Abstract
The Ginzburg–Landau (GL) equation and the Ginzburg–Landau couple system are important models in the study of superconductivity and superfluidity. This study describes the q-homotopy analysis transform method (q-HATM) as a powerful technique for solving nonlinear problems, which has been successfully used with a set of mathematical models in physics, engineering, and biology. We apply the q-HATM to solve the Ginzburg–Landau equation and the Ginzburg–Landau coupled system and derive analytical solutions in terms of the q-series. Also, we investigate the convergence and accuracy of the obtained solutions. Our results show that q-HATM is a reliable and promising approach for solving nonlinear differential equations and provides a valuable tool for researchers in the field of superconductivity. Several graphs have been presented for the solutions obtained utilizing different levels of the fractional-order derivative and at various points in time.
1. Preliminaries
The branch of fractional calculus deals with integrals and derivatives that have noninteger orders, and it has become a valuable mathematical tool for modelling intricate physical mechanisms. The study of fractional calculus has become increasingly popular because of its ability to accurately model complex systems that exhibit long-term memory, unusual propagation, and power-law characteristics. The ability to provide more accurate descriptions of a variety of physical phenomena, such as viscoelastic materials, fluid flow in porous media, and the movement of particles in complicated environments, is one of the main benefits of using fractional calculus [1, 2]. Ordinary and partial differential equations are commonly used to describe numerous natural phenomena and applications in the domains of physics and engineering. As a result, solving these equations can aid in the examination and comprehension of system dynamics, as well as the identification of the factors that impact them (see [3, 4]). Recently, one of the most important emerging fields is quantum calculus also known as q-calculus which is considered a mathematical framework that extends traditional calculus to the realm of quantum mechanics. Quantum calculus introduces novel concepts, such as q-derivatives and q-integrals, which exhibit noncommutative properties and capture the inherent uncertainty and discrete nature of quantum systems. It can also be merged with fractional calculus to solve fractional q-differential equations; for further details in this branch, see [5, 6].
The Ginzburg–Landau model was initially proposed by Ginzburg and Landau in the 1950s [7]. This model was awarded the 2003 Nobel Prize in physics. The focus of our research is to address the Ginzburg–Landau equation, a type of nonlinear partial differential equation (NPDE) that describes the behavior of the order parameter in a superconductor or superfluid system, [8, 9]. This equation takes the following form:subject towhere is the unknown complex function, is time, are material-dependent parameters, and is the fractional-order derivative . The equation predicts the existence of vortices in superconductors and superfluids.
The GL equation has been solved using several numerical techniques. Wang and Huang proposed an efficient difference scheme for solving a nonlinear GL equation, and they proved that the numerical solution is bounded and convergent [10]. Wang and Huang investigated one- and two-dimensional nonlinear GL equations using a split step quasi-compact finite-difference scheme [11]. He and Pan presented a linearized implicit finite-difference scheme, and they tested the stability and convergence of the solution [12]. Zeng and Li used the Fourier pseudospectral method for the solution and estimated the result error [13]. Sirisubtawee et al. found the solution for cubic and quintic GL equations using the modified Kudryashov method and the expansion method with the aid of conformable fractional derivatives [14]. Heydari et al. used the shifted Chebyshev polynomial and the collocation technique for solving the fractional GL equation, and the fractional derivative used was Atangana–Riemann Liouville [15]. Ouahid et al. used the extended subequation method and the unified solver method to obtain solitary solutions for the fractional GL equation using beta derivatives [16]. Saima Arshed et al. made a comparative study using the traveling wave and the Riccati equation with the aid of two types of fractional derivatives, beta and m-truncated fractional derivatives [17]. Shao-Wen Yao et al. used the natural decomposition method for solving the generalized fractional quintic GL equation [18].
