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Journal articles on the topic 'Adomian decomposition method (ADM)'

Author: Grafiati

Published: 3 June 2025

Last updated: 13 July 2025

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1

Biazar, Jafar, and Mohsen Didgar. "Numerical Solution of Riccati Equations by the Adomian and Asymptotic Decomposition Methods over Extended Domains." International Journal of Differential Equations 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/580741.

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We combine the Adomian decomposition method (ADM) and Adomian’s asymptotic decomposition method (AADM) for solving Riccati equations. We investigate the approximate global solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from Adomian’s asymptotic decomposition method for Riccati equations and in such cases when we do not find any region of overlap between the obtained approximate solutions by the two proposed methods, we connect the two approximations by the Padé approximant of the near-field approximation. We illustrate the efficiency of the technique for several specific examples of the Riccati equation for which the exact solution is known in advance.

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2

Ul Haq, Irfan. "Apply Adomian Decomposition Method (ADM) and Haar Wavelet Method (HWM) for Applications of linear Differential Equations." INTERANTIONAL JOURNAL OF SCIENTIFIC RESEARCH IN ENGINEERING AND MANAGEMENT 08, no. 12 (2024): 1–7. https://doi.org/10.55041/ijsrem39618.

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In this work, the numerical solution of the equations arising in simple harmonic motion and free oscillation is found using the Adomian decomposition method (ADM). For comparison analysis, the Haar wavelet method (HWM) is used. Some numerical examples have been performed to illustrate the accuracy of the present work. Keywords: Oscillatory motion, Adomian decomposition method (ADM), Haar wavelet method (HWM), Simple Pendulum

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3

Muhammad, Rebwar, Rostam Saeed, and Davron Juraev. "New Algorithm for Computing Adomian’s Polynomials to Solve Coupled Hirota System." Journal of Zankoy Sulaimani - Part A 24, no. 1 (2022): 38–54. http://dx.doi.org/10.17656/jzs.10868.

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In this paper, when compared to the normal Adomian decomposition approach, we updated the method of calculating Adomian's polynomial to discover the numerical solution for a non-linear coupled Hirota system (CHS) with fewer components, improved accuracy, and faster convergence (ADM). The novel algorithm offers a viable way to computing Adomian polynomials for all types of non-linearity. We can see that these two methods are both effective for solving non-linear CHS, however, the result provided by our new algorithm is superior to that obtained by the classic Adomian decomposition method. Maple 15 was utilized to do calculations in our work.

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4

Yousef, Ali Sulaiman Alsulaiman, and M. D. Al-Eybani Ahmad. "Solve the Second Order Ordinary Differential Equations by Adomian Decomposition Method." International Journal of Mathematics and Physical Sciences Research 11, no. 1 (2023): 6–9. https://doi.org/10.5281/zenodo.7907545.

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<strong>Abstract:</strong> The Adomian decomposition model (ADM) is described by Evans and Raslan (2005) as a semi-analytical model used to solve partial and ordinary non-linear differentials. ADM was designed by George Adomian between the 1970s and the 1990s (Hosseini & Nasabzadeh, 2007) whilst working at the University of Georgia’s Applied Mathematics department. <strong>Keywords:</strong> Ordinary Differential - Differential Equations - Adomian Decomposition Method. <strong>Title:</strong> Solve the Second Order Ordinary Differential Equations by Adomian Decomposition Method <strong>Author:</strong> Yousef Ali Sulaiman Alsulaiman, Ahmad M. D. Al-Eybani <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 11, Issue 1, April 2023 - September 2023</strong> <strong>Page No: 6-9</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 08-May-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7907545</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/solve-the-second-order-ordinary-differential-equations-by-adomian-decomposition-method</strong>

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Mohammed, Ali Qasem Mohsen, Abdalrahim Ahmed Dawood Arafa, and Qaid Hasan Yahya. "Solving A simple Harmonic Oscillator Equation by Adomian Decomposition Method." Solving A simple Harmonic Oscillator Equation by Adomian Decomposition Method 13, no. 1 (2025): 61–65. https://doi.org/10.5281/zenodo.15494710.

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<strong>Abstract:</strong> In this paper&sbquo; we will display Adomian decomposition method (ADM) for solving a simple harmonic oscillator equation∙ It is shown that the Adomian decomposition method (ADM) efficiency&sbquo; simple&sbquo; easy to use in solving physical equation.  The proposed method can be applied to linear problem∙ Some examples were presented to show the ability of the method for linear ordinary differential physical equations∙ <strong>Keywords:</strong> Adomian decomposition method&sbquo; harmonic oscillator schordinger equation&sbquo; physical equation. <strong>Title:</strong> Solving A simple Harmonic Oscillator Equation by Adomian Decomposition Method <strong>Author:</strong> Mohammed Ali Qasem Mohsen, Arafa Abdalrahim Ahmed Dawood, Yahya Qaid Hasan <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 13, Issue 1, April 2025 - September 2025</strong> <strong>Page No: 61-65</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 23-May-2025</strong> <strong>DOI: https://doi.org/10.5281/zenodo.15494710</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/solving-a-simple-harmonic-oscillator-equation-by-adomian-decomposition-method</strong>

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6

Mohammed, Ali Qasem Mohsen, and Abdalrahim Ahmed Dawood Arafa. "On the weight function for the Chebyshev equation of the first kind by the Adomian decomposition method." International Journal of Mathematics and Physical Sciences Research 13, no. 1 (2025): 66–69. https://doi.org/10.5281/zenodo.15494954.

