New approximate-analytical solutions to partial differential equations via auxiliary function method

Zada, Laiq; Nawaz, Rashid; Nisar, Kottakkaran Sooppy; Tahir, Muḥammad; Yavuz, Mehmet; Kaabar, Mohammed K. A.; Martínez, Francisco

doi:10.1016/j.padiff.2021.100045

Cited by 23 publications

(10 citation statements)

References 27 publications

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“…investigation are new and have been listed for the first time. The authors' suggestion for future works is employing newly well-organized methods [38][39][40][41][42][43][44] to acquire other wave structures of STOBE.…”

Section: Declaration Of Competing Interestmentioning

confidence: 99%

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

Hosseini¹,

Akbulut²,

Băleanu³

et al. 2022

Commun. Theor. Phys.

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The present paper deals with the Sharma–Tasso–Olver–Burgers equation (STOBE) and its conservation laws and kink solitons. More precisely, the formal Lagrangian, Lie symmetries, and adjoint equations of the STOBE are firstly constructed to retrieve its conservation laws. Kink solitons of the STOBE are then extracted through adopting a series of newly well-designed approaches such as Kudryashov and exponential methods. Diverse graphs in 3D postures are formally portrayed to reveal the dynamical features of kink solitons. According to the authors’ knowledge, the outcomes of the current investigation are new and have been listed for the first time.

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Section: Declaration Of Competing Interestmentioning

confidence: 99%

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

Hosseini¹,

Akbulut²,

Băleanu³

et al. 2022

Commun. Theor. Phys.

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“…It is well known that the construction of the exact solutions of fractional partial differential equations (FPDEs) is an important problem. Consequently, many scholars have introduced numerous methods to seek the solutions, such as the Lie symmetry analysis method [7][8][9], Adomian decomposition method [10], homotopy analysis transform method [11], Laplace transform collocation method [12], functional separation variables method [13], residual power series method [14], sub-equation method [15], homotopy perturbation method [16,17], invariant subspace method [18], auxiliary function method [19,20] and the classical Mittag-Leffler kernel [21].…”

Section: Introductionmentioning

confidence: 99%

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

Liu,

Gu,

Wang

2024

Commun. Theor. Phys.

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In this paper, two types of fractional nonlinear equations in Caputo sense, time fractional Newell-Whitehead equation (FNWE) and time fractional generalized Hirota-Satsuma coupled KdV system (HS-cKdVS), are investigated by means of q-homotopy analysis method (q-HAM). The approximate solutions of proposed equations are constructed in the form of convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just few terms of the series solution are required in order to obtain better approximation. For the sake of the visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.

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“…Computing the exact or approximate solutions of fractional differential equations (FDEs) is very important in all the mentioned fields, but due to the complex nature of FDEs, the exact solution is not possible in most of the cases. As a result, it is essential to compute approximate solutions through analytical or numerical methods like the fractional natural decomposition method [35], Fourier spectral method [36], Lie symmetry analysis [37], auxiliary function method [38], consistent Riccati expansion method [39], and homotopy perturbation method [40]. For better accuracy while dealing with nonlinear problems, various modifications of HPM are also employed on different equations.…”

Section: Introductionmentioning

confidence: 99%

Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

et al. 2022

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In this research, the He-Laplace algorithm is extended to generalized third order, time-fractional, Korteweg-de Vries (KdV) models. In this algorithm, the Laplace transform is hybrid with homotopy perturbation and extended to highly nonlinear fractional KdVs, including potential and Burgers KdV models. Time-fractional derivatives are taken in Caputo sense throughout the manuscript. Convergence and error estimation are confirmed theoretically as well as numerically for the current model. Numerical convergence and error analysis is also performed by computing residual errors in the entire fractional domain. Graphical illustrations show the effect of fractional parameter on the solution as 2D and 3D plots. Analysis reveals that the He-Laplace algorithm is an efficient approach for time-fractional models and can be used for other families of equations.

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New approximate-analytical solutions to partial differential equations via auxiliary function method

Cited by 23 publications

References 27 publications

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

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