The Effect of Linear Operators in Periodical Solutions of Ordinary Differential Equations

Abstract

In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, a new computational method is proposed based on the symmetric operator, namely the reproducing kernel Hilbert space (RKHS) method to obtain the interval of these solutions. This operator, as a consequence of the symmetric inner product, is a symmetric operator and it will be used to show the influence on periodic solutions. The high efficiency of the proposed strategy is presented along with some illustrative examples which demonstrate their periodic interval dealing with the choice of an appropriate continuous linear operator.

1. Introduction

Ordinary differential equations (ODEs) that have periodical solutions arise in various fields of applied sciences so that such kinds of problems are widely studied within applied and pure mathematics. In particular, numerical or semi-analytical solutions for these problems have been obtained with several methods such as the FDM [1], FEM [2,3], HAM [4,5,6], ADM [7,8,9,10] and other approaches [11]. In the last few decades, considerable attention has been focused on the periodic solutions of nonlinear oscillators, which are ubiquitous in every area of science related to oscillatory phenomena, not only in the areas of mechanics and physics, but also in other disciplines involving engineering applications [12,13,14,15]. Several computational methods have been developed for determining the periodic solutions of ODEs such as boundary shape function [16], homotopy perturbation method [17] and linearized Lindstedt-Poincaré method [18]. The continuation method of topological degree has been used in [19] to investigate the existence of periodic solutions for ODEs with sublinear impulsive effects. Zu [20] investigated the existence and uniqueness of periodic solutions for the nonlinear second-order ordinary differential problem by constructing the upper and lower boundaries and using the Leray–Schauder degree theory. More results about the existence of periodic solutions for differential equations have been discussed by several authors in [21,22,23,24,25,26].

In the last decade, a new method has been proposed by Jiang et al. [27] to solve analytically linear and nonlinear problems by the reproducing kernel Hilbert space (RKHS). We recall that a Hilbert space H of functions f on a set $\mathsf{\Omega }$ is said to be an RKHS if all the evaluation functionals ${\delta }_x\left(f\right):=f\left(x\right)$, $f\in H$, for each fixed point $x\in \mathsf{\Omega }$, are continuous. Then, in view of Riesz’s theorem [28], for each $x\in \mathsf{\Omega }$ there exists a unique element $k_x\in H$, such that

$${\delta }_x\left(f\right)=\langle f,k_x\rangle ,f\in H,$$

wherein $\langle .,.\rangle$ is the inner product in H. Let us assume $k(x,y):=\langle k_x,k_y\rangle$ for $x,y\in \mathsf{\Omega }$. Clearly, $k_y\left(x\right)=k(x,y)$, where $k(x,y)$ is called the symmetric reproducing kernel of H.

In this space, each continuous linear operator can produce correspondingly unity complete orthogonality basis in which every solution of this operator has been represented by the set of complete unity orthogonal functions [27]. Some attempts have been made in [29,30] to provide physical insight into the performance of orthogonal functions. The RKHS method which is an analytical approach, is still in progress to solve various problems [31,32,33]. In particular, the high accuracy and rapid convergence are two important features that have encouraged many authors to use this method for a variety of problems [34,35,36].

The periodic problems involve complicated situations. Because on the one hand, not all these dynamics have the possibility of periodic movements, and on the other hand, even if there exist a periodic movement in these dynamic systems, not necessarily all initial points have a periodic orbit to emanate from it, since most of the initial points do not lead to periodic orbits in the state space. In practice, the periodic conditions are the basic requirements of the numerical method for determining the unknown period of the periodic problems and their periodic conditions are highly unsuitable. For most nonlinear dynamical systems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, the main aim of this paper is to introduce the RKHS method for solving a class of second-order ODEs, that, without loss of generality, have been assumed on interval $[0,\infty ]$, with the following form:

$$y{}^{″}+f\left(y\right)=0,y\left(0\right)=A,y{}^{\prime }\left(0\right)=B,$$

(1)

where $f\left(y\right)$ is a continuous function and $A,B$ are real numbers. By assuming that L is a continuous linear operator and f is a continuous non-linear operator in H, from (1) we obtain:

$$L\left[u\right]=f\left(u\right).$$

(2)

