Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

: The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate 𝑟 -level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter 𝑟 are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at 𝑟 = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter 𝑟 = 1 . The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.


Introduction
Fuzzy differential equations have attracted a lot of attention, because of their active role in modeling various phenomena under uncertainty that arise in applied science with many physical applications, including civil engineering, quantum field theory, population, acoustics, hydraulic, optics, and chaotic dynamical systems. The fuzziness appears as an explicit form when the physical problem is formulated in one or several uncertain parameters [1−5]. In any case, the Duffing oscillator model was first proposed while studying the motion of electronics of a dynamic system at the beginning of the last century by the German electrical engineer Georg Duffing in 1918. This model is one of the most important examples of nonlinear second-order differential equations that provide a superb model to search nonlinear oscillations, where the "Duffing" term refers to any oscillating problem that includes a cubic stiffness term. Duffing oscillator has been used to describe dwindling oscillatory motion with more complex capabilities than simple harmonic motion in the physical sense, to show the chaotic behaviors of nonlinear dynamic systems, and to display vibration jumps in the changing frequency phases of the periodically forced oscillator with nonlinear elasticity, along with many applications, including optimal control problems, robotics, electromagnetic pulses, and fuzzy modeling [6−9]. However, serious studies have been conducted to solve the Duffing equation, such as the study of a flexible pendulum motion that has a stiff spring that does not follow Hooke's law, and the study of non-harmonic external perturbations [10−12]. In most cases where the entries may be precise (crisp) or imprecise (fuzzy), the variables, parameters, or conditions were considered in crisp terms. However, in fact, some actual practices of different applications of the Duffing equation intersect with uncertainty, leading to a fuzzy Duffing equation. To beat this uncertainty environment, one may use the concept of fuzziness over coefficients, variables, and initial-boundary conditions instead of crisp ones. So, it is necessary to have some mathematical tools to understand this uncertainty [13−16]. In this regard, the term "crisp" identifies a formal logic class with indicator function, sometimes called binary-valued logic or standard logic, where the statement is either true or false but not both. While the term "fuzzy" captures the degree of membership to which something is true, it is a continuous-valued logic with a membership function. Fuzzy logic was developed based on fuzzy set theory for the need to model the type of vague or ill-defined systems that are difficult to deal with using standard binary logic.
The basic motivation of this analysis is to present and applied the residual power series (RPS) method to produce the fuzzy analytical solution for the following fuzzy Duffing equation: where : [0,1] → ℝ ℱ is a continuous fuzzy-valued function, , , and are real parameters, such that ≥ 0 is the amount of viscous damping, is the stiffness of the spring, is the amount of nonlinearity in restoring force, is the cubic stiffness term, ( ) is a continuous real-valued function represent a loading external force term, and , ∈ ℝ ℱ . If = 0, this model reduces to simple damped harmonic oscillator. For the undamped and unforced Duffing equation = = 0, the model can show chaotic dynamical behaviors. Here, ℝ ℱ refers to the set of all fuzzy numbers defined on the real line. In this light, it is assumed that fuzzy initial value problems (FIVPs) (1) and (2) have unique and sufficiently smooth solutions in the domain of interest.
Investigation about the fuzzy Duffing equation, numerically or analytically, is scarce and missing. For example, the Laplace transform decomposition method [17] has been used for solving a class of fuzzy Duffing equation. Meanwhile, the variational iteration method [18] has been applied to obtain the approximate fuzzy analytic solution for such a fuzzy Duffing equation. In 2013, the RPS method was first proposed and developed by Jordanian mathematician Abu Arqub [19] as an efficient and accurate analytical-numerical method in solving first and second FIVPs. It has been successfully used to establish reliable approximate solutions of many physical and engineering problems, including crisp initial value problems, differential algebraic equations system, singular initial value problems of nonlinear systems, and a fractional stiff system [20−23]. This approach aims to construct series solutions expansion, by minimizing the residual functions in computing the desirable unknown coefficients of these solutions, which typically produces the solutions in rapidly convergent series forms with no need linearization or any limitation on the nature of the problem and its classification [24−33].
This analysis is arranged as follows. In Section 2, some fundamental concepts, definitions, and results in fuzzy calculus theory are given. In Section 3, the formulation of a cubic fuzzy Duffing equation is presented under the concept of strongly generalized differentiability. In Section 4, the RPS algorithm is implemented to handle some illustrated problems. At the end of the article, some concluding remarks are given.

