0}^{\infty }{C}_n{(z^{\ast }-z)}^{\frac{n}{2}}.$$

Thus, we come to the conclusion that $z^{\ast }$ is a moving singular point of the solution $y\left(z\right)$ to the Cauchy problem (3) and (4). □

Theorem 3. 

(Point criterion for the existence of movable singular points.) For $z^{\ast }$ to be a movable singular point of the function $y\left(z\right)$, the solution to the Cauchy problem (3) and (4) is necessary and sufficient in order to solve the function $z\left(w\right)$, which is the inverse function of the solution to the inverse Cauchy problem (5) and (6), satisfied by the following conditions:

$$z\left(0\right)=z^{\ast },z^{\prime }\left(0\right)=-\frac{2}{\sqrt[3]{15}},z^{″}\left(0\right)=z^{‴}\left(0\right)=0.$$

(10)

Proof. Necessity. 

Let $z^{\ast }$ be a moving singular point of the function $y\left(z\right)$, a solution to the Cauchy problem (3) and (4). Let us prove that the function $z\left(w\right)$, which is the inverse function of the solution to the inverse Cauchy problem (5) and (6), satisfies the following conditions:

$$z\left(0\right)=z^{\ast },z^{\prime }\left(0\right)=-\frac{2}{\sqrt[3]{15}},z^{″}\left(0\right)=z^{‴}\left(0\right)=0.$$

Taking into account the replacement $y^2\left(z\right)=\frac{1}{w\left(z\right)}$, let us represent the function $w\left(z\right)$ as a regular series:

$$w\left(z\right)=D_0(z^{\ast }-z)+D_1{(z^{\ast }-z)}^{\frac{3}{2}}+D_2{(z^{\ast }-z)}^2+\cdots ,$$

(11)

$$D_0=\frac{1}{C_0^2},D_1=D_2=\cdots =D_6=0.$$

Taking into account that $w\left(z^{\ast }\right)=0$ and $D_0=\frac{1}{C_0^2},D_1=D_2=\cdots =D_6=0$, and based on the Lagrange theorem on series inversion [16], we obtain:

$$z^{\ast }-z=\sum _{n=0}^{\infty }{B}_n·w^{\frac{n+2}{2}}=B_{0}w+B_7{w}^{\frac{9}{2}}+\dots .$$

(12)

If $w=0$, we obtain $z\left(0\right)=z^{\ast }$. Then, differentiating (12) by w, we get:

$$z^{\prime }=-B_0-\frac{9}{2}{B}_7{w}^{\frac{7}{2}}-5B_8{w}^4-\dots$$

(13)

Based on the Lagrange theorem on series inversion and (13), we obtain $z^{\prime }\left(0\right)=-B_0=-D_0=-\frac{2}{\sqrt[3]{15}}$. Differentiating the last expression by w, we obtain:

$$z^{″}=-\frac{63}{4}{B}_7{w}^{\frac{5}{2}}-20B_8{w}^3-\dots$$

(14)

From there, we obtain $z^{″}\left(0\right)=0.$ Differentiating (14) by w, we obtain:

$$z^{‴}=-\frac{315}{8}{B}_7{w}^{\frac{3}{2}}-60B_8{w}^2-\dots$$

As a result, we get the required $z\left(0\right)=z^{\ast },z^{\prime }\left(0\right)=-\frac{2}{\sqrt[3]{15}},z^{″}\left(0\right)=z^{‴}\left(0\right)=0.$ □

Proof. Sufficiency. 

According to the condition of the theorem, the function $z\left(w\right)$, which is the inverse function of the solution to the inverse Cauchy problem (5) and (6), satisfies the following conditions: $z\left(0\right)=z^{\ast },z^{\prime }\left(0\right)=-\frac{2}{\sqrt[3]{15}},z^{″}\left(0\right)=z^{‴}\left(0\right)=0.$ Let us prove that the original function $y\left(z\right)$ has a moving singular point of algebraic type.

It follows from the theorem conditions that the function is represented by a regular series:

$$z\left(w\right)=B_0+B_{1}w+B_2{w}^2+...$$

(15)

It is also in accordance with the condition of the theorem $B_0=z^{\ast }$. By differentiating (15), we obtain:

$$z^{\prime }=B_1+2B_{2}w+3B_3{w}^2+...$$

(16)

From (13), it follows that $z^{\prime }\left(0\right)=B_1=-\frac{2}{\sqrt[3]{15}}$. Further, by differentiating (16), we have:

$$z^{″}=2B_2+6B_{3}w+12B_4{w}^2+...$$

(17)

whence it follows that $z^{″}\left(0\right)=B_2=0$.

Further, by differentiating (17), we have:

$$z^{‴}=6B_3+24B_{4}w+\dots$$

whence it follows that $z^{‴}\left(0\right)=B_3=0$.

Thus, for $z\left(w\right)$, we get the expansion:

$$z\left(w\right)=z^{\ast }+B_{1}w+{\tilde{B}}_4{w}^4+{\tilde{B}}_5{w}^5+\dots ,$$

or

$$z^{\ast }-z\left(w\right)=-B_{1}w-{\tilde{B}}_4{w}^4-{\tilde{B}}_5{w}^5-\dots .$$

(18)

Based on the series inversion theorem [10], it follows from (18):

$$w\left(z\right)=D_0(z^{\ast }-z)+D_3{(z^{\ast }-z)}^4+D_4{(z^{\ast }-z)}^5+\cdots ,,$$

where $D_0=\frac{2}{\sqrt[3]{15}}.$

By virtue of the applied replacement (5), we obtain the following representation for the function $y\left(z\right)$:

$$y\left(z\right)=\frac{1}{\sqrt{w\left(z\right)}}=\frac{1}{\sqrt{D_0}}{(z^{\ast }-z)}^{-\frac{1}{2}}+{\tilde{D}}_1{(z^{\ast }-z)}^0+{\tilde{D}}_2{(z^{\ast }-z)}^{\frac{1}{2}}+\cdots$$

□

3. Discussion

This study uses series with fractional negative degrees, which at the moment are not provided with specific terminology. The proven interval criteria are the basis for compiling an algorithm for searching for moving singular points. The difference between Theorem 1 and Theorem 2 is that the movement along the regular line, based on Theorem 1, is associated with an additional solution to the optimization problem due to the discretization of the regular line, which is absent if we apply Theorem 2, where the movement occurs along an irregular line. Thus, Theorem 2 makes it possible to find a moving singular point with the lowest number of calculations, and, accordingly, in less time. Theorem 3 is used to control the performed calculations.

The latter proves that $y\left(z\right)$, which is the solution to the Cauchy problem (3) and (4), has a moving singular point of algebraic type $z^{\ast }$. Let us consider the Cauchy problems (3) and (4), where $r\left(z\right)=0;y\left(0\right)=1/4;y^{\prime }\left(0\right)=i;y^{″}\left(0\right)=1$. In the manual version, the value of the movable singular point is calculated using MATLAB $z^{\ast }=2.652717$, which has $Im\left(z^{\ast }\right)=0.$ The calculations are confirmed using the point criterion, the theorem of the existence and uniqueness of the solution [15].

4. Conclusions

This article presents theoretical material—exact criteria for the existence of moving singular points of the considered class of nonlinear differential equations for a complex domain. In the formulations and proofs of the theorems, phase spaces, the technology of the method of regularization of moving singular points, the theory of series inversion, as well as the theorem of the existence and uniqueness of the solution of the class of nonlinear differential equations under consideration, previously proved by the authors, were considered. The obtained results are necessary for the development of a program for the search of moving singular points with any accuracy.