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<strong>Abstract:</strong> In this paper, we study the use of the Adomian decomposition method (ADM) to solve the Chebyshev equation of the first kind. We show that the ADM can be used to obtain a series solution for the equation, and that the weight function for the series can be chosen to improve the accuracy of the solution. We studied the weight of Chebyshev differential equations using the Adomian decomposition method and show that the ADM is a more efficient and accurate method. <strong>Keywords:</strong> Adomian decomposition method, Chebyshev equation from first kind, weight function, Chebyshev polynomial. <strong>Title:</strong> On the weight function for the Chebyshev equation of the first kind by the Adomian decomposition method <strong>Author:</strong> Mohammed Ali Qasem Mohsen&sbquo; Arafa Abdalrahim Ahmed Dawood <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 13, Issue 1, April 2025 - September 2025</strong> <strong>Page No: 66-69</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 23-May-2025</strong> <strong>DOI: </strong><strong>https://doi.org/10.5281/zenodo.15494954</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/on-the-weight-function-for-the-chebyshev-equation-of-the-first-kind-by-the-adomian-decomposition-method</strong>

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7

Pandey, Kanti. "Adomian Decomposition Method- a brief introduction." Anusandhaan - Vigyaan Shodh Patrika 7, no. 01 (2019): 92–98. https://doi.org/10.22445/avsp.v7il.17.

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In present article, an attempt is made to give a brief introduction of Adomian Decomposition Method(ADM) for solving differential equations. Advantage and accuracy of the method is shown by solving some examples.

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8

Turkyilmazoglu, Mustafa. "Accelerating the convergence of Adomian decomposition method (ADM)." Journal of Computational Science 31 (February 2019): 54–59. http://dx.doi.org/10.1016/j.jocs.2018.12.014.

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9

ANSARI, ASIYA, and NAJMUDDIN AHMAD. "NUMERICAL ACCURACY OF FREDHOLM LINEAR INTEGRO-DIFFERENTIAL EQUATIONS BY USING ADOMIAN DECOMPOSITION METHOD, MODIFIED ADOMIAN DECOMPOSITION METHOD AND VARIATIONAL ITERATION METHOD." Journal of Science and Arts 23, no. 3 (2023): 625–38. http://dx.doi.org/10.46939/j.sci.arts-23.3-a05.

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In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4.

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10

Bildik, Necdet, and Mustafa Inc. "A Comparison between Adomian Decomposition and Tau Methods." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/621019.

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We present a comparison between Adomian decomposition method (ADM) and Tau method (TM) for the integro-differential equations with the initial or the boundary conditions. The problem is solved quickly, easily, and elegantly by ADM. The numerical results on the examples are shown to validate the proposed ADM as an effective numerical method to solve the integro-differential equations. The numerical results show that ADM method is very effective and convenient for solving differential equations than Tao method.

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11

Manaa, Saad A., and Nergiz M. Mosa. "Adomian Decomposition and Successive Approximation Methods for Solving Kaup-Boussinesq System." Science Journal of University of Zakho 7, no. 3 (2019): 101–7. http://dx.doi.org/10.25271/sjuoz.2019.7.3.582.

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The Kaup-Boussinesq system has been solved numerically by using two methods, Successive approximation method (SAM) and Adomian decomposition method (ADM). Comparison between the two methods has been made and both can solve this kind of problems, also both methods are accurate and has faster convergence. The comparison showed that the Adomian decomposition method much more accurate than Successive approximation method.

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12

Alomari, Saleh Ali, and Yahya Qaid Hasan. "Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems." Asian Research Journal of Mathematics 20, no. 11 (2024): 26–41. http://dx.doi.org/10.9734/arjom/2024/v20i11857.

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This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

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13

Ahmad, M. D. Al-Eybani. "A Comparative Analysis of the Differential Transform Method and the Adomian Decomposition Method for Solving the Burgers Equation." International Journal of Mathematics and Physical Sciences Research 13, no. 1 (2025): 20–25. https://doi.org/10.5281/zenodo.15303937.