In this space, the Fourier’s multiplier $B_i$, for all $x\in \mathsf{\Omega }$, are given by some determined point-wise linear functionals [27]. If ${\mathsf{\Omega }}_D$ is an arbitrary dense subset of $\mathsf{\Omega }$ and ${\left\{{\mathsf{\Phi }}_i\right\}}_{i\in {\mathsf{\Omega }}_D}$ is a unitary complete orthogonal basis, generated by L, then each solution of (2) has the following form:

$$u\left(x\right)=\sum _{i\in {\mathsf{\Omega }}_D}{B}_i{\mathsf{\Phi }}_i\left(x\right).$$

(3)

Moreover, since every linear operator is continuous for all $x\in \mathsf{\Omega }$, there exists $C_x>0$, for all $x\in \mathsf{\Omega }$, such that:

$$|{\delta }_x\left(f\right)|\le C_x{\parallel f\parallel }_H,\forall f\in H.$$

If ${\left\{C_x\right\}}_{x\in \mathsf{\Omega }}$ is bounded by a positive real number C, then we have:

$$|{\delta }_x{\left(f\right)|\le C\parallel f\parallel }_H,\forall f\in H.$$

(4)

The condition (4) implies that each Cauchy sequence has a uniform convergence and

$${\left|\right|f\left|\right|}_{\infty }\le C{\parallel f\parallel }_H,\forall f\in H.$$

(5)

As an example of these RKHS spaces, we have the Sobolev space with suitable order on a Lipschitz domain in ${\mathbb{R}}^n$ (see, e.g., [37]).

Assuming that (5) holds true. Then, one of the most important property of RKHS is that the approximate solution (3) is equal point-wise, for all $x\in \mathsf{\Omega }$, or equivalently we have:

$$u\left(x\right)=\sum _{i\in {\mathsf{\Omega }}_D}{B}_i{\mathsf{\Phi }}_i\left(x\right),\forall x\in \mathsf{\Omega }.$$

Another advantage of this method is that there is a freedom to choose the continuous linear operator and the sample set point of the domain. In this paper, we investigate the influence of the chosen continuous linear operator used in RKHS. In the future, it would be interesting to also study the influence of the choice of a sample set point for this method.

The rest of this paper is organized as follows. The next section contains a short outlook of the proposed method. Section 3 describes some examples. The first example is a linear ODE with an initial condition in which we compare the approximate solution with the exact solution. The second and third examples are nonlinear ODEs. We obtain the numerical solution for them via the RKHS method and compared it with Runge–Kutta (RK) method. From the perspective of these results, it is obvious that these problems have a periodical solution due to the fact that the choice of a linear operator affects the solutions of these examples. Finally, the concluding remarks of this study are summarized in Section 4.

2. The Method of Solution

Let us first consider the following theorem where it is shown that Equation (1) has a periodical solution of period $\tau$.

Theorem 1.

Let f be the primitive F. Assume $a=\frac{1}{2}{B}^2+F\left(A\right)$ and $A\in [A_1,A_2]$ with $A_1,A_2$ two roots of $a-F\left(y\right)$. Furthermore, F is positive (or its sign changes on the neighborhood of both roots) and the sign of f changes on $[A_1,A_2]$. Then, Equation (1) has a unique periodic solution with period:

$$\frac{\tau }{2}=\underset{A_1}{\overset{A_2}{\int }}\frac{dy}{\sqrt{a-F\left(y\right)}}.$$

(6)

Proof. 

The proof of existence and uniqueness of the solution is given by the Picard theorem [37]. Therefore, we focus only on the periodicity of the solution.

