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21 pages, 661 KiB

Open AccessArticle

Semi-Analytical Solutions of the Rayleigh Oscillator Using Laplace–Adomian Decomposition and Homotopy Perturbation Methods: Insights into Symmetric and Asymmetric Dynamics

by Emad K. Jaradat, Omar Alomari, Audai A. Al-Zgool and Omar K. Jaradat

Symmetry 2025, 17(7), 1081; https://doi.org/10.3390/sym17071081 - 7 Jul 2025

Viewed by 195

Abstract

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the [...] Read more.

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica’s NDSolve function. We perform a detailed error analysis across different damping parameter values ε and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions. Full article

(This article belongs to the Section Physics)

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20 pages, 1790 KiB

Open AccessArticle

Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations

by Fang Wang, Qing Fang and Yanyan Hu

Fractal Fract. 2025, 9(4), 253; https://doi.org/10.3390/fractalfract9040253 - 16 Apr 2025

Viewed by 493

Abstract

This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series [...] Read more.

This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter ℏ, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter

ℏ = - 1

, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing. Full article

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21 pages, 5722 KiB

Open AccessArticle

Analytical Solutions of Time-Fractional Navier–Stokes Equations Employing Homotopy Perturbation–Laplace Transform Method

by Awatif Muflih Alqahtani, Hamza Mihoubi, Yacine Arioua and Brahim Bouderah

Fractal Fract. 2025, 9(1), 23; https://doi.org/10.3390/fractalfract9010023 - 31 Dec 2024

Cited by 2 | Viewed by 1085

Abstract

The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional [...] Read more.

The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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25 pages, 1009 KiB

Open AccessArticle

Solution for Time-Fractional Coupled Burgers Equations by Generalized-Laplace Transform Methods

by Hassan Eltayeb and Said Mesloub

Fractal Fract. 2024, 8(12), 692; https://doi.org/10.3390/fractalfract8120692 - 25 Nov 2024

Viewed by 783

Abstract

In this work, nonlinear time-fractional coupled Burgers equations are solved utilizing a computational method, which is called the double and triple generalized-Laplace transform and decomposition method. We discuss the proof of triple generalized-Laplace transform for a Caputo fractional derivative. We have given four [...] Read more.

In this work, nonlinear time-fractional coupled Burgers equations are solved utilizing a computational method, which is called the double and triple generalized-Laplace transform and decomposition method. We discuss the proof of triple generalized-Laplace transform for a Caputo fractional derivative. We have given four examples to show the precision and adequacy of the suggested approach. The results show that this method is easy and accurate when compared to the A domain decomposition method (ADM), homotopy perturbation method (HPM), and generalized differential transform method (GDTM). Finally, we have sketched the graphics for all these examples. Full article

(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)

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25 pages, 507 KiB

Open AccessArticle

Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations

by Musa Rahamh GadAllah and Hassan Eltayeb Gadain

Symmetry 2024, 16(9), 1232; https://doi.org/10.3390/sym16091232 - 19 Sep 2024

Cited by 2 | Viewed by 1211

Abstract

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included [...] Read more.

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. Full article

(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)

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21 pages, 786 KiB

Open AccessArticle

A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Fractal Fract. 2024, 8(5), 286; https://doi.org/10.3390/fractalfract8050286 - 10 May 2024

Cited by 7 | Viewed by 1344

Abstract

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear [...] Read more.

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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41 pages, 619 KiB

Open AccessArticle

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

by Mawia Osman, Muhammad Marwan, Syed Omar Shah, Lamia Loudahi, Mahvish Samar, Ebrima Bittaye and Altyeb Mohammed Mustafa

Fractal Fract. 2023, 7(12), 851; https://doi.org/10.3390/fractalfract7120851 - 29 Nov 2023

Cited by 2 | Viewed by 1676

Abstract

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local [...] Read more.