We are also interested in examining the coupled fractional Ginzburg–Landau equations:where and are the unknown complex functions that describe the behavior of the physical system, and are parameters which connected the parts of the system, and are the second-order term spatial derivative coefficients, which describe the diffusion of the order parameters, and describe the nonlinearity of the diffusion, and are the coefficients of the nonlinear terms that couple and , which describe the effect of one function on the other, and are the coefficients of the nonlinear self-interaction terms for and , respectively, and are the coefficients of the cross-interaction terms for and , which describe the effect of one function on the self-interaction of the other, and and are the coefficients of the external forces acting on and , respectively, which represent the effect of external fields or other environmental factors on the behavior of the complex functions.
The fractional coupled GL equations attract the attention of researchers due to their importance in engineering applications. Meng Li and Chengming Huang used a midpoint difference scheme to construct numerical solutions for fractional coupled GL equations using the Riesz fractional derivative [19]. Heydari et al. used the shifted Vieta–Lucas polynomials to construct a numerical solution for the fractional coupled GL system, and they tested the convergence of the solution [20]. Zaky et al. used the Galerkin spectral scheme to find an approximate solution, and they established the error estimates of the obtained solution [21]. Our contribution to the research is the solution of the proposed equation and system using a semianalytical approach (q-HATM), which is the first time it has been addressed using this method.
The structure of this paper is organized in the following manner: In Section 2, a summary of fractional derivatives and their properties is provided. Section 3 outlines the methodology of q-HATM and investigates the convergence. A brief explanation of the steps for solving the fractional GL equation and its fractional couple system is given in Section 4. In Section 5, the results of the section four solutions are represented graphically. Lastly, Section 6 offers concluding remarks for the study.
2. Fractional Derivatives
Most of investigations in fractional calculus focus on examining the Riemann–Liouville and Caputo types of fractional derivatives, which have distinct definitions. The Caputo definition has gained more popularity in simulating actual problems because it offers two advantages. First, the derivative of a constant is equal to zero, which makes the Caputo fractional derivative bounded. Second, the initial conditions can be expressed in terms of an integer-order derivative. It is worth noting that Caputo’s definition only applies to differentiable functions, see [22–24].
In this work, we motivate to solve the time-fractional GL equation and fractional coupled GL equation using the Caputo fractional derivative definition.
Definition 1. The definition of the Riemann–Liouville fractional integral of a function with order , where , can be stated as follows:where denotes the collection of real numbers that are positive.
The fractional integral of Riemann–Liouville conforms to the subsequent property:
Definition 2. The Riemann–Liouville fractional integral of a function has two defined sides, with the left- and right-hand sides being described as follows:The characterization of the Riemann–Liouville fractional derivative of a function is described as follows: the order of the derivative is denoted by :The fractional derivative of Riemann–Liouville does not exhibit the characteristic of having a derivative of zero for a constant. However, it does fulfill the relation:where represents a constant value.
Definition 3. The Caputo fractional derivative for a function can be stated as follows:The Caputo fractional derivative satisfies the following properties:
Definition 4. The Caputo fractional derivative has the following Laplace transform rule:For further details, see [25–27].
3. Methodology
In this section, we outline the steps involved in implementing q-HATM. The procedure of a homotopy technique entails developing a link between linear and nonlinear differential equations of significance. The proposed technique is considered a combination between the Laplace transform and the homotopy analysis method (HAM) [28–31].
Considering the nonlinear, nonhomogeneous fractional differential equation,where denotes the linear operator, is the nonlinear operator, refer to the external source term, and is the Caputo fractional derivative for the unknown function . Applying Laplace transform to all terms of (12), using formula (11), we get
Equation (13), after simplification, is
The nonlinear operator according to the HAM is defined as follows:where .
We construct homotopy aswhere is the embedding parameter which falls within the range of , denotes a nonzero auxiliary or supplementary parameter, and is an auxiliary function.
Evidently, if , then , and if , then . As the value of transitions from 0 to , the solution varies continuously from the initial estimate to the actual solution .