Let us multiply both sides of (1) in $y^{\prime }$. By integrating and using the given initial conditions in (1), there immediately follows:

$$\frac{1}{2}{\left(y{}^{\prime }\right)}^2+F\left(y\right)=\frac{1}{2}{B}^2+F\left(A\right).$$

(7)

Since $F\left(y\right)$ is positive on $[A_1,A_2]$ and $y^{\prime }$ is bounded on half-infinity interval, the solution of Equation (1) is bounded. Suppose $B\ge 0$ (the proof is similar for $B\le 0$) and y increase to the greater root of $a-F\left(y\right)$, that is $A_2$. Let us call the corresponding argument as $x_2$. From Equation (7), $y^{\prime }$ is zero, and then $x_2$ is the maximum value of y. If y does not increase, it decreases and obtains the lower root of $a-F\left(y\right)$ that is $A_1$. So, we can obtain the minimum value of y and name the corresponding argument with $x_1$.

When we go from $A_2$ to $A_1$, the sign of $y^{\prime }$ does not change on $[x_2,x_1]$. The previous demonstration can be implemented sequentially in this process through its domain. There follows that the solution of Equation (1) is periodic. From Equation (7), since its sign does not change when it converts into (6), the periodicity is clearly shown. □

Moreover, if $B=0$, then A is one of the roots of Equation (1). Therefore, it is the minimum or maximum value for the solution of Equation (1). To apply the RKHS method for Equation (1), we should set to zero its initial condition. Hence, we need to define a special Hilbert space associated with a kernel that reproduces (via an inner product) each function in the space. Therefore, we introduce the reproducing kernel spaces $W_2^3[0,1]$ and ${}^0{W}_2^3[0,1]$ as follows:

$$W_2^3[0,1]=\left\{v\left(x\right)|v\left(x\right),v^{\prime }\left(x\right)and{v}^{″}\left(x\right)\mathrm{is}\mathrm{absolutely}\mathrm{continuous},v^{\left(3\right)}\left(x\right)\in L^2[0,1]\right\},$$

and

$${}^0{W}_2^3[0,1]=\left\{v\left(x\right)\in W_2^3[0,1]|v\left(0\right)=0,v^{\prime }\left(0\right)=0\right\}.$$

The inner product and norm in $W_2^3[0,1]$ and ${}^0{W}_2^3[0,1]$ are given, respectively, by:

$${\langle u,v\rangle }_{W_2^3[0,1]}=\sum _{i=0}^2{u}^{\left(i\right)}\left(0\right)v^{\left(i\right)}\left(0\right)+{\int }_0^1{u}^{\left(3\right)}\left(x\right)v^{\left(3\right)}\left(x\right)\mathrm{d}x,{\parallel v\parallel }_{W_2^3}=\sqrt{{\langle v,v\rangle }_{W_2^3}},$$

and

$${\langle u,v\rangle }_{{}^0{W}_2^3[0,1]}=u^{\left(2\right)}\left(0\right)v^{\left(2\right)}\left(0\right)+{\int }_0^1{u}^{\left(3\right)}\left(x\right)v^{\left(3\right)}\left(x\right)\mathrm{d}x,{\parallel v\parallel }_{{}^0{W}_2^3}=\sqrt{{\langle v,v\rangle }_{{}^0{W}_2^3}}.$$

The reproducing kernel functions of $W_2^3[0,1]$ and ${}^0{W}_2^3[0,1]$, are defined, respectively, as follows (see, e.g., [27]):

$$r_y\left(x\right)=\left\{\begin{array}{cc}\frac{1}{120}(x^5-5x^{4}y+10x^3{y}^2+30x^2{y}^2)+xy+1,& x<y\\ \frac{1}{120}(10x^2{y}^3+30x^2{y}^2-5x{y}^4+y^5)+xy+1,& x\le y,\end{array}\right.$$

and

$$R_y\left(x\right)=\left\{\begin{array}{cc}\frac{1}{120}\left(x^5-5x^{4}y+10x^3{y}^2+30x^2{y}^2\right)+xy,& x<y\\ \frac{1}{120}\left(10x^2{y}^3+30x^2{y}^2-5x{y}^4+y^5\right)+xy,& x\le y.\end{array}\right.$$