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<strong>Abstract:</strong> The Burgers equation, a fundamental partial differential equation (PDE) in applied mathematics and physics, is widely studied due to its applications in fluid dynamics, gas dynamics, traffic flow, and nonlinear wave propagation. Named after Johannes Martinus Burgers, this equation combines nonlinear convection and diffusion, making it a simplified model for complex phenomena like turbulence and shock waves. The one-dimensional Burgers equation is typically expressed as: where   represents the velocity field,   is the kinematic viscosity,  is time, and  is the spatial coordinate. The nonlinear term   introduces complexity, making analytical solutions challenging except in specific cases. As a result, semi-analytical and numerical methods have been developed to approximate solutions to the Burgers equation. Among these methods, the Differential Transform Method (DTM) and the Adomian Decomposition Method (ADM) are two powerful semi-analytical techniques that have gained attention for their ability to handle nonlinear PDEs efficiently. This article provides a detailed comparison of DTM and ADM when applied to the Burgers equation, examining their theoretical foundations, implementation procedures, advantages, limitations, and performance in terms of accuracy, computational efficiency, and applicability. <strong>Keywords:</strong> Differential Transform Method (DTM), Adomian Decomposition Method (ADM), Burgers equation. <strong>Title:</strong> A Comparative Analysis of the Differential Transform Method and the Adomian Decomposition Method for Solving the Burgers Equation <strong>Author:</strong> Ahmad M. D. Al-Eybani <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 13, Issue 1, April 2025 - September 2025</strong> <strong>Page No: 20-25</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 29-April-2025</strong> <strong>DOI: https://doi.org/10.5281/zenodo.15303937</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/a-comparative-analysis-of-the-differential-transform-method-and-the-adomian-decomposition-method-for-solving-the-burgers-equation</strong>

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14

Islam, Danish Ul, and Anjali Shirivatasava. "Solving Linear Second Kind Non-Homogenous Volterra Integral Equations By Numerical Methods VIM, ADM And TSM." Indian Journal Of Science And Technology 17, no. 31 (2024): 3165–73. http://dx.doi.org/10.17485/ijst/v17i31.2099.

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Objectives: For the purpose of evaluating and determining the solution of the integral equations, various numerical methods is used to obtain the optimum solution of such equations. Methods: We developed numerical methods VIM, ADM and TSM to solve the linear second kind non-homogenous Volterra Integral Equations. Findings: Based on the findings of the study, implementing numerical methods VIM, ADM, and TSM are very helpful to obtain the solutions of linear second kind non-homogenous Volterra Integral Equations. Further numerical examples illustrate the accuracy of the Adomian Decomposition Method, Variational Iteration Method and Taylors Series Method. Novelty: Our research highlights the importance of the numerical methods VIM, ADM add TSM as these methods are quick convergence of solutions. Furthermore, our analysis led us to the determination that any integral equations can be solved by these methods easily and all the methods led us approximately at one point. Keywords: Volterra Integral Equation, NonHomogenous Volterra Integral Equations, Adomian Decomposition Method, Variational Iteration Method, Taylors Series Method

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15

Ujlayan, Amit, and Mohit Arya. "Approximate Solution of Riccati Differential Equation via Modified Greens Decomposition Method." Defence Science Journal 70, no. 4 (2020): 419–24. http://dx.doi.org/10.14429/dsj.70.14467.

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Riccati differential equations (RDEs) plays important role in the various fields of defence, physics, engineering, medical science, and mathematics. A new approach to find the numerical solution of a class of RDEs with quadratic nonlinearity is presented in this paper. In the process of solving the pre-mentioned class of RDEs, we used an ordered combination of Green’s function, Adomian’s polynomials, and Pade` approximation. This technique is named as green decomposition method with Pade` approximation (GDMP). Since, the most contemporary definition of Adomian polynomials has been used in GDMP. Therefore, a specific class of Adomian polynomials is used to advance GDMP to modified green decomposition method with Pade` approximation (MGDMP). Further, MGDMP is applied to solve some special RDEs, belonging to the considered class of RDEs, absolute error of the obtained solution is compared with Adomian decomposition method (ADM) and Laplace decomposition method with Pade` approximation (LADM-Pade`). As well, the impedance of the method emphasised with the comparative error tables of the exact solution and the associated solutions with respect to ADM, LADM-Pade`, and MGDMP. The observation from this comparative study exhibits that MGDMP provides an improved numerical solution in the given interval. In spite of this, generally, some of the particular RDEs (with variable coefficients) cannot be easily solved by some of the existing methods, such as LADM-Pade` or Homotopy perturbation methods. However, under some limitations, MGDMP can be successfully applied to solve such type of RDEs.

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16

BAGAYOGO, Moussa, Youssouf MINOUNGOU, NEBIE Abdoul Wassiha, and Youssouf PARE. "Solving viscid Burgers equation by Adomian decomposition Method (ADM), Regular Pertubation Method (RPM) and Homotopy Perturbation Method (HPM)." International Journal of Applied Mathematical Research 13, no. 2 (2024): 54–68. http://dx.doi.org/10.14419/gxb78b77.