Expanding around by employing the Taylor series,where
Assuming suitable choices of the supplementary element , the initial approximation , and the auxiliary function , the series outlined in equation (12) can be expressed in a manner such that it converges at :
We define the vector as
Taking the -th derivative of equation (16) with respect to and evaluating it at , we get the following result:where
Taking inverse Laplace of both sides of equation (21),wherewhere is the homotopy polynomial:
Hence,
Finally, the series solution is
3.1. Convergence Analysis
The utilization of the q-HATM results in a truncated power series that possesses the subsequent form:
In order to demonstrate the convergence of the solution, we will refer to the following theorem:
Theorem 5. If we approximate the solution by truncating the series up to terms, i.e., , then the maximum absolute error of this truncated series can be given as follows:
In this case, is a constant with a value between 0 and 1. It is selected in a way that is less than or equal to times , i.e.,
Proof. We have
4. Implementation
This section will provide a concise demonstration of the steps involved in utilizing q-HATM to solve the time-fractional GL equation (1) according to [32] and the time-fractional coupled GL system (2) according to [19].
4.1. Time-Fractional GL Equation
We considersubject towhereand are constants.
The exact solution at iswhere .
By taking Laplace transform of all terms of (32),
We construct homotopy as
Taking and following steps (19) to (26) to obtain the iterative scheme for the GL equation,where
We set :
We set :
We set :where
After simplification,
With the aid of Mathematica software, we can continue in the same manner, but we truncate at due to huge results; hence, the approximate series solution becomes
Table 1 represents the absolute error results from solving the time-fractional GL equation at different values of , at , and at with different values of .
Figures 1–3 represent the obtained solutions in two- and three-dimensional profiles and also the graph of the exact solution. We note that when the fractional-order derivative becomes close to 1 (integer-order derivative), the solution approaches the exact solution. Also, we note that when becomes smaller, the solitary wave decays faster. The graphs match the estimated results obtained in Table 1.
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To improve the results we obtained, we should find the interval of that makes the solution converge fast. Figure 4 represents the curves of that are plotted at various values. From the curves, we observe that when , the solution converges fast.
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Table 2 outlines the absolute error at distinct values of at the same parameters chosen in Table 1, but with , the results improve and the error decreases.
Table 3 compares the suggested method to the Galerkin finite element method published in [32] in order to increase the proposed method’s effectiveness.
From Table 3, we notice that the accuracy of the Galerkin method depends on the step size (as the step size decreases, accuracy increases), but in q-HATM, accuracy depends on the value of the optimal parameter (when the value of falls in the region of convergence, accuracy increases).
4.2. Time-Fractional GL Coupled System
Example 1. Considering the time-fractional coupled Ginzburg–Landau system,Subject to the initial conditions,The exact solution at isThe source terms and are selected to match the exact solution. The same steps should be applied that we implemented on the GL equation. By taking Laplace transform of all terms of equation (46),We construct the homotopy for the couple system:Taking , the iterative scheme for the coupled GL system iswhereSetting (for simplicity, assume , and ),whereand therefore,whereand hence,Setting ,whereand therefore,Similarly,We can continue in the same manner, but we truncate at and due to huge results; hence, the approximate solution of the couple system will be in the form:Table 4 represents the absolute error obtained for the couple system unknown functions and for different values of and the optimal parameter , and we notice that as decreases, accuracy increases. This is in line with Figure 5 ( curve) which indicates to the region of accuracy which is very close to zero.
From Table 5, we notice that the error reduced from the implicit midpoint method and q-HATM is nearly similar although the implicit midpoint method is a numerical scheme and q-HATM is considered a semianalytical method. Also, we notice that the parameter that affects the accuracy of the solution in the implicit midpoint scheme is the step size parameter, while the parameter that controls the efficiency and accuracy of the q-HATM is the optimal parameter .