Now, let us show how we nullify the initial condition of Equation (1). We need to define a linear differential operator $\mathbf{L}$ on a suitable space as follows:

$$\mathbf{L}:{}^0{W}_2^3[0,1]⟶W_2^1[0,1],$$

where $\mathbf{L}y=y{}^{″}$. So that the initial condition of ODE (1) is converted into the following equivalent form:

$$\begin{array}{cc}\hfill \mathbf{L}\mathbf{z}& =\mathbf{f}(x,\mathbf{z},{\mathbf{z}}^{\prime }),\hfill \\ \hfill \mathbf{z}\left(0\right)& ={\mathbf{z}}^{\prime }\left(0\right)=0,\hfill \end{array}$$

where $y\left(x\right)=\mathbf{z}\left(x\right)+\psi \left(x\right)$ and $\psi$ is the first order minimal characteristic polynomial with unknown coefficients $\alpha =({\alpha }_{\circ },{\alpha }_1)$. One can easily show that $\mathbf{L}$ is a bounded linear operator from ${}^0{W}_2^3[0,1]$ to $W_2^1[0,1]$ (see [38]).

Suppose ${\left\{{\overline{\varphi }}_i\left(x\right)\right\}}_{i=1}^{\infty }$ is a bi-orthogonal system of the completed set ${\left\{{\varphi }_i\left(x\right)\right\}}_{i=1}^{\infty }$ in ${}^0{W}_2^3[0,1]$. Then, it follows that:

$${({\overline{\varphi }}_i\left(x\right),{\varphi }_j\left(x\right))}_3={\delta }_{i,j}=\left\{\begin{array}{cc}1\hfill & i=j,\hfill \\ 0\hfill & i\ne j,\hfill \end{array}\right.$$

where ${\varphi }_i\left(x\right)={\mathbf{L}}_y^{*}{R}_y\left(x_i\right)$, with ${\mathbf{L}}^{*}$ adjoint operator of $\mathbf{L}$, and the collocation points ${\left\{x_i\right\}}_{i=1}^{\infty }$ are dense on $\left[0,1\right]$.

Functions ${\overline{\varphi }}_i\left(x\right)$’s are called the canonical dual frame of ${\varphi }_i\left(x\right)$’s in the frame theory. Moreover, using the properties of the frame basis functions, we can extend the representation of $u\in {}^0{W}_2^3[0,1]$ with some properties of adjoint linear operator as follows:

$$u\left(x\right)=\sum _{i=1}^{\infty }{(u,{\varphi }_i)}_3{\overline{\varphi }}_i\left(x\right)=\sum _{i=1}^{\infty }{(u,{\mathbf{L}}_y^{*}{R}_y\left(x_i\right))}_3{\overline{\varphi }}_i\left(x\right)=\sum _{i=1}^{\infty }{({\mathbf{L}}_{y}u,R_y\left(x_i\right))}_1{\overline{\varphi }}_i\left(x\right)=\sum _{i=1}^{\infty }{u}^{\left(2\right)}\left(x_i\right){\overline{\varphi }}_i\left(x\right).$$

(8)

If $u\left(x\right)$ satisfies in Equation (2), then the solution of Equation (1) can be represented in the following form:

$$u\left(x\right)=\sum _{i=1}^{\infty }\mathbf{f}(x_i,u\left(x_i\right),u^{\prime }\left(x_i\right)){\overline{\varphi }}_i\left(x\right).$$

(9)

In addition, the derivatives are defined as:

$$u^{\left(k\right)}\left(x\right)=\sum _{i=1}^{\infty }\mathbf{f}(x_i,u\left(x_i\right),u^{\prime }\left(x_i\right)){\overline{\varphi }}_i^{\left(k\right)}\left(x\right),$$

(10)

for all $k=1,2$.

