Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (50)

Search Parameters:
Keywords = Adomian (Laplace) decomposition method

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
25 pages, 3362 KiB  
Article
The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics
by Oswaldo González-Gaxiola
AppliedMath 2025, 5(3), 98; https://doi.org/10.3390/appliedmath5030098 (registering DOI) - 1 Aug 2025
Viewed by 47
Abstract
The objective of this investigation is to obtain numerical solutions for a variety of mathematical models in a wide range of disciplines, such as chemical kinetics, neurosciences, nonlinear optics, metallurgical separation/alloying processes, and asset dynamics in mathematical finance. This research features numerical simulations [...] Read more.
The objective of this investigation is to obtain numerical solutions for a variety of mathematical models in a wide range of disciplines, such as chemical kinetics, neurosciences, nonlinear optics, metallurgical separation/alloying processes, and asset dynamics in mathematical finance. This research features numerical simulations conducted with a remarkably low error measure, providing a visual representation of the examined models in these areas. The proposed method is the double Laplace–Adomian decomposition method, which facilitates the numerical acquisition and analysis of solutions. This paper presents the first report of numerical simulations employing this innovative methodology to address these problems. The findings are expected to benefit the natural sciences, mathematical modeling, and their practical applications, representing the innovative aspect of this article. Additionally, this method can analyze many classes of partial differential equations, whether linear or nonlinear, without the need for linearization or discretization. Full article
Show Figures

Figure 1

21 pages, 661 KiB  
Article
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 238
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)
Show Figures

Figure 1

20 pages, 1115 KiB  
Article
A Novel Computational Framework for Time-Fractional Higher-Order KdV Models: CLADM-Based Solutions and Comparative Analysis
by Priti V. Tandel, Anant Patel and Trushitkumar Patel
Axioms 2025, 14(7), 511; https://doi.org/10.3390/axioms14070511 - 1 Jul 2025
Viewed by 236
Abstract
This study applies the Conformable Laplace Adomian Decomposition Method (CLADM) to solve generalized time-fractional Korteweg–de Vries (KdV) models, including seventh- and fifth-order models. CLADM combines the conformable fractional derivative and Laplace transform with the Adomian decomposition technique, offering analytic approximate solutions. Numerical and [...] Read more.
This study applies the Conformable Laplace Adomian Decomposition Method (CLADM) to solve generalized time-fractional Korteweg–de Vries (KdV) models, including seventh- and fifth-order models. CLADM combines the conformable fractional derivative and Laplace transform with the Adomian decomposition technique, offering analytic approximate solutions. Numerical and graphical results, generated using MATLAB R2020a 9.8.0.1323502, validate the method’s efficiency and precision in capturing fractional-order dynamics. Fractional parameters ϱ significantly influence wave behavior, with higher orders yielding smoother profiles and reduced oscillations. Comparative analysis confirms CLADM’s superiority over existing methods in minimizing errors. The versatility of CLADM highlights its potential for studying nonlinear wave phenomena in diverse applications. Full article
(This article belongs to the Special Issue Fractional Calculus and Applied Analysis, 2nd Edition)
Show Figures

Figure 1

20 pages, 1790 KiB  
Article
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 513
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 , 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 =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
Show Figures

Figure 1

27 pages, 1500 KiB  
Article
An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator
by Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan and Fahad Aljuaydi
Fractal Fract. 2025, 9(4), 199; https://doi.org/10.3390/fractalfract9040199 - 25 Mar 2025
Cited by 2 | Viewed by 564
Abstract
The development of numerical or analytical solutions for fractional mathematical models describing specific phenomena is an important subject in physics, mathematics, and engineering. This paper’s main objective is to investigate the approximation of the fractional order Caudrey–Dodd–Gibbon (CDG) nonlinear [...] Read more.
The development of numerical or analytical solutions for fractional mathematical models describing specific phenomena is an important subject in physics, mathematics, and engineering. This paper’s main objective is to investigate the approximation of the fractional order Caudrey–Dodd–Gibbon (CDG) nonlinear equation, which appears in the fields of laser optics and plasma physics. The physical issue is modeled using the Caputo derivative. Adomian and homotopy polynomials facilitate the handling of the nonlinear term. The main innovation in this paper is how the recurrence relation, which generates the series solutions after just a few iterations, is handled. We examined the assumed model in fractional form in order to demonstrate and verify the efficacy of the new methods. Moreover, the numerical simulation is used to show how the physical behavior of the suggested method’s solution has been represented in plots and tables for various fractional orders. We provide three problems of each equation to check the validity of the offered schemes. It is discovered that the outcomes derived are close to the accurate result of the problems illustrated. Additionally, we compare our results with the Laplace residual power series method (LRPSM), the natural transform decomposition method (NTDM), and the homotopy analysis shehu transform method (HASTM). From the comparison, our methods have been demonstrated to be more accurate than alternative approaches. The results demonstrate the significant benefit of the established methodologies in achieving both approximate and accurate solutions to the problems. The results show that the technique is extremely methodical, accurate, and very effective for examining the nature of nonlinear differential equations of arbitrary order that have arisen in related scientific fields. Full article
Show Figures