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In this paper the Adomian decomposition Method (ADM), Regular Pertubation Method (RPM) and the Homotopy Perturbation Method (HPM) are used to study Burgers equation. Then we compare the solutions obtained by these three methods.

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17

Turut, Veyis, and Nuran Güzel. "Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/746401.

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Two tecHniques were implemented, the Adomian decomposition method (ADM) and multivariate Padé approximation (MPA), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

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18

Danish, Ul Islam, and Shirivatasava Anjali. "Solving Linear Second Kind Non-Homogenous Volterra Integral Equations By Numerical Methods VIM, ADM And TSM." Indian Journal of Science and Technology 17, no. 31 (2024): 3165–73. https://doi.org/10.17485/IJST/v17i31.2099.

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Abstract <strong>Objectives:</strong> For the purpose of evaluating and determining the solution of the integral equations, various numerical methods is used to obtain the optimum solution of such equations. <strong>Methods:</strong> We developed numerical methods VIM, ADM and TSM to solve the linear second kind non-homogenous Volterra Integral Equations. <strong>Findings:</strong> Based on the findings of the study, implementing numerical methods VIM, ADM, and TSM are very helpful to obtain the solutions of linear second kind non-homogenous Volterra Integral Equations. Further numerical examples illustrate the accuracy of the Adomian Decomposition Method, Variational Iteration Method and Taylors Series Method. <strong>Novelty:</strong> Our research highlights the importance of the numerical methods VIM, ADM add TSM as these methods are quick convergence of solutions. Furthermore, our analysis led us to the determination that any integral equations can be solved by these methods easily and all the methods led us approximately at one point. <strong>Keywords:</strong> Volterra Integral Equation, NonHomogenous Volterra Integral Equations, Adomian Decomposition Method, Variational Iteration Method, Taylors Series Method

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19

Mohammed, Ali Qasem Mohsen, and Qaid Hasan Yahya. "Adomian Decomposition Method to Solve the Second Order Ordinary Differential Equations with Constant Coefficient." International Journal of Mathematics and Physical Sciences Research 10, no. 2 (2023): 94–99. https://doi.org/10.5281/zenodo.7683761.

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<strong>Abstract:</strong> In this paper, we will display Adomian decomposition method for solving second-order ordinary differential equations with constant coefficient. The Adomian decomposition method(ADM) is a creative and effective method for exact solution. It is important to note that a lot of research work has been devoted to the application of the Adomian decomposition method to a wide class of linear and non-linear problems. Some examples were presented to show the ability of method for linear and non-linear ordinary differential equations. <strong>Keywords:</strong> Adomian decomposition method; second-order ordinary differential equations. <strong>Title:</strong> Adomian Decomposition Method to Solve the Second Order Ordinary Differential Equations with Constant Coefficient <strong>Author:</strong> Mohammed Ali Qasem Mohsen, Yahya Qaid Hasan <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 10, Issue 2, October 2022 - March 2023</strong> <strong>Page No: 94-99</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 28-February-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7683761</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/adomian-decomposition-method-to-solve-the-second-order-ordinary-differential-equations-with-constant-coefficient</strong>

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Agom, E. U., F. O. Ogunfiditimi, E. V. Bassey, and C. Igiri. "REACTION-DIFFUSION FISHER’S EQUATIONS VIA DECOMPOSITION METHOD." Journal of Computer Science and Applied Mathematics 5, no. 2 (2023): 145–53. http://dx.doi.org/10.37418/jcsam.5.2.7.

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The effect of the source, initial or boundary conditions in the use of Adomian decomposition method (ADM) on nonlinear partial differential equation or nonlinear equation in general is enormous. Sometimes the equation in question result to continuous exact solution in series form, other times it result to discrete approximate analytical solutions. In this paper, we show that continuous exact solitons can be obtained on application of ADM to the Fisher's equation with the deployment Taylor theorem to the terms(s) in question. And, the resulting series is split into the integral equations during the solution process. Resulting to multivariate Taylor's series of the exact solitons with the help of Adomian polynomials of the nonlinear reaction term correctly calculated. More physical results are further depicted in 2D, 3D and contour plots.

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21

Abdallah, Habila Ali, and Yousif Elriyah Abrar. "Using Adomian decomposition methods for solving systems of nonlinear partial differential equations." International Journal of Mathematics and Physical Sciences Research 10, no. 2 (2022): 14–22. https://doi.org/10.5281/zenodo.7182781.