By using the representation (10), and by taking into account that Equation (1) has the unique solution for any arbitrary parameters, we conclude that ${\overline{\varphi }}_i^{\left(2\right)}\left(x_j\right)={\delta }_{i,j}$. Therefore, from Equation (9), the vector form of the solution at the collocation points is defined as follows:

$$\mathbf{U}=\mathbf{F}\left(\mathbf{U}\right),$$

(11)

where the arguments of

$$\mathbf{U}=\left(\begin{array}{c}{\mathbf{u}}_1\\ {\mathbf{u}}_2\\ {\mathbf{u}}_3\\ ⋮\end{array}\right)$$

are the unknown coefficientand determined in such a way that they are the solution of the following algebraic equation:

$$u\left(x\right)=\sum _{i=1}^{\infty }{\mathbf{u}}_i{\overline{\varphi }}_i\left(x\right),$$

(12)

and $\mathbf{F}$ is the infinity dimensional vector function:

$${\mathbf{F}}_i\left(\mathbf{U}\right)=\mathbf{f}(x_i,{\mathbf{a}}_{\circ }^i\mathbf{U},{\mathbf{a}}_1^i\mathbf{U},\cdots ,{\mathbf{a}}_{n-1}^i\mathbf{U}),$$

where ${\mathbf{a}}_k$’s are the infinity dimensional invertible matrix, and the $(i,j)$—th array is ${\overline{\varphi }}_i^{\left(k\right)}\left(x_j\right)$, for all $k=0,1,2$ and ${\mathbf{a}}_k^i$ is $i$—th column of ${\mathbf{a}}_k$.

In view of the relation (9) we have:

$${\mathbf{u}}_1=\mathbf{f}\left(0\right).$$

So, the iterative method to obtain the solution of (11) is obtained as:

$${\mathbf{U}}^{k+1}={\psi }_{k+1}{\pi }_{k+1}\mathbf{F}\left({\mathbf{U}}^k\right),$$

(13)

where ${\mathbf{U}}^0=(0,0,0,\cdots )$ and ${\pi }_k$ and ${\psi }_k$ are the reduction and expansion map for the $k$—first arguments on ${\mathbf{l}}^2$ [39], respectively.

In the next section, we will see the influence of the choice of linear operators in the solution of Equation (1) by using the RKHS method via some examples.

3. Numerical Examples

In this section, the RKHS method will be applied to three examples. First, we choose the linear operators and then we show the corresponding plots with the results.

Example 1.

As a first example let us consider the following second-order ODE:

$$y^{″}+y=0,s.t.y\left(0\right)=1,y^{\prime }\left(0\right)=0.$$

(14)

This differential problem fulfills the conditions of Theorem 1. Now, in order to apply the RKHS method, we solve ODE (14) by using three linear operators, $L_1\left[y\right]=y^{″}+y$, $L_2\left[y\right]=y^{″}$ and $L_3\left[y\right]=y^{″}+4y$ with $f_1\left(y\right)=0$, $f_2\left(y\right)=-y$ and $f_3\left(y\right)=3y$, respectively. The correspondent solutions are obtained in the following space:

$$\begin{array}{c}\hfill W_0^3[0,2\pi ]=\{u\in W^3[0,2\pi ]:u\left(0\right)=0=u^{\prime }\left(0\right)\},\end{array}$$

where

$$\begin{array}{c}\hfill W^3[0,2\pi ]=\{u\in C^2[0,2\pi ]:u^{\left(3\right)}\left(x\right)\in L^2[0,2\pi ]\}.\end{array}$$

Figure 1, Figure 2 and Figure 3 show the comparison between the exact and approximate solutions. These figures show that the obtained results for $L_1\left[y\right]$ can produce a better approximation of the solution and is more effective than the other two operators $L_2\left[y\right]$ and $L_3\left[y\right]$.

