Abstract
The nonlinear fractional-order cubic-quintic-heptic Duffing problem will be solved through a new numerical approximation technique. The suggested method is based on the Pell-Lucas polynomials’ operational matrix in the fractional and integer orders. The studied problem will be transformed into a nonlinear system of algebraic equations. The numerical expansion containing unknown coefficients will be obtained numerically via applying Newton’s iteration method to the claimed system. Convergence analysis and error estimates for the introduced process will be discussed. Numerical applications will be given to illustrate the applicability and accuracy of the proposed method.
1 Introduction
Mathematicians, physicists, and engineers have undertaken a multidisciplinary endeavor to obtain a new instrument for describing many complex problems [1–3]. One of the most important outcomes of these efforts is the fractional calculus field [4]. This branch of science enables researchers to create a flow of ideas for solving and describing many real-world problems [5,6]. Therefore, there are accurate descriptions for a lot of application problems in terms of fractional-order differential equations in a wide range of fields, such as physics [7], biology [8], mechanics [9], medical [10], astrophysics [11], engineering [12], and chemistry [13].
Other than modeling fractional-order differential equations, solutions to these models can be considered one of the important aspects as well. Several numerical techniques are used for solutions for fractional differential equations such as wavelet method [14], a domain decomposition technique [15], spectral Legendre method [16], Laplace transform approach [17], Chebyshev collocation [18], Tau procedure [19], variational iteration method and differential transformation technique [20], Homotopy analysis approach [21], operational matrix approach [22], finite difference method [23], nonstandard finite difference [24], and other techniques [25–29]. The major characteristic of using the spectral Tau method and operational matrix method is that it reduces the fractional-order problems to a system of algebraic equations. Moreover, the advantage of using operational matrices and the Tau method is their simple procedure, rapid convergence, and easy computation.
As we know, the nonlinear differential equations appear to detail many physical phenomena located around us [21]. One of these equations is the Duffing equation, whose general form is given by
subject to the conditions
Equation (1) is called cubic, if
As a result of the importance and appearance of the Duffing equations in many applications, the researchers have studied the fractional form of this equation [37], but there are not many other works on this topic. Therefore, this article will introduce a numerical treatment for the general formula of the nonlinear Duffing equation (cubic-quintic-heptic equation). Consider the following formula for this equation:
subject to the conditions
where
This work makes three main significant contributions: first, it introduces a new method for numerically solving a nonlinear fractional-order Duffing equation of various orders using operational matrices of fractional-order derivatives of Pell-Lucas polynomials. Second, it presents an algorithm that combines the use of the spectral and Tau methods to solve a fractional-order cubic-quintic-heptic Duffing problem. Third, pay close attention to the convergence analysis that results from the suggested Pell-Lucas expansion. Finally, it illustrates that a variety of fractional-order differential equation issues can be solved using the created operational matrix and methodology.
The organization of this work is as follows: In Section 2, briefly, some tools of the fractional calculus, in addition to definitions and mathematical formulae of Pell-Lucas polynomials, are presented. In Section 3, integer and fractional-order operational matrices in terms of Pell-Lucas polynomials are constructed. In Section 4, expression of the problem in terms of derived matrices and claimed numerical solution is given. In Section 5, a global error estimate and convergence analysis for the suggested Pell-Lucas expansion will be derived. Section 6 demonstrates the accuracy and efficiency of the proposed method by introducing some test examples. Section 7 gives the concluding remarks.
2 Preliminaries and principal formulae
Some principles of fractional calculus theory are presented in this section and will be helpful throughout the rest of the article. Additionally, a description of Pell-Lucas polynomials is provided, along with a few new formulas related to them.
2.1 Fractional and integral operators
Definition 2.1
[4] The fractional-order integral operator
The Riemann-Liouville integration operator meets the following criteria:
where
Definition 2.2
[4] The fractional-order derivative of order
Definition 2.3
[26] Consider the function
This is known as the Caputo differential operator, where
The following relations are satisfied by the operator
where the ceiling notation is
2.2 Pell-Lucas polynomial overview
Generalized Lucas polynomials have also been widely studied and used in various areas of mathematics, such as the study of Diophantine equations, number theory, and the solution of fractional- and integer-order differential equations [38–42]. Pell-Lucas polynomials are a specific case of generalized Lucas polynomials [43]. However, the properties, applications, and uses of both types of polynomials are sometimes different depending on the properties and the application area because the properties of Pell-Lucas polynomials are relatively simple and well-known, whereas the properties of generalized Lucas polynomials are more complex [43].
Definition 2.4
The following power expression defines Pell-Lucas polynomials of degree
where
Also, Pell-Lucas polynomials,
with the starting functions
Additionally, the polynomials defined by
The generating functions for
Theorem 1
The power function
where
Theorem 2
The following relation can be employed to illustrate how the original functions of Pell-Lucas polynomials and their first derivative are related:
Presently, the analytical description of Pell-Lucas polynomials that were stated in equation (9) can be rephrased as the following congruent formula:
where
Also, equation (13) is equivalent to
The two latter equations will be applied within some of the suggestion theorems in this article. For more details and knowledge about Pell-Lucas polynomials and their associated characteristics, see [44,45,46].
3 Operational matrices of derivatives for Pell-Lucas polynomials
In this section, we look into the operational matrices of Pell-Lucas polynomials for both the integer and fractional orders of derivatives.
