A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

Fang, Jiahua; Nadeem, Muhammad; Habib, Mustafa; Karim, Shazia; Wahash, Hanan A.

doi:10.1155/2022/5365810

Cited by 13 publications

(6 citation statements)

References 29 publications

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“…Te clique polynomial is regarded as a fundamental function for operational matrices in this method. For more details, refer to the following references: [30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Deresse,

Dufera

2024

Applied Computational Intelligence and Soft Computing

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The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

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“…Te clique polynomial is regarded as a fundamental function for operational matrices in this method. For more details, refer to the following references: [30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Deresse,

Dufera

2024

Applied Computational Intelligence and Soft Computing

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“…Indeed, during the development of fractional calculus, numerous scientists such as Euler, Riemann-Riouville, Leibniz, Bernoulli, L'Hospital and Wallis have contributed significantly to the research of this field (Hilfer, 2000;Sabatier et al, 2007;Kai, 2010). In addition to the commonly used methods for solving partial differential Application of LADM for FFPE and TFCBBEs equations (PDEs), many new methods have been used to solve PDEs such as new iterative method (Fang et al, 2022), Laplace residual power series method (Luo and Nadeem, 2023a, b), Mohand homotopy transform scheme (Luo and Nadeem, 2023a, b), Elzaki homotopy perturbation transform scheme (Nadeem et al, 2023a, b), fractional optimal control approach (Nadeem et al, 2023a, b). Laplace Adomian decomposition method (LADM) is a combination of the Laplace transform method and the Adomian decomposition method (ADM) (Yan, 2013;Adomian, 1988;Odibat and Momani, 2007;Wazwaz, 1998;Duan et al, 2012), which was first proposed by Khuri to solve nonlinear differential equations (Khuri and Suheil, 2001).…”

Section: Introductionmentioning

confidence: 99%

“…, 2007; Kai, 2010). In addition to the commonly used methods for solving partial differential equations (PDEs), many new methods have been used to solve PDEs such as new iterative method (Fang et al. , 2022), Laplace residual power series method (Luo and Nadeem, 2023a, b), Mohand homotopy transform scheme (Luo and Nadeem, 2023a, b), Elzaki homotopy perturbation transform scheme (Nadeem et al.…”

Section: Introductionmentioning

confidence: 99%

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Zhang,

Wang

2024

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PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

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“…Therefore, many scholars focused on using different methods to find fractional solitons. For instance, the first integral method [9], the fractional sub-equation method [10], the homotopy perturbation method [11][12][13] and its modifications, Mohand transform-homotopy perturbation method [14,15], two-scale transform--homotopy perturbation method [16], Laplace transform-homotopy perturbation method [17], Li-He's modified homotopy perturbation method [18][19][20], the tanh-function method [21,22] and its modification-tanh function expansion method [23]-and modified extended tanh-function method [24,25]. It is worth mentioning that fractional complex transform was first proposed by [26]; it can convert fractional differential equations directly into ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fractional solitons: New phenomena and exact solutions

et al. 2023

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The fractional solitons have demonstrated many new phenomena, which cannot be explained by the traditional solitary wave theory. This paper studies some famous fractional wave equations including the fractional KdV–Burgers equation and the fractional approximate long water wave equation by a modified tanh-function method. The solving process is given in details, and new solitons can be rigorously explained by the obtained exact solutions. This paper offers a new window for studying fractional solitons.

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A New Iterative Method for the Approximate Solution of Klein-Gordon and Sine-Gordon Equations

Cited by 13 publications

References 29 publications

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Exploring Physics‐Informed Neural Networks for the Generalized Nonlinear Sine‐Gordon Equation

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

Fractional solitons: New phenomena and exact solutions

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