New iterative approach for the solutions of fractional order inhomogeneous partial differential equations

Zada, Laiq; Nawaz, Rashid; Ahsan, Sumbal; Nisar, Kottakkaran Sooppy; Băleanu, Dumitru

doi:10.3934/math.2021084

Cited by 13 publications

(5 citation statements)

References 16 publications

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“…It starts with an initial estimate and improves it through a succession of well-planned steps. Due to its usefulness in various scientific and technical fields, NIM is widely used by researchers and professionals seeking a dependable and adaptable approach to getting approximate solutions to complex mathematical models [56][57][58][59]. Through numerical analysis, its use has considerably improved our knowledge of and response to complicated real-world occurrences.…”

Section: Introductionmentioning

confidence: 99%

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

Yasmin,

Alkhezi,

Alhamad

2023

Symmetry

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In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis.

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Section: Introductionmentioning

confidence: 99%

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

Yasmin,

Alkhezi,

Alhamad

2023

Symmetry

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“…To handle these problems, the researchers used numerical, analytical, and some homotopy-based methods to approximate such a nonlinear problem. Some famous methods which handle the DEs of fractional and integer order are the residual power series method [9,10], Laplace decomposition method (LDM) [11], q-homotopy analysis method (q-HAM) [12], Adomian decomposition method (ADM) [13], reduced differential transform method (RDTM) [14], variational iteration method (VIM) [15], optimal homotopy asymptotic method (OHAM) [16,17], homotopy perturbation method (HPM) [18], homotopy analysis method (HAM) [19], etc. Besides these methods, many researchers have introduced many numerical methods to handle differential equations [20].…”

Section: Introductionmentioning

confidence: 99%

A Novel Analysis of Generalized Perturbed Zakharov–Kuznetsov Equation of Fractional‐Order Arising in Dusty Plasma by Natural Transform Decomposition Method

Alhazmi

Abdelmohsen

Alyami

et al. 2022

Journal of Nanomaterials

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The natural transform decomposition method (NTDM) is a relatively new transformation method for finding an approximate differential equation solution. In the current study, the NTDM has been used for obtaining an approximate solution of the fractional-order generalized perturbed Zakharov–Kuznetsov (GPZK) equation. The method has been tested for three nonlinear cases of the fractional-order GPZK equation. The absolute errors are analyzed by the proposed method and the q-homotopy analysis transform method (q-HATM). 3D and 2D graphs have shown the proposed method’s accuracy and effectiveness. NTDM gives a much-closed solution after a few terms.

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“…The proposed techniques have been recently applied by Nawaz et al for solving noninteger order differential equations [13]. Many other researchers have applied NIM and natural transform for handling the FDEs [14][15][16][17]. In this article, we will consider fractional biological population model (FBPM) as [18]…”

Section: Introductionmentioning

confidence: 99%

Analytical Approximate Solution of the Fractional Order Biological Population Model by Using Natural Transform

Ali

Nawaz

Zada

et al. 2022

Journal of Nanomaterials

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In the present work, the natural transform iterative method (NTIM) is implemented to solve the biological population model (BPM) of fractional order. The method is tested for three nonlinear examples. The NTIM is a combination of a new iterative method and natural transform. We see that the solution pattern converges to the exact solution in a few iterations. The method handles an extensive range of differential equations of both fractional and integer order. The fractional order derivative is considered in Caputo’s sense. For mathematical computation, Mathematica 10 is used.

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New iterative approach for the solutions of fractional order inhomogeneous partial differential equations

Cited by 13 publications

References 16 publications

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

A Novel Analysis of Generalized Perturbed Zakharov–Kuznetsov Equation of Fractional‐Order Arising in Dusty Plasma by Natural Transform Decomposition Method

Analytical Approximate Solution of the Fractional Order Biological Population Model by Using Natural Transform

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