3.1 Integer-order operational matrix of derivatives for Pell-Lucas polynomials
Consider a square Lebesgue function
As a result of the approximation theory, we are able to truncate all terms except for the first (
where
and
Let the first derivative of the vector
where
For example, for
Hence, with the aid of Theorem 2, together with the two equations (23) and (24), we can produce an integer-order operational matrix of derivatives in the generalized description for Pell-Lucas polynomials as follows:
where
3.2 Fractional-order operational matrix of derivatives for Pell-Lucas polynomials
Theorem 3
Assume
where
The elements entries
where
Proof
Caputo operator
We can accomplish the following if we continue with the explanation in equation (30) and carry out some extensive algebraic calculations:
and
Equation (31) can be alternatively rewritten as the equivalence vector formula
Moreover, we can write
The intended outcome is reached by assembling equations (32) and (33).□
4 Numerical treatment of fractional-order Duffing equation
Consider equation (3), then using equation (20) in addition to equation (25) and Theorem 3, we obtain the following matrix form:
The residual of equation (34) can be computed through the following formula:
By means of Tau method implementation (see, e.g., [26]) we have
Additionally, the approximate solution in matrix form (20) and the integer-order matrix of derivatives for Pell-Lucas polynomials (23) are applied on the initial conditions that are given in equation (4) to have the next description
Using equations (34) and (37), a system of nonlinear algebraic equations are created to represent the unknown expansion coefficients
Pell-Lucas Tau algorithm for equation (3)
Step 1. Provided
Step 2. Determine
Step 3. Evaluate
Step 4. Calculate the results of equation (36).
Step 5. Join (Output 4, equation (37)).
Step 6. Solve numerically (Results of 5).
5 Convergence and error estimate discussion
This section will discuss the error estimate and convergence analysis of the proposed methodology. Following is an introduction to certain lemmas that are regarded essential for achieving this goal in the sequel:
Lemma 5.1
Assume at the point
Proof
In the beginning, according to Taylor series expansion, for any infinitely differential function
Inserting equation (18) in equation (39), we have
where
After expanding equation (40), combine the identical terms on its right-hand side. From there, the following formula can be produced:
hence Lemma 5.1 is proved.□
Lemma 5.2
[47] By using the well-known modified first kind Bessel function of order
Lemma 5.3
[48]
Lemma 5.4
Pell-Lucas polynomials satisfy the following property:
Proof
We claim to prove through induction on order
Using equation (10), we have
since
therefore,
Theorem 4
Let the function
Moreover, absolute convergence of the series
Proof
Lemma 5.1 in addition to the hypothesis of Theorem 4 enable us to write
In virtue of Lemma 5.2, we have
Through using Lemma 5.3, we have
Hence, part one of the proof for Theorem (4) is completed.
Second, to prove
Lemma 5.4 is applied here to gain
since
Consequently, the proof of Theorem 4 is complete.
Theorem 5
Let
where
Proof
Theorem 4 implies that
where
where the symbol
Thus, the proof of Theorem 5 is completed.□
6 Numerical applications
In this section, we used the Pell-Lucas Tau operational matrix method to numerically solve the fractional-order nonlinear Duffing problem. These numerical tests are provided to demonstrate the precision, applicability, and effectiveness of the suggested method as well as to validate the theoretical findings.
Example 6.1
Consider the nonlinear cubic-quintic-heptic Duffing equation of the fractional-order as follows:
subject to the conditions
where
Case one: Consider the fractional orders as integer numbers
Table 1 lists the numerical results that were obtained using the Pell-Lucas Tau spectral method for three distinct types of the integer-order Duffing equation (IDE). This table reports the absolute error of 6.1 case one for the three types of Duffing equations
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Case two: Consider several cases of the fractional-orders
The numerical results using the suggested technique for Case two are plotted in Figure 3. This figure presents the numerical solution for the several choices of the fractional-order parameters. The plotted figure indicates that the numerical solutions for distinct values in the fractional-order case have behavior similar to that in the integer-order case. Also, these results supported the accuracy and applicability of our proposed technique for solving various linear and nonlinear fractional-order differential equations.
The numerical results using the suggested technique for Case two are plotted in Figure 3. This figure presents the numerical solution for the various selections of the fractional-order parameters
Example 6.2
Consider the following nonlinear fractional-order cubic Duffing equation:
subject to the conditions
where
The approximate solutions for Example 6.2 using the introduced technique are reported in Table 2. This table lists these results for multi-choice terms for the power series approximation via Pell-Lucas polynomials
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Example 6.3
Consider the following nonlinear quintic Duffing fractional-order differential equation:
subject to the conditions
where
We implement the presented method for different values of
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7 Conclusion
In this article, a nonlinear cubic-quintic-heptic Duffing equation of the fractional-order is solved numerically via a systematic technique. The method under investigation is based on developing new operational matrices of the integer/fractional-order derivatives of Pell-Lucas polynomials in conjunction with the use of the appropriate spectral Tau method. The fractional-order is described by the Caputo sense operator. The convergence and error estimates are examined using the new suggested technique. We solved the examples via the proposed technique with multiple possibilities for the fractional parameters
Acknowledgments
The author appreciates the anonymous reviewers’ attentive reading of the work and their insightful remarks, which helped in improving the article to its form.
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Funding information: Not applicable.
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Author contributions: The study was done by the author. Also, he has agreed to take full responsibility for this manuscript’s content and has given his approval for its submission.
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Conflict of interest: The author states no conflict of interest.
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Ethical approval: The research being done has nothing to do with using humans or animals.
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Informed consent: Not applicable.
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Data availability statement: This article contains all the necessary information and materials.
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