Figure 1

19 pages, 1606 KiB  
Article
Chaos in Fractional-Order Glucose–Insulin Models with Variable Derivatives: Insights from the Laplace–Adomian Decomposition Method and Generalized Euler Techniques
by Sayed Saber, Emad Solouma, Rasmiyah A. Alharb and Ahmad Alalyani
Fractal Fract. 2025, 9(3), 149; https://doi.org/10.3390/fractalfract9030149 - 27 Feb 2025
Cited by 7 | Viewed by 810
Abstract
This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate [...] Read more.
This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (λ1=0.123) in the uncontrolled system to negative values (λ1=0.045) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems. Full article
Show Figures

Figure 1

22 pages, 425 KiB  
Article
Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations
by Rodolphe Heyd
Mathematics 2024, 12(22), 3612; https://doi.org/10.3390/math12223612 - 19 Nov 2024
Cited by 1 | Viewed by 854
Abstract
A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and [...] Read more.
A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters. Full article
Show Figures

Figure 1

16 pages, 537 KiB  
Article
Generalized Laplace Transform with Adomian Decomposition Method for Solving Fractional Differential Equations Involving ψ-Caputo Derivative
by Mona Alsulami, Mariam Al-Mazmumy, Maryam Ahmed Alyami and Asrar Saleh Alsulami
Mathematics 2024, 12(22), 3499; https://doi.org/10.3390/math12223499 - 8 Nov 2024
Cited by 3 | Viewed by 1317
Abstract
In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in [...] Read more.
In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the ψ-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various ψ-fractional differential equations. Full article
(This article belongs to the Section E: Applied Mathematics)
Show Figures

Figure 1

21 pages, 665 KiB  
Article
A Note on Fractional Third-Order Partial Differential Equations and the Generalized Laplace Transform Decomposition Method
by Hassan Eltayeb and Diaa Eldin Elgezouli
Fractal Fract. 2024, 8(10), 602; https://doi.org/10.3390/fractalfract8100602 - 15 Oct 2024
Viewed by 1003
Abstract
This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling [...] Read more.
This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations. Full article
Show Figures

Figure 1

12 pages, 6478 KiB  
Article
Solving a Novel System of Time-Dependent Nuclear Reactor Equations of Fractional Order
by Doaa Filali, Mohammed Shqair, Fatemah A. Alghamdi, Sherif Ismaeel and Ahmed Hagag
Symmetry 2024, 16(7), 831; https://doi.org/10.3390/sym16070831 - 2 Jul 2024
Cited by 2 | Viewed by 1693
Abstract
Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with delayed neutrons in cylindrical reactors of fractional order. [...] Read more.
Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with delayed neutrons in cylindrical reactors of fractional order. The method of solving used integrates the technique of residual power series (RPS) with the Laplace transform (LT) method. Anomalous neutron behavior is explained by examining the non-Gaussian scenario with various fractional parameters α. The LRPSM Laplace transform and residual power series method employed in this approach eliminates the complex difficulties. This simplicity makes the method particularly coherent with different fractional calculus applications. To validate the proposed method, numerical simulations are conducted with two different initial conditions representing distinct scenarios. The obtained results are presented in suitable tables and figures. It should be emphasized that this system is solved for the first time utilizing fractional calculus techniques. The outcomes are consistent with those achieved using the Adomian decomposition method. Full article
(This article belongs to the Section Mathematics)
Show Figures