Example 2.

The second example is the standard nonlinear oscillation problem of Duffing:

$$y^{″}+y+ϵy^3=0,y\left(0\right)=1,y^{\prime }\left(0\right)=0,$$

(15)

where ϵ is a positive perturbation parameter.

This problem has been solved numerically with the Renormalization Group Method [40]. Moreover, Bender and Orszag [41] obtained an asymptotic solution for Equation (15) up to the third order in $ϵ$. Later, Liang and Liu produced an approximate solution by the HAM [5,17,42].

Like Example 1, in this case, the choice of linear operator will be effective for the analytical solution of this problem via RKHS. According to Theorem 1, one can easily check that also this problem has a periodical solution. Now, we compute the period of solution for Equation (15) as $a=1+\frac{ϵ}{2}$ and $\tau$. On the other hand, $-1,1$ are real roots of equation $a-y^2-\frac{ϵ}{2}{y}^4=0$. So, it follows that the solution of Equation (15) has the following periodicity:

$$\frac{\tau }{2}=\underset{-1}{\overset{1}{\int }}\frac{dy}{\sqrt{a-y^2-\frac{ϵ}{2}{y}^4}}.$$

The solution of Equation (15) has been computed by the RKHS method and 25 terms of the truncated series. Figure 4 shows the influence of the choices of the continuous linear operator by the RKHS method. For $ϵ$= 20, set $L_i\left[y\right]=y^{″}+b_{i}y,f_i\left(y\right)=-ϵy^3+(b_i-1)y$ with $b_i=-0.001,0,0.001,0.01$. In this figure, the computed results of RKHS for $L_i$ and $b_i$, $i=1,2,3,4$ (which are marked as blue curves), are compared with RK45 (the red curve).

The last example is a strongly nonlinear problem. We consider a problem that can be converted into the Casimir equation [43]. Then, we obtain the wave solution of the traveling ODE which depends on two parameters (for more details see [44]).

Example 3.

Let us consider the following initial value problem:

$$\left\{\begin{array}{c}{\displaystyle y^{″}+y-\frac{a}{y}+\frac{1}{y^3}=0}\hfill \\ y\left(0\right)=A,y^{\prime }\left(0\right)=0,\hfill \end{array}\right.$$

(16)

where $a\in \mathbb{R}$ and $A>0$ are given parameters, suitably chosen to ensure the existence and periodicity of the solutions of (16).

According to Theorem 1, we show that problem (16) has an unique and periodic solution in $[0,\infty )$ with the following period:

$$\begin{array}{c}\hfill \frac{\tau }{2}={\int }_{y_{min}(A,a)}^{y_{max}(A,a)}\frac{s}{\sqrt{1-2as+I(A,a)s^2-s^4}}ds,\end{array}$$

(17)

where the upper and lower bounds of the integral are the roots of the denominator and ${\displaystyle I(A,a)=A^2-\frac{1}{A^2}+2\frac{a}{A}}$. The approximation has been obtained by considering the first 25 terms of the truncated series and we have assumed $A=0.7$, $a=2$. In Figure 5, the blue and red curves show the approximate solutions for $L_1\left[y\right]$ and $L_2\left[y\right]$ that have been chosen similarly to the Example 1, respectively. These curves are compared with the cyan curve that refers to RK45. From the obtained results, we conclude that in the RKHS method, the choice of linear operator influences the solution of problems having periodical solutions.

4. Conclusions

In this paper we have shown that by using the RKHS method for the solution of problems having periodical solutions, the choice of the linear operator influences the solution. For this purpose, a new computational method based on the symmetric operator RKHS, which is very effective in finding periodic solutions, was discussed to obtain the distance of these solutions. The validity of the proposed numerical method is testified in some examples. Due to the high accuracy and rapid convergence of this method, a perspective for future works in this field has also been provided for periodic solutions of partial differential equations. In the future, it would be interesting to also study the influence of the choice of fractional operators for this method.