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives. Full article

(This article belongs to the Special Issue Application of Anomalous Diffusion Modeling Based on Fractal and Fractional Derivatives)

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17 pages, 1739 KiB

Open AccessArticle

A New Extension of Optimal Auxiliary Function Method to Fractional Non-Linear Coupled ITO System and Time Fractional Non-Linear KDV System

by Rashid Nawaz, Aaqib Iqbal, Hina Bakhtiar, Wissal Audah Alhilfi, Nicholas Fewster-Young, Ali Hasan Ali and Ana Danca Poțclean

Axioms 2023, 12(9), 881; https://doi.org/10.3390/axioms12090881 - 14 Sep 2023

Cited by 6 | Viewed by 1451

Abstract

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature [...] Read more.

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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14 pages, 345 KiB

Open AccessArticle

An Efficient Approach to Solving the Fractional SIR Epidemic Model with the Atangana–Baleanu–Caputo Fractional Operator

by Lakhdar Riabi, Mountassir Hamdi Cherif and Carlo Cattani

Fractal Fract. 2023, 7(8), 618; https://doi.org/10.3390/fractalfract7080618 - 11 Aug 2023

Cited by 4 | Viewed by 1792

Abstract

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an [...] Read more.

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

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10 pages, 807 KiB

Open AccessArticle

On the Modified Laplace Homotopy Perturbation Method for Solving Damped Modified Kawahara Equation and Its Application in a Fluid

by Noufe H. Aljahdaly and Alhanouf M. Alweldi

Symmetry 2023, 15(2), 394; https://doi.org/10.3390/sym15020394 - 2 Feb 2023

Cited by 13 | Viewed by 2056

Abstract

The manuscript solves a modified Kawahara equation (mKE) within two cases with and without a damping term by applying the Laplace homotopy perturbation method (LHPM). Since the damped mKE is non-integrable (i.e., it does not have analytic integrals) and does not have exact [...] Read more.

The manuscript solves a modified Kawahara equation (mKE) within two cases with and without a damping term by applying the Laplace homotopy perturbation method (LHPM). Since the damped mKE is non-integrable (i.e., it does not have analytic integrals) and does not have exact initial conditions, this challenge makes many numerical methods fail to solve non-integrable equations. In this article, we suggested a new modification at LHPM by setting a perturbation parameter and an embedding parameter as the damping parameter and using the initial condition for mKE as the initial condition for non-damped mKE. The results proved that this mathematical approach is an effective method for solving damped mKE. Thus, we believe that the presented method will be helpful for solving many non-integrable equations that describe phenomena in sciences, such as nonlinear symmetrical wave propagation in plasma. Full article

(This article belongs to the Special Issue On the Analytical and Numerical Methods for Modeling (A)symmetrical Nonlinear Waves and Oscillations in a Plasma)

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11 pages, 951 KiB

Open AccessArticle

A Computational Scheme for the Numerical Results of Time-Fractional Degasperis–Procesi and Camassa–Holm Models

by Muhammad Nadeem, Hossein Jafari, Ali Akgül and Manuel De la Sen

Symmetry 2022, 14(12), 2532; https://doi.org/10.3390/sym14122532 - 30 Nov 2022

Cited by 3 | Viewed by 1616

Abstract

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (

L

T) and homotopy perturbation method (HPM) to formulate the [...] Read more.

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (

L

T) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (

L

HPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed scheme. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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18 pages, 1660 KiB

Open AccessArticle

A Fractional Order Investigation of Smoking Model Using Caputo-Fabrizio Differential Operator

by Yasir Nadeem Anjam, Ramsha Shafqat, Ioannis E. Sarris, Mati ur Rahman, Sajida Touseef and Muhammad Arshad

Fractal Fract. 2022, 6(11), 623; https://doi.org/10.3390/fractalfract6110623 - 26 Oct 2022

Cited by 31 | Viewed by 2481

Abstract

Smoking is a social trend that is prevalent around the world, particularly in places of learning and at some significant events. The World Health Organization defines smoking as the most important preventable cause of disease and the third major cause of death in [...] Read more.