Overview of Fuzzy Calculus Theory
In this section, necessary definitions and main results concerning the theory of fuzzy calculus are presented briefly. For more details, refer to [34−39]. The set ℝ ℱ , stands to the set of fuzzy numbers are defined on ℝ, which is a convex, normal, completely supported, and upper semi-continuous variable.
The -level representation of a fuzzy number ∈ ℝ ℱ , is defined as: Let , , ∊ ℝ and ∈ ℝ, the following are some of interesting properties of on ℝ ℱ : 1.

Remark 1.
If is differentiable for any point ∈ ( , ), then we say that is differentiable on ( , ). Moreover, if is differentiable in terms of the first condition of Definition 3, where its derivative at is given by ′( ) = ( ), then we say that is (1)-differentiable on ( , ). As well, if is differentiable in terms of second condition of Definition 3, where its derivative at is given by ′( ) = ( ), then we say that; is (2)-differentiable on ( , ). However, if ( ) exists, then ( ) does not exist.  (4) Consequently, we choose the desirable type of ( , )-solution, and then we convert the FDE into a crisp system of second-order differential equations. In any case, by solving the converted system corresponding to ( , )-system and checking the validity of -level cut, the solution of IVPs (3) and (4) can be obtained. In this regard, four systems are possible of the second-order fuzzy Duffing's equation, which are: Case one: (1,1)-system as follows Case two: (1,2)-system as follows Case three: (2,1)-system as follows Case four: (2,2)-system as follows  (1) and (2). The following algorithm shows the RPS strategy, which will be presented in the next section, for solving FIVPs (3) and (4), within the -level representation, converted to a crisp system of ordinary differential equations (ODEs). Algorithm 1. To obtain the fuzzy solution ( ) for the FIVPs (3) and (4), four cases are considered according to the kinds of ( , )-differentiability as follows: (3) and (4) can be converted into the crisp system given in Equation (5). Consequently, the following actions should be taken: • A1: Solve the crisp system (5) using the procedures of the RPS algorithm. Case II: If ( ) is (1,2)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (6). Consequently, the following actions should be taken: • B1: Solve the crisp system (6) using the procedures of the RPS algorithm. Case III: If ( ) is (2,1)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (7). Consequently, the following actions should be taken: • C1: Solve the crisp system (7) using the procedures of RPS algorithm. • C3: Obtain the (2,1)-solution ( ) whose -level representation is [ ( ), ( )].
Case IV: If ( ) is (2,2)-differentiable, then FIVPs (3) and (4) can be converted into the crisp system given in Equation (8). Consequently, the following actions should be taken: • D1: Solve the crisp system (8) using the procedures of the RPS algorithm. • D3: Obtain the (2,2)-solution ( ) whose -level representation is [ ( ), The former formulation for the FIVPs (3) and (4), together with Theorems 2 and 3, exhibits how to deal with the solution of the second-order fuzzy Duffing equation, allowing the consideration of four cases. For each case, the original fuzzy Duffing equation can be switched to an equivalent crisp system of ODEs. As a result, the proposed method can be used directly to solve the crisp system obtained, without having to be formulated in an uncertain sense [13−16].

The RPS Method for fuzzy Duffing oscillator
In this section, the procedure of RPS method is presented to construct (1,1)-solution, through a PS expansion, based on its truncated residual functions. Meanwhile, the same procedure can be performed for other cases. To do that, let and are (1)-differentiable, that is, ( ) and , ( ) exists, therefore, according to the RPS approach [25−29], the solutions of the converted crisp system (5) at = 0 can be given by the following forms: Using the initial conditions (0) = = , (0) = = , and (0) = = , (0) = = as initial iterative data, the expansion (9) can be written as Consequently, the -truncated series solutions of ( ) and ( ) can be given by In order to determine the unknown coefficients and for = 1,2, … , , we define the following th -truncated residual functions: where the ∞ th -residual functions are given by where is the radius of convergence, that is, the ∞ th residual functions are infinitely differentiable functions about = 0. Moreover,  Thus, by solving , ( ) │ = 0 and , ( ) │ = 0, we observe that . Hence, the 3 rd -RPS approximations, , ( ) and , ( ), are obtained. For = 4, if we substitute the 4 th -truncated series solutions of (11) into the 4 th -residual functions , ( ) and , ( ) of Equation (12), and solve , (0) = , (0) = 0 , the 4 th coefficients and can be obtained, such that . By applying the same procedure until an arbitrary order , the desirable unknown coefficients and of the series solutions (11) can be obtained, and then the -truncated series solutions , ( ) and , ( ) of (1,1)-system are also given. On the other hand, more iterations of lead to more accurate solutions [28−30].