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<strong>Abstract:</strong> In this paper, we apply the Adomian decomposition method (ADM) and Modified decomposition method (MDM) on two different types of nonlinear partial differential equations (PDEs), has been solved by using the homotopy perturbation method combined with new transform (NTHPM). But after solved by (MADM) we found (MADM) has less of computational work than (NTHPM), more effective, powerful and simple than (NTHPM). <strong>Keywords:</strong> Systems of nonlinear partial differential equations, Adomian decomposition method, Modified decomposition method, homotopy perturbation method combined with new transform. <strong>Title:</strong> Using Adomian decomposition methods for solving systems of nonlinear partial differential equations <strong>Author:</strong> Abdallah Habila Ali, Abrar Yousif Elriyah <strong>International Journal of Mathematics and Physical Sciences Research   </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 10, Issue 2, October 2022 - March 2023</strong> <strong>Page No: 14-22</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 10-October-2022</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7182781</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/using-adomian-decomposition-methods-for-solving-systems-of-nonlinear-partial-differential-equations</strong>

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22

Chen, Qin. "Adomian decomposition method for solving fuzzy fractional Volterra-Fredholmintegro-differential equations." Mathematical Modeling and Algorithm Application 2, no. 2 (2024): 36–42. http://dx.doi.org/10.54097/evxhvq18.

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This paper mainly studies the fuzzy nonlinear fractional Volterra-Fredholm integro-differential equations based on fuzzy Caputo derivative under the generalized Hukuhara difference. By usingSchauder fixed point theorem, the existence of solutions are proved. Because of the good convergence and convenient calculation of the Adomian decomposition method (ADM), we expand the nonlinear part of the equation into the Adomian polynomial of infinite series, and then construct the iterative sequence of the numerical solution of the equation. The effectiveness and applicability of ADM are verified by numerical examples.

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23

Jimoh, AbdulAzeez Kayode, and Aolat Olabisi Oyedeji. "On Adomian decomposition method for solving nonlinear ordinary differential equations of variable coefficients." Open Journal of Mathematical Sciences 4, no. 1 (2020): 476–84. http://dx.doi.org/10.30538/oms2020.0138.

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This paper considers the extension of the Adomian decomposition method (ADM) for solving nonlinear ordinary differential equations of constant coefficients to those equations with variable coefficients. The total derivatives of the nonlinear functions involved in the problem considered were derived in order to obtain the Adomian polynomials for the problems. Numerical experiments show that Adomian decomposition method can be extended as alternative way for finding numerical solutions to ordinary differential equations of variable coefficients. Furthermore, the method is easy with no assumption and it produces accurate results when compared with other methods in literature.

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24

Richard, Metomou, and Weidong Zhao. "Padé-Sumudu-Adomian Decomposition Method for Nonlinear Schrödinger Equation." Journal of Applied Mathematics 2021 (March 5, 2021): 1–19. http://dx.doi.org/10.1155/2021/6626236.

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The main purpose of this paper is to solve the nonlinear Schrödinger equation using some suitable analytical and numerical methods such as Sumudu transform, Adomian Decomposition Method (ADM), and Padé approximation technique. In many literatures, we can see the Sumudu Adomian decomposition method (SADM) and the Laplace Adomian decomposition method (LADM); the SADM and LADM provide similar results. The SADM and LADM methods have been applied to solve nonlinear PDE, but the solution has small convergence radius for some PDE. We perform the SADM solution by using the function P L / M · called double Padé approximation. We will provide the graphical numerical simulations in 3D surface solutions of each application and the absolute error to illustrate the efficiency of the method. In our methods, the nonlinear terms are computed using Adomian polynomials, and the Padé approximation will be used to control the convergence of the series solutions. The suggested technique is successfully applied to nonlinear Schrödinger equations and proved to be highly accurate compared to the Sumudu Adomian decomposition method.

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Biazar, Jafar, and Kamyar Hosseini. "A modified Adomian decomposition method for singular initial value Emden-Fowler type equations." International Journal of Applied Mathematical Research 5, no. 1 (2016): 69. http://dx.doi.org/10.14419/ijamr.v5i1.5666.

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<p>Traditional Adomian decomposition method (ADM) usually fails to solve singular initial value problems of Emden-Fowler type. To overcome this shortcoming, a new and effective modification of ADM that only requires calculation of the first Adomian polynomial is formally proposed in the present paper. Three singular initial value problems of Emden-Fowler type with alpha=1, 2, and &gt;2, have been selected to demonstrate the efficiency of the method. </p>

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Şentürk, Erman, Safa Bozkurt Coşkun, and Mehmet Tarık Atay. "Solution of jamming transition problem using adomian decomposition method." Engineering Computations 35, no. 5 (2018): 1950–64. http://dx.doi.org/10.1108/ec-12-2016-0437.

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Purpose The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM). Design/methodology/approach In this study, the jamming transition is presented as a result of spontaneous deviations of headway and velocity that is caused by the acceleration/breaking rate to be higher than the critical value. Dissipative dynamics of traffic flow can be represented within the framework of the Lorenz scheme based on the car-following model in the one-lane highway. Through this paper, an analytical approximation for the solution is calculated via ADM that leads to a solution for headway deviation as a function of time. Findings A highly nonlinear differential equation having no exact solution due to JTP is considered and headway deviation is obtained implementing a number of different initial conditions. The results are discussed and compared with the available data in the literature and numerical solutions obtained from a built-in numerical function of the mathematical software used in the study. The advantage of using ADM for the problem is presented in the study and discussed on the basis of the results produced by the applied method. Originality/value This is the first study to apply ADM to JTP.

