The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model

This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the optimal homotopy asymptotic method (OHAM). These represent the ε-approximate OHAM solutions. A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure is suitable to be applied for dynamical systems with certain geometrical properties.

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“…Using the functions given by Equation ( 15), we recall some types of approximate solutions of Equation ( 7) from [46].…”

Section: Basic Ideas Of the Oham Techniquementioning

confidence: 99%

Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method

et al. 2022

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“…Te homotopy perturbation method (HPM) provides a semianalytical algorithm for solving both linear and nonlinear ordinary/partial diferential equations [42]. It is also applied to diferential system of equations [43]. In order to solve diferential equations in fractional form more accurately, many modifcations of HPM have been introduced.…”

Section: Introductionmentioning

confidence: 99%

New Solutions of Time- and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

Qayyum,

Ahmad

2024

Journal of Mathematics

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The current study explores the space and time-fractional Black–Scholes European option pricing model that primarily occurs in the financial market. To tackle the complexities associated with solving models in a fractional environment, the Aboodh transform is hybridized with He’s algorithm. This facilitates in improving the efficiency and applicability of the classical homotopy perturbation method (HPM) by ensuring the rapid convergence of the series form solution. Three cases that are time-fractional scenario, space-fractional scenario, and time-space-fractional scenario are observed through graphs and tables. 2D graphical analysis is performed to depict the behaviour of a given option pricing model for varying time, stock price, and fractional parameters. Solutions of the European option pricing model at various fractional orders are also presented as 3D plots. The results obtained through these graphs unfold the interchange between time- and space-fractional derivatives, presenting a comprehensive study of option pricing under fractional dynamics. The competency of the proposed scheme is illustrated via solutions and errors throughout the fractional domain in tabular form. The validity of the He-Aboodh results is exhibited by comparison with existing errors. Analysis shows that the proposed methodology (He-Aboodh algorithm) is a valuable scheme for solving time-space-fractional models arising in business and economics.

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“…Paşca et al [6] propose the least-squares homotopy perturbation as a straightforward and accurate method for obtaining approximate analytical solutions for systems of ordinary differential equations. The technique is used to resolve a problem involving the laminar flow of a viscous fluid in a semi-porous channel, which may be used to simulate blood flow through a blood vessel while taking the effects of a magnetic field into account.…”

mentioning

confidence: 99%

Preface to the Special Issue on “Advances in Differential Dynamical Systems with Applications to Economics and Biology”

2022

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The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model

Cited by 5 publications

References 33 publications

Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method

Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method

New Solutions of Time- and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

Preface to the Special Issue on “Advances in Differential Dynamical Systems with Applications to Economics and Biology”

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