Iterative Multistep Reproducing Kernel Hilbert Space Method for Solving Strongly Nonlinear Oscillators

AnewalgorithmcalledmultistepreproducingkernelHilbertspacemethodisrepresentedtosolvenonlinearoscillator’smodels.The proposedschemeisamodificationofthereproducingkernelHilbertspacemethod,whichwillincreasetheintervalsofconvergence fortheseriessolution.Thenumericalresultsdemonstratethevalidityandtheapplicabilityofthenewtechnique.Averygood agreementwasfoundbetweentheresultsobtainedusingthepresentedalgorithmandtheRunge-Kuttamethod,whichshowsthat themultistepreproducingkernelHilbertspacemethodisveryefficientandconvenientforsolvingnonlinearoscillator’smodels.


Introduction
Nonlinear oscillators have several applications in many fields of physics, engineering, and biology [1][2][3][4].In general, nonlinear oscillator's problems are sometimes too complicated to be solved exactly, so several numerical methods are proposed by many authors such as harmonic balance method, multiple scale method, Adomian decomposition method, homotopy perturbation method, homotopy analysis method, and differential transform method.The reader is kindly requested to go through [5][6][7][8][9][10][11][12][13][14][15][16][17] in order to know more details about these methods, including their history, their kinds and types, their modification for use, their scientific applications, and their characteristics.
Reproducing kernel theory has important applications in numerical analysis, differential equations, integral equations, integrodifferential equations, and probability and statistics [18][19][20].Recently, a lot of research work has been devoted to the applications of RKHS method for wide classes of stochastic and deterministic problems involving operator equations, differential equations, integral equations, and integrodifferential equations.The RKHS method was successfully used by many authors to investigate several scientific applications side by side with their theories.The reader is kindly requested to go through [21][22][23][24][25][26][27][28][29][30][31][32] in order to know more details about RKHS method, including its history, its modification for use, its scientific applications, its kernel functions, and its characteristics.
The new algorithm is a simple modification of the RKHS method, for finding approximate solutions to the linear and nonlinear oscillator's equations in large intervals.It is found that the corresponding RKHS method is valid only for short intervals, but, by using multistep RKHS method, more valid and accurate solutions over large intervals can be obtained.The new method has the following characteristics; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear case; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions will be applicable; fourth, the method does not require discretization of the variables, and it is not affected by computation round-off errors and one 2 Advances in Mathematical Physics is not faced with necessity of large computer memory and time; fifth, the proposed approach does not resort to more advanced mathematical tools; that is, the algorithm is simple to understand and implement and should be thus easily accepted in the mathematical and engineering application's fields.
This paper is comprised of four sections including the Introduction.In Section 2 we describe the multistep RKHS method.In Section 3 we present four examples to show the efficiency and simplicity of the method.The conclusions are given in Section 4.

Multistep Reproducing Kernel Hilbert Space Method
In functional analysis, the RKHS is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional.Equivalently, they are spaces that can be defined by reproducing kernels.In this section, we utilize the reproducing kernel concept to construct two reproducing kernel Hilbert spaces and to find out their representation of reproducing functions for solving second-order oscillator equation via RKHS technique.Let us consider the following second-order nonlinear oscillator equation: subject to the initial conditions () = ,   () = .It is worth mentioning that the reproducing kernel  of a Hilbert space H is unique, and the existence of  is due to the Riesz representation theorem, where  completely determines the space H.Moreover, every sequence of functions  1 ,  2 , . . .,   , . . .which converges strongly to a function  in H converges also in the pointwise sense.This convergence is uniform on every subset on  on which  → (, ) is bounded.In this occasion, these spaces have wide applications including complex analysis, harmonic analysis, quantum mechanics, statistics, and machine learning.Subsequently, the space  3  2 [, ] is constructed in which every function satisfies the initial conditions () = ,   () =  and then utilized the space  1 2 [, ].For the theoretical background of reproducing kernel Hilbert space theory and its applications, we refer the reader to [18][19][20].
where   =  −1 + (  −  −1 )(( − 1)/( − 1)) and The spaces  3 2 [, ] and  1 2 [, ] are complete Hilbert with some special properties.So, all the properties of the Hilbert space will hold.Further, these spaces possess some special and better properties which could make some problems be solved easier.For instance, many problems studied in  2 [, ] space, which is a complete Hilbert space, require large amount of integral computations and such computations may be very difficult in some cases.Thus, the numerical integrals have to be calculated at the cost of losing some accuracy.However, the properties of  3 2 [, ] and  1 2 [, ] require no more integral computation for some functions, instead of computing some values of a function at some nodes.In fact, this simplification of integral computation not only improves the computational speed, but also improves the computational accuracy.