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AL-Refaidi, A., N. ALZaid, Huda Bakodah, and M. AL-Mazmumy. "An Efficient Numerical Approach Based on the Adomian Chebyshev Decomposition Method for Two-Point Boundary Value Problems." European Journal of Pure and Applied Mathematics 17, no. 3 (2024): 1497–515. http://dx.doi.org/10.29020/nybg.ejpam.v17i3.5232.

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The current manuscript devises an efficient numerical method for solving two-point nonhomogeneous Boundary Value Problems (BVPs) with Dirichlet conditions. The method is based on the application of the celebrated Adomian Decomposition Method (ADM) and, the Chebyshev polynomials. This method which refers to ”Adomian Chebyshev Decomposition Method” (ACDM) is further proved to be a robust numerical method as the associated nonhomogeneous terms are successfully reinstated with a reliable Chebyshev series. Lastly, a comparative study between the acquired numerical results and the existing exact solutions of the test problems has been established to demonstrate the salient features of the devised method

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Alwan, Zainab Mohammed. "Solution Non Linear Partial Differential Equations By ZMA Decomposition Method." WSEAS TRANSACTIONS ON MATHEMATICS 20 (December 28, 2021): 712–16. http://dx.doi.org/10.37394/23206.2021.20.75.

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In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

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Wazwaz, Abdul-Majid, Randolph Rach, and Lazhar Bougoffa. "Dual solutions for nonlinear boundary value problems by the Adomian decomposition method." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 8 (2016): 2393–409. http://dx.doi.org/10.1108/hff-10-2015-0439.

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Purpose The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions. Design/methodology/approach The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions. Findings The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions. Originality/value This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.

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Che Hussin, Che Haziqah, and Adem Kiliçman. "On the Solutions of Nonlinear Higher-Order Boundary Value Problems by Using Differential Transformation Method and Adomian Decomposition Method." Mathematical Problems in Engineering 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/724927.

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We study higher-order boundary value problems (HOBVP) for higher-order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

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O, Okai J., M. S. Adamu, Cornelius M., et al. "A Hybrid Approach of the Variational Iteration Method and Adomian Decomposition Method for Solving Fractional Integro-Differential Equations." YASIN 5, no. 4 (2025): 2681–706. https://doi.org/10.58578/yasin.v5i4.5720.

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In this study, we propose a hybrid analytical technique that integrates the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve both linear and nonlinear integro-differential equations of integer and fractional orders. This approach extends and refines the Odibat Decomposition Method (ODM) by addressing key limitations inherent in ADM and VIM—specifically, the reliance on linearization, Adomian polynomials, and Lagrange multipliers. By circumventing these computational complexities, the proposed method enables the direct and efficient construction of series solutions with improved convergence properties. The hybrid scheme is designed for broader applicability and enhanced computational simplicity, making it a powerful tool for analyzing complex integro-differential systems. Its effectiveness and robustness are demonstrated through a range of illustrative examples, confirming the method’s capability to provide accurate analytical approximations with minimal computational overhead.

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GANJI, D. D., N. JAMSHIDI, and Z. Z. GANJI. "HPM AND VIM METHODS FOR FINDING THE EXACT SOLUTIONS OF THE NONLINEAR DISPERSIVE EQUATIONS AND SEVENTH-ORDER SAWADA–KOTERA EQUATION." International Journal of Modern Physics B 23, no. 01 (2009): 39–52. http://dx.doi.org/10.1142/s0217979209049607.

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In this paper, nonlinear dispersive equations and seventh-order Sawada–Kotera equation are solved using homotopy perturbation method (HPM) and variational iteration method (VIM). The results obtained by the proposed methods are then compared with that of Adomian decomposition method (ADM). The comparisons demonstrate that the two obtained solutions are in excellent agreement. The numerical results calculated show that the methods can be accurately implemented to these types of nonlinear equations. The results of HPM and VIM confirm the correctness of those obtained by Adomian decomposition method.

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Irfan Ul Haq, Sandeep Kumar Tiwari, Pradeep Porwal, and Naveed Ul Haq. "Solving Simple Hormonic Problems Using Adomian Decomposition Method." International Journal of Scientific Research in Science, Engineering and Technology 11, no. 4 (2024): 66–70. http://dx.doi.org/10.32628/ijsrset241145.

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In this research paper, we propose classical numerical technique for solving some simple harmonic problems arising in some applications of science. Adomian decomposition method (ADM) are used. Some numerical examples have been solved to illustrate the accuracy and efficiency of this numerical method.