Figure 1

18 pages, 2830 KiB  
Article
Analysis of Caputo Fractional-Order Co-Infection COVID-19 and Influenza SEIR Epidemiology by Laplace Adomian Decomposition Method
by Annamalai Meenakshi, Elango Renuga, Robert Čep and Krishnasamy Karthik
Mathematics 2024, 12(12), 1876; https://doi.org/10.3390/math12121876 - 16 Jun 2024
Cited by 1 | Viewed by 1385
Abstract
Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. [...] Read more.
Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. Fractional differential equations are currently used worldwide to model physical and biological phenomena. Our comprehension of complicated phenomena is improved when fractional-order derivatives are used to model systems with memory effects and long-range interactions. Mathematical depictions of infectious disease dynamics and dissemination across communities are provided by epidemiological models, which are essential resources for understanding and controlling infectious diseases. These models support informed decision making to prevent outbreaks, evaluate intervention measures, and help researchers and policymakers understand how diseases spread. A subclass of epidemiological models called co-infection models focuses on studying the dynamics of several infectious illnesses that occur in the same population at the same time. They are especially useful in situations where people are simultaneously susceptible to or infected with several infections. Co-infection models provide information on the development of effective control techniques, the progression of disease, and the interactions between several pathogens. The qualitative study via stability analysis is discussed at equilibrium, the reproduction number R0 is computed, and the results are simulated using the Laplace Adomian Decomposition Method (LADM) for Fractional Differential Equations. We employ MATLAB R2023a for graphical presentations and numerical simulations. Full article
Show Figures

Figure 1

17 pages, 297 KiB  
Article
A Note on the Application of the Double Sumudu–Generalized Laplace Decomposition Method and 1+1- and 2+1-Dimensional Time-Fractional Boussinesq Equations
by Hassan Eltayeb and Said Mesloub
Symmetry 2024, 16(6), 665; https://doi.org/10.3390/sym16060665 - 28 May 2024
Viewed by 833
Abstract
The current paper concentrates on discovering the exact solutions of the singular time-fractional Boussinesq equation and coupled time-fractional Boussinesq equation by presenting a new technique known as the double Sumudu–generalized Laplace and Adomian decomposition method. Here, two main theorems are addressed that are [...] Read more.
The current paper concentrates on discovering the exact solutions of the singular time-fractional Boussinesq equation and coupled time-fractional Boussinesq equation by presenting a new technique known as the double Sumudu–generalized Laplace and Adomian decomposition method. Here, two main theorems are addressed that are very useful in this work. Moreover, the mentioned method is effective in solving several problems. Some examples are presented to check the precision and symmetry of the technique. The outcomes show that the proposed technique is precise and gives better solutions as compared to existing methods in the literature. Full article
(This article belongs to the Special Issue Discussion of Properties and Applications of Integral Transform)
21 pages, 786 KiB  
Article
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 1362
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
Show Figures

Figure 1

11 pages, 297 KiB  
Article
Two Different Analytical Approaches for Solving the Pantograph Delay Equation with Variable Coefficient of Exponential Order
by Reem Alrebdi and Hind K. Al-Jeaid
Axioms 2024, 13(4), 229; https://doi.org/10.3390/axioms13040229 - 30 Mar 2024
Cited by 2 | Viewed by 1350
Abstract
The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact [...] Read more.
The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients. Full article
Show Figures

Figure 1

13 pages, 1850 KiB  
Article
Investigation of a Spatio-Temporal Fractal Fractional Coupled Hirota System
by Obaid J. Algahtani
Fractal Fract. 2024, 8(3), 178; https://doi.org/10.3390/fractalfract8030178 - 21 Mar 2024
Viewed by 1473
Abstract
This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that [...] Read more.
This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that the suggested technique offers a systematic and effective method to solve complex nonlinear systems. Employing the Banach contraction theorem, it is confirmed that the LADM leads to a convergent solution. The numerical analysis of the solutions demonstrates the confinement of the carrier wave and the presence of confined wave packets. The dispersion nonlinear parameter reduction equally influences the wave amplitude and spatial width. The localized internal oscillations in the solitary waves decreased the wave collapsing effect at comparatively small dispersion. Furthermore, it is also shown that the amplitude of the solitary wave solution increases by reducing the fractal derivative. It is evident that decreasing the order α modifies the nature of the solitary wave solutions and marginally decreases the amplitude. The numerical and approximation solutions correspond effectively for specific values of time (t). However, when the fractal or fractional derivative is set to one by increasing time, the wave amplitude increases. The absolute error analysis between the obtained series solutions and the accurate solutions are also presented. Full article
Show Figures

Figure 1

Back to TopTop