Numerical Examples and Graphical Results
Numerical techniques are widely used by scientists and engineers to solve their problems.A major advantage for numerical techniques is that a numerical answer can be obtained even when a problem has no analytical solution.However, result from numerical analysis is an approximation, in general, which can be made as accurate as desired.Because a computer has a finite word length only a fixed number of digits are stored and used during computation.In order to demonstrate the applicability and effectiveness of the proposed algorithm, four examples will be solved numerically in this section.
In this example, we apply the proposed algorithm on the interval [0, 15] and choose to divide the interval into subintervals with time step size Δ = 1.In fact, assume that the interval [0, 15] is divided into 15 subintervals [ −1 ,   ],  = 1, 2, . . ., 15, of equal step size ℎ = 1/( − 1).Anyhow, we apply RKHS method with  = 26 in each IVP: The numerical results at some selected points in [0, 15] are given in Table 1, while, on the other aspect as well, Figure 1 shows that the results of our computations are in excellent agreement with the exact solution.It is observed that the increase in the number of node results in a reduction in the absolute error and correspondingly an improvement in the accuracy of the obtained solution.This goes in agreement with the known fact; the error is monotone decreasing, where more accurate solutions are achieved using an increase in the number of nodes.
In this example, we apply the proposed algorithm on the interval [0, 150] and choose to divide the interval [0, 150] to subintervals with time step size Δ = 1.Similarly, assume that the interval [0, 150] is divided into 150 subintervals [ −1 ,   ],  = 1, 2, . . ., 150, of equal step size ℎ = 1/( − 1).Anyhow, we apply RKHS method with  = 26 in each IVP: Figure 2(a) shows that the results of our computations are in excellent agreement with the results obtained by  the numerical solution of [11] using multistep differential transform method.In Figure 2(b) we give a comparison between the multistep RKHS method and RK method for the problem.
This procedure can be repeated till the arbitrary order coefficients of the multistep RKHS solution are obtained.Moreover, higher accuracy can be achieved by evaluating more components of the solution.
In this example, we apply the proposed algorithm on the interval [0, 60] and choose to divide the interval [0, 60] to subintervals with time step size Δ = 1.In fact, assume that the interval [0, 60] is divided into 60 subintervals [ −1 ,   ],  = 1, 2, . . ., 60, of equal step size ℎ = 1/( − 1).Anyhow, we apply RKHS method with  = 26 in each IVP: As in the last example, Figure 3(a) shows that the results of our computations are in excellent agreement with the results obtained by the numerical solution of [11] using multistep differential transform method.On the other hand, in Figure 3(b), we give a comparison between the multistep RKHS method and RK method for the problem.
In this example, we apply the proposed algorithm on the interval [0, 100] and choose to divide the interval [0, 100] to subintervals with time step size Δ = 1.Similarly, assume that the interval [0, 100] is divided into 100 subintervals [ −1 ,   ],  = 1, 2, . . ., 100, of equal step size ℎ = 1/( − 1).Anyhow, we apply RKHS method with  = 26 in each IVP: Advances in Mathematical Physics As a result, Figure 4(a) shows that the results of our computations are in excellent agreement with the results obtained by the numerical solution of [11] using multistep differential transform method.Anyhow, in Figure 4(b), we give a comparison between the multistep RKHS method and RK method for the problem.

Conclusions
In this study, a new algorithm is proposed for finding a numerical solution of linear and nonlinear oscillators, namely, multistep reproducing kernel Hilbert space method.The main characteristic feature of the multistep RKHS method is that the global approximation can be established on the whole solution domain, in contrast with other numerical methods like one step and multistep methods, and the convergence is uniform.Indeed, the present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case.On the other aspect as well, the derivatives of the approximate solutions are also uniformly convergent.Comparison results between multistep RKHS method solution and RK method are discussed; the results show that this method is accurate for solving this kind of equations.

Figure 1 :
Figure 1: (a) Plots of displacement of  versus time: solid line the multistep RKHS method and dashed line exact solution.(b) Phase plane diagram of Example 1.

Figure 2 :
Figure 2: (a) Plots of displacement of  versus time.(b) Phase plane diagram of Example 2: solid line the multistep RKHS method and dotted line RK method.

Figure 3 :
Figure 3: (a) Plots of displacement of  versus time.(b) Phase plane diagram of Example 3: solid line the multistep RKHS method and dashed line RK method.

Figure 4 :
Figure 4: (a) Plots of displacement of  versus time.(b) Phase plane diagram of Example 4: solid line the multistep RKHS method and dashed line RK method.

Table 1 :
Numerical results for Example 1.