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Eltayeb, Hassan, and Adem Kılıçman. "Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/412948.

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We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

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Appadu, Appanah Rao, and Abey Sherif Kelil. "Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations." Demonstratio Mathematica 54, no. 1 (2021): 377–409. http://dx.doi.org/10.1515/dema-2021-0039.

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Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

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Singh, Randhir, Gnaneshwar Nelakanti, and Jitendra Kumar. "Approximate Solution of Urysohn Integral Equations Using the Adomian Decomposition Method." Scientific World Journal 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/150483.

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We apply Adomian decomposition method (ADM) for obtaining approximate series solution of Urysohn integral equations. The ADM provides a direct recursive scheme for solving such problems approximately. The approximations of the solution are obtained in the form of series with easily calculable components. Furthermore, we also discuss the convergence and error analysis of the ADM. Moreover, three numerical examples are included to demonstrate the accuracy and applicability of the method.

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Muley, Yogesh. "Application of Laplace Decomposition Method to Solve Boundary Value Problems." ISAR - International Journal of Mathematics and Computing Techniques 3, no. 6 (2020): 1–5. https://doi.org/10.5281/zenodo.14614872.

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In this paper, Laplace DecompositionMethod (LDM) is applied to nonlinearpartial differential equations with boundaryconditions. Numerical results obtained byusing LDM compared with the resultsobtained by using Adomian decompositionMethod (ADM) and Variational iterationmethod (VIM).

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Alharbi, A., and E. S. Fahmy. "APPROXIMATE SOLUTION FOR TIME-DELAYED CONVECTIVE FISHER EQUATION BY ADM-PADÉ TECHNIQUE." Asian-European Journal of Mathematics 03, no. 02 (2010): 221–33. http://dx.doi.org/10.1142/s1793557110000155.

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We present an approximate solution to the time-delayed convective Fisher equation using ADM-Padé technique which is a combination of Adomian decomposition method and Padé approximation. This technique gives the approximate solution with faster convergence and higher accuracy than using ADM alone.

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Eltayeb, Hassan, Adem Kılıçman, and Said Mesloub. "Application of Sumudu Decomposition Method to Solve Nonlinear System Volterra Integrodifferential Equations." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/503141.

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We develop a method to obtain approximate solutions for nonlinear systems of Volterra integrodifferential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled Volterra integrodifferential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples and results of the present technique have close agreement with approximate solutions which were obtained with the help of Adomian decomposition method (ADM).

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Pedro, Pablo Cárdenas Alzate, Gerardo Cardona José, and María Rojas Luz. "A Survey of the Noise Terms Phenomenon in Adomian Method: A Special Case." European Journal of Advances in Engineering and Technology 7, no. 8 (2020): 27–30. https://doi.org/10.5281/zenodo.10667577.

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<strong>ABSTRACT</strong> In this work, we apply an iterative method (Adomian Decomposition Method) for solving a special case of nonlinear PDEs. The efficiency of this method is illustrated by investigating the convergence results for this equation. We show that the noise terms are conditional for nonhomogeneous equations and the numerical results show the reliability and accuracy of the ADM.

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LIN, YINWEI, TZON-TZER LU, and CHA’O-KUANG CHEN. "LARGE INTERVAL SOLUTION OF THE EMDEN–FOWLER EQUATION USING A MODIFIED ADOMIAN DECOMPOSITION METHOD WITH AN INTEGRATING FACTOR." ANZIAM Journal 56, no. 2 (2014): 192–208. http://dx.doi.org/10.1017/s1446181114000340.

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AbstractWe propose a new Adomian decomposition method (ADM) using an integrating factor for the Emden–Fowler equation. With this method, we are able to solve certain Emden–Fowler equations for which the traditional ADM fails. Numerical results obtained from testing our linear and nonlinear models are far more reliable and efficient than those from existing methods. We also present a complete error analysis and a convergence criterion for this method. One drawback of the traditional ADM is that the interval of convergence of the Adomian truncated series is very small. Some techniques, such as Pade approximants, can enlarge this interval, but they are too complicated. Here, we use a continuation technique to extend our method to a larger interval.

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Moniem, A. A., and J. Satouri. "Solution of Prey–Predator System by ADM." International Journal of Analysis and Applications 23 (February 3, 2025): 29. https://doi.org/10.28924/2291-8639-23-2025-29.

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A prey-predator system with an abundance of nutrients is considered. Utilizing Adomian decomposition method to numerate and approximate the solution of that governing system. Providing many examples to obtain some numerical simulation solutions and plot the results for the prey and predator populations versus time.

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Bougoffa, Lazhar, Randolph Rach, Abdul-Majid Wazwaz, and Jun-Sheng Duan. "On the Adomian decomposition method for solving the Stefan problem." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 4 (2015): 912–28. http://dx.doi.org/10.1108/hff-05-2014-0159.

