ARA-residual power series method for solving partial fractional differential equations

Applied Computational Intelligence and Soft Computing

2023

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This article presents the solution of the fractional SIR epidemic model using the Laplace residual power series method. We introduce the fractional SIR model in the sense of Caputo’s derivative; it is presented by three fractional differential equations, in which the third one depends on the first coupled equations. The Laplace residual power series method (LRPSM) is implemented in this research to solve the proposed model, in which we present the solution in a form of convergent series expansion that converges rapidly to the exact one. We analyze the results and compare the obtained approximate solutions to those obtained from other methods. Figures and tables are illustrated to show the efficiency of the LRPSM in handling the proposed SIR model.

“…Operating the inverse Laplace transform on each equation in (32), we obtain the kth solution of system (12) as follows:…”

Section: Applied Computational Intelligence and Soft Computingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Analytical Solution of Fractional SIR Epidemic Model

Qazza

Applied Computational Intelligence and Soft Computing

2023

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“…Tere are various analytical and numerical methods available for handling various forms of ffth-order KdV-type equations in the literature. Some of them are the Adomian decomposition technique [35], modifed Adomian decomposition method [36], Laplace decomposition approach [37], diferential transform technique [38,39], Hirota's bilinear techniques [40], inverse scattering algorithm [41], He's semiinverse scheme [42], extended Tanh method [43], homotopy analysis technique [14,44], fractional homotopy analysis transform algorithm [45], modifed homotopy perturbation technique [46], variational iteration technique [47], homotopy perturbation method [48,49], homotopy perturbation transform method [50], hyperbolic and exponential ansatz methods [51], multiple exp-function method [52], and others [53][54][55]. Moreover, many methods are available to solve the fractional-order KdV equations.…”

Section: Introductionmentioning

confidence: 99%

A New Computational Technique for Analytic Treatment of Time‐Fractional Nonlinear Equations Arising in Magneto‐Acoustic Waves

Prakasha

Mathematical Problems in Engineering

Kachhia

et al. 2023

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This paper presents the study of time-fractional nonlinear fifth-order Korteweg–de Vries equations by utilizing an adequate novel technique, namely, the q-homotopy analysis transform method. The fifth-order Korteweg–de Vries equation has got its importance in the study of magneto-sound propagation in plasma, capillary gravity water waves, and the motion of long waves under the influence of gravity in shallow water. To justify the effectiveness and pertinence of the contemplated technique, we take a look at three examples of the time-fractional fifth-order Korteweg–de Vries equations. The q-homotopy analysis transform method offers us to modulate the range of convergence of the series solution using ℏ, called the auxiliary parameter or convergence control parameter. The study of the fractional behaviour of the considered equations expresses the originality of the presented work. There is a visible variation in the obtained solutions for different fractional orders and which can lead to different consequences for future work. As a future research direction, readers can use the hybrid methodologies merging with our projected scheme to achieve better consequences. Additionally, to validate the precision and reliability of the proposed method, we organized suitable numerical simulations. The obtained findings show that the proposed method is very gratifying and examines the complex nonlinear challenges that arise in science and innovation.

“…Several analytical approaches have been developed by numerous researchers to solve linear and nonlinear partial diferential equations, such as the Homotopy analysis method [14], Adomian decomposition method [15], the variational iteration method [16], power series method [17], residual power series method [18], Laplace residual power series [19], ARA residual power series method [20], and others [21,22].…”

Section: Introductionmentioning

confidence: 99%

A Generalized Approach of Triple Integral Transforms and Applications

2023

Journal of Mathematics

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In this study, we introduce a novel generalization of triple integral transforms, which is called a general triple transform. We present the definition of the new approach and prove the main properties related to the existence, uniqueness, shifting, scaling, and inverse. Moreover, relations between the new general triple transform and other transforms are presented, and new results related to partial derivatives and the triple convolution theorem are established. We apply the general triple transform to solve some applications of various types of partial differential equations. The strength of the new approach is that it covers almost all integral transforms of order one, two, and three, and hence no need to find new formulas of triple integral transforms or to study the basic properties.