Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu-Caputo model of the Lienard's equation

Abstract. In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard's model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana-Baleanu-Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on Fourier approximation theorem to generate required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of non fractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.


Preface and Presentation
Mathematical simulation of nonlinear physical and abstract models is a very vital process for predicting the solution behavior of fractional differential equations (FDEs) corresponding to different applications in science and engineering. Whilst, computational models of physical systems are typically triggered through computer simulations, symbolic programming, and approximate numerical algorithms for emerging nonlinear problems, which often include a set of corresponding FDEs. Solutions of such nonlinear FDEs issues are great importance for describing the dynamic and asymptotic behavior of materials in many applications such as electric circuit, fluid mechanics, viscoelastic damping, electromagnetic, and electrochemistry [1][2][3][4][5]. In this light, there exists no classic, precise method that yields an analytical solution within a closed-form in terms of spatial and temporal parameters to deal with these types of nonlinear evolution models. Therefore, there is an urgent need for effective and sophisticated methods and algorithms for exploring analytical solutions of such challenges.
In the current work, the application of the piecewise optimal fractional reproducing kernel method (FRKM) is extended to provide a convenient methodology to derive approximate numerical solution for model of the fractional Lienard's equation (FLE) arising in oscillating circuits with ABC derivative. To achieve our goal, we consider the subordinate model: (1) To assign more on model of the FLE: indicates the damping force, denotes the restoring force, and stands for the external force. In the development of radio and vacuum tube technology, FLE is extremely investigated to describe the oscillating circuits. In reality, the FLE contains the damped pendulum or a damped spring-mass system as a particular case. It is also very useful as nonlinear models of various scientific fields while considering various selection of , , and . For paradigm, the choice , , and progresses to Eq. (1) to the fractional Van der Pol equation used as a nonlinear model of electronic oscillation. More details about elaborated investigations of FLE model with its special properties one can refer to [27][28][29][30][31].
Extremely, we will symbolize the following: ; ; ; ; and . Exceedingly, we will write to sign the ABC fractional derivative of in over of order and is realized as (2) in which is a base point acquaint at and . Whereas, with and is the Mittag-Leffler function and is the Sobolev space of order on the domain and is realized as The standard RKM main area topic is in mathematical modelling and numerical simulation of multidimensional issues arising in engineering and physical sciences [32][33][34]. Later, the FRKM has been used in generating numerical solutions for strongly nonlinear integral-differential operators in the form of a rapidly convergent series with a minimum size of calculations without any restrictive hypotheses. So, this adaptive has been used as an alternative technique in solving several nonlinear and discontinus shapes problems arising in engineering and physics as utilized in [35][36][37][38][39][40].
In addition to the preface and presentation cutter, the residual synopsis of the article is structured in the following subsidiary sections, simultaneously. Preliminaries and assumptions in the FRKM: requirements, principles, and tools. Impersonation solution of FLE in the ABC sense: formulation and solutions. Convergence and error conductance: behavior and characterization theorems. Numerical applications and computational results: FRKM steps, test applications, and discussions. Finally, Summary and outline are utilized on record.

Preliminaries and Assumptions in the FRKM
This section is dedicated to briefly present some definitions and properties of the FRKM and its corresponding hypothesis. At all events the FRKM is an effective approach for solving wide class of differential-integral operators in the fractional emotion and provided a general numeric scheme to handle the solution behaviors. Extremely, we will symbolize to denotes the set of absolutely continuous functions on . Let be a Hilbert space of functions defined on . A function is a reproducing kernel of if it fulfills the subsequent requirments: Definition 1 [18] The Hilbert space and its functional structure are arranged as (5) Definition 2 [18] The Hilbert space and its functional structure are arranged as (6) Theorem 1 [18] The Hilbert space is a complete reproducing kernel with kernel function (7) Theorem 2 [18] The Hilbert space is a complete reproducing kernel with kernel function (8) In applicability of the FRKM, we divide the compact set into uniform splits encoded by . This assumed that the gained set will be dense in . Anyhow, we attempt to cover the set as well as the numerical process should end up in finite phases.
For independency, if is fixed according to ; takeover as to , then for . De facto, this showed that is linearly independent . To generalize more, this will generate the subordinate result: the system is linearly independent in .
3 Impersonation Solution of FLE in the ABC Sense The main tools in the FRKM are follows: building appropriate Hilbert spaces, generating corresponding kernel functions, defining appropriate differential linear operator, fitting orthonormal function systems, collecting symbolic computations and data sets, and software mathematical package solver. Hither, the FRKM is developing to construct highly efficient numerical solutions for the FLE that arise in the engineering and physical sciences and combine the implementations in a user-friendly framework.
Hither, based on applied the auxiliary parametrized technique and to fixed the suggested solution in ; the subsequent replacement ) will convert FLE of Eq. (1) into homogenous one. Anyhow, we still signify the conversion solution by as (9) Essentially, define the linear operator and its map as Based on this, if , then Eq. (9) can be converted into the form of (11) Following, we will organize and construct system of orthogonal functions using the subsequent steps: put and , where is the adjoint operator of , and is dense on .
Generally, the Gram-Schmidt orthogonalization process used to generating the systems of orthonormal functions on . Aught, if one refer to the orthogonalization coefficients by in the indexes , and , then Proof. The convergent in the emotion of will gives (15) This due to the fact that and is the Fourier series amplification around . Ultimately, for numerical computations, we truncated the series in Eq. (13) and creating theterm numerical solution of . ■