Smoking is a social trend that is prevalent around the world, particularly in places of learning and at some significant events. The World Health Organization defines smoking as the most important preventable cause of disease and the third major cause of death in humans. In order to analyze this matter, this study typically emphasizes analyzing the dynamics of the fractional order quitting smoking model via the Caputo-Fabrizio differential operator. For the numerical solution of the considered model, the Laplace transform with the Adomian decomposition method (LADM) and Homotopy perturbation method (HPM) is applied, and the comparison of both the achieved numerical solutions is presented. Moreover, numerical simulation for the suggested scheme has been presented in various fractional orders with the aid of Matlab and the numerical results are supported by illustrative graphics. The simulation reveals the aptness of the considered model. Full article

(This article belongs to the Special Issue Fractional Order Viral Epidemic Models and Their Applications)

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15 pages, 1216 KiB

Open AccessArticle

A Reliable Technique for Solving Fractional Partial Differential Equation

by Azzh Saad Alshehry, Rasool Shah, Nehad Ali Shah and Ioannis Dassios

Axioms 2022, 11(10), 574; https://doi.org/10.3390/axioms11100574 - 20 Oct 2022

Cited by 20 | Viewed by 3077

Abstract

The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional [...] Read more.

The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional partial differential equations, is introduced in this study. The residual-power-series method (RPSM), a well-known technique, and the Laplace transform (LT) are elegantly combined in the suggested technique. This innovative approach computes the fractional derivative in the Caputo sense. The proposed method for handling fractional partial differential equations is provided in detail, along with its implementation. The novel approach yields a series solution to fractional partial differential equations. To validate the simplicity, effectiveness, and viability of the suggested technique, the provided model is tested and simulated. A numerical and graphical description of the effects of the fractional order

γ

on approximating the solutions is provided. Comparative results show that the suggested method approximates more precisely than current methods such as the natural homotopy perturbation method. The study showed that the aforementioned method is straightforward, trustworthy, and suitable for analysing non-linear engineering and physical issues. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)

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19 pages, 6848 KiB

Open AccessArticle

New Soliton Solutions of Time-Fractional Korteweg–de Vries Systems

by Mubashir Qayyum, Efaza Ahmad, Muhammad Bilal Riaz, Jan Awrejcewicz and Syed Tauseef Saeed

Universe 2022, 8(9), 444; https://doi.org/10.3390/universe8090444 - 26 Aug 2022

Cited by 12 | Viewed by 2160

Abstract

Model construction for different physical situations, and developing their solutions, are the major characteristics of the scientific work in physics and engineering. Korteweg–de Vries (KdV) models are very important due to their ability to capture different physical situations such as thin film flows [...] Read more.

Model construction for different physical situations, and developing their solutions, are the major characteristics of the scientific work in physics and engineering. Korteweg–de Vries (KdV) models are very important due to their ability to capture different physical situations such as thin film flows and waves on shallow water surfaces. In this work, a new approach for predicting and analyzing nonlinear time-fractional coupled KdV systems is proposed based on Laplace transform and homotopy perturbation along with Caputo fractional derivatives. This algorithm provides a convergent series solution by applying simple steps through symbolic computations. The efficiency of the proposed algorithm is tested against different nonlinear time-fractional KdV systems, including dispersive long wave and generalized Hirota–Satsuma KdV systems. For validity purposes, the obtained results are compared with the existing solutions from the literature. The convergence of the proposed algorithm over the entire fractional domain is confirmed by finding solutions and errors at various values of fractional parameters. Numerical simulations clearly reassert the supremacy and capability of the proposed technique in terms of accuracy and fewer computations as compared to other available schemes. Analysis reveals that the projected scheme is reliable and hence can be utilized with other kernels in more advanced systems in physics and engineering. Full article

(This article belongs to the Special Issue Research on Optical Soliton Perturbation)

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12 pages, 1276 KiB

Open AccessArticle

A Numerical Strategy for the Approximate Solution of the Nonlinear Time-Fractional Foam Drainage Equation

by Fenglian Liu, Jinxing Liu and Muhammad Nadeem

Fractal Fract. 2022, 6(8), 452; https://doi.org/10.3390/fractalfract6080452 - 19 Aug 2022

Cited by 5 | Viewed by 1684

Abstract

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy [...] Read more.

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions. Full article

(This article belongs to the Special Issue Numerical and Analytical Methods for Differential Equations and Systems)

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