Numerical Applications
This section purposes to illustrate the accuracy, simplicity and applicability of the RPS approach by constructing approximation solution for the fuzzy Duffing oscillator. The RPS methodology is directly employed, without using discretization, transformation, and restrictive assumptions. The numerical computations are performed using Mathematica codes. Application 1. Consider the following nonlinear fuzzy Duffing's oscillator [17,18]: This model can be regarded as an example of a nonlinear damped driven nonharmonic oscillating problem with soft spring, which can display chaos in a second-order non-autonomous periodic system. In particular, the analytic solution of this model at = 1 is ( ) = sin( ).
According to Algorithm 1, the fuzzy Duffing oscillator (15) and (16) According to the RPSM algorithm, the solution details for each case of Equations (15) and (16) can be found in Appendix. However, to show the effectiveness of the proposed algorithm, numerical outcomes of lower and upper bounds of the fuzzy solutions of fuzzy Duffing oscillator (15) and (16) compared with the variational iteration method (VIM) [18], Maples numerical solution (MNS) [18], and the exact solutions are summarized in Table 1 Table 2 shows the absolute and relative errors of the 5 th -RPS solutions for Equations (15) and (16)  From these results, we conclude that the results are in good agreement with each other in our two cases and are closer to solutions at = 1, as values increase. In any case, to demonstrate the 5 th -order RPS solutions behavior of Equations (15) and (16), the coupled surface has been plotted in a 3-dim graph for all possible ( , )-systems of Equations (15) and (16) for each ∈ [0,1] and ∈ [0,1], as shown in Figure 1, whereas blue and yellow correspond to upper and lower bounds of the fuzzy solutions, respectively.     Table . The -level RPS solutions for system (18)

Application 2.
Consider the following nonlinear fuzzy Duffing's oscillator [18]: This model can be regarded as an example of a nonlinear, nonharmonic, damped driven oscillator system with a nonlinear elasticity, whose restoring force can be written as = − ( ) − ( ). However, this model does not have an exact analytical solution.
According to Algorithm 1, the fuzzy Duffing oscillator (21) and (22) According to the RPSM algorithm, the solution details for each case of FVIPs (21) and (22) can be found in Appendix. However, since this model does not have an exact solution, so to illustrate the efficiency and accuracy of RPS algorithm, the following residual error is defined ( ; ) = | ( ; ) + ( ; ) + ( ( ; )) |.
To demonstrate the effectiveness of the RPS method for solving Equations (21) and (22), the lower and upper bounds of 10 th -RPS approximate solutions of (1,1)-system, with their residual errors, are computed and listed in Table 5 for = 0.1 and ∈ [0,1] with step size 0.25, which can illustrate the efficiency of the proposed method. Symmetry is generally found in many mathematical works involving the fuzzy systems and fuzzy logic, if not most of them. When a fuzzy system is modeled, it naturally tends to choose symmetric features, the fuzzy membership functions that often designed regularly and symmetrically over the universal convex domain. Unfortunately, this does not always happen if features are automatically generated. So, researchers usually continue to study symmetrical fuzzy systems, using a deliberate lack of symmetry. In any case, Figure 4 shows the solution behavior oflevel 6 th -RPS approximate solutions for each possible ( , )-systems of Equations (21) and (22)

Conclusions
In this article, attractive RPS strategy has been extended and implemented to explore analyticfuzzy approximate solution of damped driven nonlinear fuzzy Duffing oscillator occurring in nonharmonic oscillating phenomena. This method was used directly, by selecting appropriate fuzzy initial guesses to obtain the approximation solution in the series formula with precisely computed constructs, without the need for unphysical restrictive assumptions, linearization, or perturbation. Graphical and numerical results have illustrated the pattern behavior at different values of andlevel. The results indicated that the approximate solutions are coinciding with each other for the selected nods and parameters. Based on the graphical results, it can be concluded that the opposite boundaries of fuzzy solutions form a somewhat symmetric triangular around the central symmetry at = 1. Meanwhile, the r-level of the symmetrical fuzzy-valued solutions are smooth in the same direction without any intersections. From our results, we can also conclude that the proposed method is a systematic and suitable scheme to address many initial value problems under uncertainty and solve them with great potential in scientific and physical applications. In particular, the lower and upper approximate solutions of the fuzzy Duffing equation (15) and (16) meet at = 1 such that ( ) = ( ) . Thus, the RPS solution ( ) = − + − + + ⋯ matches the exact solution ( ) = sin ( ) that is given in [17].
• Following the proposed method in solving the fuzzy Duffing oscillator (21) and (22)