Convergence and Error Conductance
This section is devoted to descripe the convergence of the FRKM for solving FLE in the ABC sense beneath the compatibility conditions of the existence, uniqueness, and adequate differentiability in the Hilbert domain . To achieve our goal, we undertake is bounded whenever and is dense on . To ensure that the error will decreasing for sufficiently large ; its apparent that (16) Consequently, is decreasing in . Using the convergent fact on yields that whenever as long as and are extracted from Eqs. (13) and (14), simultaneously. Visibly, this will generate the subordinate result; the sequence of error is decreasing in and whenever .
Proof. Hither, we will consider exclusively. Perception that and integrated it from to , the result is (17) Integrate again from to , yields that (18) Taking the ususl metric functions length yields that (19) After rearranging the mathematical formulation, one find (20) Using H l er' i equ lity Eq 5), we can note the dependency Proof. Firstly, we will prove that , . Since, we can note that (24) where in between i and It pursue that whenever . By means of it implies the demanded. ■ Therafter, we will symbolize . In fact, this allows one to put as (25) Theorem 5 In the iterative formula of Eqs. (25), one has whenever .
Proof. From Eq. (25), we deduce that . From orthogonality of one acquired (26) This gives and such that , which glimpse that . In order to (27) it succeed for that (28) whereas, . So, whenever , one has and by the completeness such that as in the emotion of . ■ In analogous route, the modality form is . By the density condition, ; such that whenever or . Let one get . Also, since , then satisfies Eq. (11). Eventually, the uniqueness of solution on Eq. (11) utilizes the required score.

Numerical Applications and Computational Results
This section utilizes some numerical applications with different fractional order parameters for model of FLE in the ABC fractional sense. The numerical results are discussed graphically and quantitatively to verify the validity and applicability of the proposed FRKM in obtaining accurate numerical solutions for those models. Mathematica 10 software package is used in all computational process.

The FRKM Steps
The attached steps focussing on the computational steps required using an approperiate software package for solving Eq. (11) using FRKM. In these Procedure and Algorithm, some subsequent inputs are needed so as to deduce the numerical reckoning of in .
Procedure 1 Steps of the orthonormal Gram-Schmidt process: Step 1: For , and , do the following: Output: The orthogonalization coefficients .
Algorithm 1 Steps of FRKM for numerical reckonings model of FLE in ABC approach.
Step I: Fixed in and do Phases 1 and 2: Phase 1: Set in the indices .
Phase 2: Set in the indices . Output: the orthogonal function system .
Step II: In the indices and do Procedure 1. Output: the orthogonalization coefficients .
Step III: Set in the indices . Output: the orthonormal function system .
Step IV: Set and in the index do Phases 1, 2, and 3: Phase 1: Set . Phase 2: Set .
Phase 3: Set . Output: The -term numerical reckoning of .

Test Applications
To fill our present results in form of realistic and tangible models; three applications for model of FLE in the ABC fractional sense containes forcing term in its nonhomogeneous part are discussed. The readers should be note thither that in the next three applications and are known and may not homogeneous.
Application 1 Essentially, let allow us behold the subsequent model of FLE in ABC approach: (33) with forcing term i and the exact solution when agree well with i ■ (34) Application 2 Now, let allow us behold the subsequent model of FLE in ABC approach: (35) with forcing term and the exact solution when agree well with ■ Application 3 Finally, let allow us behold the subsequent model of FLE in ABC approach: (37) with forcing term and the exact solution when agree well with i ■ (38) In those three applications, the researchers should note that the absence of analytic solutions does not have much effect on the gained results; because we will obtain and plot the numerical solutions at different values of .

Results and Discussions
Occupancy in the indices on in and then applying Procedure 1 and Algorithm 1.
Anyhow, we apply the FRKM to discudded the previous three aplications in which , . Following, numerical validations for different values of grid points when will be exhibited. For this purpose, Tables 1, 2, and 3 tabulates the evolution of exact solutions , numerical solutions , absolute errors , and relative errors for Applications 1, 2, and 3, simultaneously, where  . It is obvious from the Tables that, there is good harmony and agreement between the presented data whether for or . Table 4. Numerical solutions of Application 1 when using Algorithm 1. Table 5. Numerical solutions of Application 2 when using Algorithm 1. Table 6. Numerical solutions of Application 3 when using Algorithm 1.
Of the data gained, it can be observed that the columns tables almost match, similar in their digits, and in good agreement with each other, especially when considering the integer-order derivative. Indeed, the ABC fractional orders have strong effects on the model profiles, which tend to lead to unusual behaviors in the event of a significant departure from the integer value of . The surfaces plots of dynamic and geometric behaviors along the memory and heritage characteristics of the FRKM are given next. Following, the geometrical validations of the approximate solutions for different values of grid points and are exhibited as presented in Figure 1 (a,b,c) for Applications 1, 2, and 3, simultaneously. Finally, as the saying goes, it ended with a refreshing scent; the attitude evaluation of is examined. Aught, at various and when ; Figure 2 (a,b,c) gives the relevant plots data of the FRKM elements for Applications 1, 2, and 3, simultaneously. It is seen that the expansion in the quantity of hub brings about a decrease in the total errors and correspondingly an improvement in the exactness of the got solutions.

Summary and Outline
This article targets to extend the application of the FRKM to explore numerical solutions for model of FLE in the ABC fractional derivative. This target has been successfully achieved using the suggested approach that directly implemented to provide suitable approximations that easily determinable components without any need for transformation or discretization. Some efficacious experiments are presented to validate the capacity and reliability of the approach, where the FRKM results are compared with exact solutions when , and with each other for different values of fractional ABC levels, which are found in good agreement with each other even after computing a few iterations. From the numerical and graphical results, it can be concluded that the proposed method is a systematic, powerful, and suitable tool for different nonlinear models, valid for a long time with great potential in scientific applications.