Example 2. Considering the time-fractional coupled Ginzburg–Landau homogeneous system,According to the starting guess,The same steps of Example 1 should be applied. Assuming that , and ), we obtain:Similarly,We truncate at and ; hence, the approximate solution of the homogeneous couple system will be
5. Graphical Illustrations
Graphs, either in 2D or in 3D, offer a pioneering illustration of the system’s behavior being studied. They facilitate a side-by-side comparison of the precise and estimated solutions, allowing researchers to evaluate the precision of the numerical method used to produce the approximation solution. Multiple graphs were generated in this study, depending on the initial conditions applied to the system. Figure 1 represents the estimated solution of the GL equation at specific values of all parameters, and the most important parameter that affects the soliton wave is . We note that as decreased, the wave began to damp (see the 3D profile of Figure 1 at , Figure 2 at , Figure 3 at , and Figure 4 at ), and damping the wave is very clear. Damping of soliton waves may be caused by the dissipation of energy through the flow of normal fluid or the interaction of the soliton wave with defects in the medium. Figure 1(a) represents the 2D profile of the solution one time at different values of the fractional-order parameter at fixed time, and the other is the approximate solution at with various steps of time. Figure 1(b) represents the 3D profile of the approximate solution and the exact solution at . Figure 2(a) represents the 2D approximate solution at and . Figure 2(b) shows the three-dimensional representation of the estimated solution. Figures 3(a) and 3(b) show the 2D and 3D profiles of the solution at and . Figure 4 shows the approximate solution at . The previous figures clarify the occurrence of damping waves which is very necessary to ensure that the wave reaches a stable state. If damping is not taken into account, the wave may continue to oscillate indefinitely, leading to an unstable solution. Figure 6 clarifies the curve that is very important to specify the region in which the solution converges fast. In this case study, , which is clear from the error estimated in Tables 1 and 2.
Figure 7 represents the approximate and exact 2D profiles of the time-fractional nonhomogeneous couple GL system, and it is clear that the approximate solution is very close to the exact solution. Figure 8 clarifies the 3D of the exact and approximate unknown functions and . Figure 9(a) shows the approximate solution of the nonhomogeneous couple system at time for distinct values of the fractional-order parameter , Figure 9(b) shows the approximate solution for at different steps of time. Figure 5 represents the curve that clarifies the region in which the system converges; from the figure, the region is approximately in the range of [−0.2, 0].
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When solving the time-fractional homogeneous couple GL system (Example 2), there are three representations being presented for each figure. The first one shows the 2D profile of the solution at different fractional-order parameters but at fixed time. The second representation shows an approximate solution at but with different time steps, and the third one is the 3D profile that shows the behavior of the obtained waves and at . In each case, we show the interaction between the two waves and . Figure 10 has been drawn for specific values of all parameters and at . Figure 11 shows the approximate solution at . Figure 12 shows the representation at . Figure 13 clarifies the estimated solution in two and three dimensions at . The representation of the approximate solutions clarifies the occurrence of damping waves as decreases.
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6. Conclusion
In this paper, the q-homotopy analysis transform method (q-HATM) has been successfully applied to solve the Ginzburg–Landau equation and the Ginzburg–Landau couple system either homogeneous or nonhomogeneous. Analytical solutions have been obtained in terms of approximate series, and the convergence and accuracy according to the numerical results of the solutions have been investigated. The results demonstrate that q-HATM is a powerful and reliable technique for solving nonlinear differential equations and has the potential to be an important tool for researchers in applied fields. In addition, several graphs have been presented to visualize the solutions obtained at various values of the fractional-order derivative and for several steps of time.
6.1. Future Work
There are several avenues for future work following this study. One possible direction is to extend the q-HATM to more complex models in applied physics and mathematics; it can be expanded to solve boundary value problems. In addition, it may be interesting to compare the performance of the q-HATM with other numerical techniques for solving nonlinear differential equations. Another potential area for future research is to investigate the physical implications of the solutions obtained using q-HATM and how they can be applied to real-world systems, especially in biomedical fields and modelling. Overall, q-HATM is a potentially effective method for solving nonlinear problems and presents a valuable opportunity for further research in this field.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors declare that this study was realized in collaboration with equal responsibility. All the authors read and approved the final manuscript.
Acknowledgments
The authors would to like acknowledge the Deanship of Scientific Research, Taif University, for funding this work.