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Purpose – The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method. Design/methodology/approach – The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem. Findings – The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat. Originality/value – The paper presents a new solution algorithm for the Stefan problem.

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Ebaid, Abdelhalim. "Approximate Analytical Solution of a Nonlinear Boundary Value Problem and its Application in Fluid Mechanics." Zeitschrift für Naturforschung A 66, no. 6-7 (2011): 423–26. http://dx.doi.org/10.1515/zna-2011-6-707.

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Although the decomposition method and its modified form were used during the last two decades by many authors to investigate various scientific models, a little attention was devoted for their applications in the field of fluid mechanics. In this paper, the Adomian decomposition method (ADM) is implemented for solving the nonlinear partial differential equation (PDE) describing the peristaltic flow of a power-law fluid in a circular cylindrical tube under the effect of a magnetic field. The numerical solutions obtained in this paper show the effectiveness of Adomian’s method over the perturbation technique

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Feng, Yu, and Dexiang Ma. "Numerical Solution for a Fractional Differential Equation Arising In Optics." Journal of Physics: Conference Series 2597, no. 1 (2023): 012011. http://dx.doi.org/10.1088/1742-6596/2597/1/012011.

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Abstract A new recursive algorithm (named as Abel-ADM) is given to obtain approximate solution for a class of nonlinear fractional differential equation arising in optics. Abel-ADM is a method that combines the generalized Abel equation with the Adomian decomposition method (ADM). The effectiveness of the method was verified through numerical examples.

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Mao, Qi Bo, Yan Ping Nie, and Wei Zhang. "Vibration Analysis of a Stepped Beam by Using Adomian Decomposition Method." Applied Mechanics and Materials 160 (March 2012): 292–96. http://dx.doi.org/10.4028/www.scientific.net/amm.160.292.

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The free vibrations of a stepped Euler-Bernoulli beam are investigated by using the Adomian decomposition method (ADM). The stepped beam consists two uniform sections and each section is considered a substructure which can be modeled using ADM. By using boundary condition and continuity condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions are presented. Comparing the results using ADM to those given in the literature, excellent agreement is achieved.

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Haq, I., and I. Singh. "Solving Some Oscillatory Problems using Adomian Decomposition Method and Haar Wavelet Method." Journal of Scientific Research 12, no. 3 (2020): 289–302. http://dx.doi.org/10.3329/jsr.v12i3.44287.

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In this research, we presented two classical numerical techniques to solve some oscillatory problems arised in various applications of sciences and engineering. Adomian decomposition method (ADM) and Haar wavelet method (HWM) are utilized for this purpose. Some numerical examples have been performed to illustrate the accuracy of the present methods.

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CHU, HONGMEI, and YINPING LIU. "THE NEW ADM–PADÉ TECHNIQUE FOR THE GENERALIZED EMDEN–FOWLER EQUATIONS." Modern Physics Letters B 24, no. 12 (2010): 1237–54. http://dx.doi.org/10.1142/s0217984910023268.

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In this paper, the Emden–Fowler equations are investigated by employing the Adomian decomposition method (ADM) and the Padé approximant. By using the new type of Adomian polynomials proposed by Randolph C. Rach in 2008, our obtained solution series converges much faster than the regular ADM solution of the same order. Meanwhile, we note that the solutions obtained by using the new ADM–Padé technique have much higher accuracy and larger convergence domain than those obtained by using the regular ADM together with the Padé technique. Finally, comparison of our new obtained solutions are given with those existing exact ones graphically to illustrate the validity and the promising potential of the new ADM–Padé technique for solving nonlinear problems.

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BAGAYOGO, Moussa, Youssouf PARE, and Youssouf MINOUNGOU. "An Approached Solution of Wave Equation with Cubic Damping by Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian Decomposition Method (ADM)." Journal of Mathematics Research 10, no. 2 (2018): 166. http://dx.doi.org/10.5539/jmr.v10n2p166.

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In this study, we consider the wave equation with cubic damping with its initial conditions. Homotopy Perturbation Method (HPM), Regular Pertubation Method (RPM) and Adomian decomposition Method (ADM) are applied to this equation. Then, the solution yielding the given initial conditions is gained. Finally, the solutions obtained by each method are compared.

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G. Thunibat, Reem. "DEMONSTRATING NON-LINEAR RLC CIRCUIT EQUATION BY INVOLVING FRACTIONAL ADOMIAN DECOMPOSITION METHOD BY CAPUTO DEFINITION." Advanced Mathematical Models & Applications 9, no. 3 (2024): 387–400. https://doi.org/10.62476/amma93387.

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This research displays nonlinear RLC circuit equation applying the fractional Adomian Decomposition Method (ADM) to investigate an approximate analytical solution. Where the fractional derivative described here as in the Caputo definition. The result behavior obtained by ADM is displayed graphically and numerically where it indicates a great consist compared with those obtained by other methods

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