The Fractional Series Solutions for the Conformable Time‐Fractional Swift‐Hohenberg Equation through the Conformable Shehu Daftardar‐Jafari Approach with Comparative Analysis

Liaqat, Muhammad Imran; Okyere, Eric

doi:10.1155/2022/3295076

Cited by 4 publications

(1 citation statement)

References 40 publications

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“…Although fractional derivatives can be defned in a variety of ways, not all of them are generally used. Te Atangana-Baleanu, Riemann-Liouville (R-L), Caputo-Fabrizio, Caputo, and conformable operators are the most frequently used [6][7][8][9][10][11][12]. In some cases, fractional derivatives are preferable to integerorder derivatives for modeling because they can simulate and analyze complicated systems having complicated nonlinear processes and higher-order behaviors.…”

Section: Introductionmentioning

confidence: 99%

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Liaqat

Okyere

2023

Journal of Mathematics

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The residual power series method is effective for obtaining solutions to fractional-order differential equations. However, the procedure needs the n − 1 ϖ derivative of the residual function. We are all aware of the difficulty of computing the fractional derivative of a function. In this article, we considered the simple and efficient method known as the Laplace residual power series method (LRPSM) to find the analytical approximate and exact solutions of the time-fractional Black–Scholes option pricing equations (BSOPE) in the sense of the Caputo derivative. This approach combines the Laplace transform and the residual power series method. The suggested method just needs the idea of an infinite limit, so the computations required to determine the coefficients are minimal. The obtained results are compared in the sense of absolute errors against those of other approaches, such as the homotopy perturbation method, the Aboodh transform decomposition method, and the projected differential transform method. The results obtained using the provided method show strong agreement with different series solution methods, demonstrating that the suggested method is a suitable alternative tool to the methods based on He’s or Adomian polynomials.

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Section: Introductionmentioning

confidence: 99%

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Liaqat

Okyere

2023

Journal of Mathematics

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Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

Liaqat,

Din,

Albalawi

et al. 2024

MATH

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<abstract><p>In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.</p></abstract>

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Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

Albalawi,

Liaqat,

Ud Din

et al. 2024

MATH

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<abstract><p>One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results.</p></abstract>

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The Fractional Series Solutions for the Conformable Time‐Fractional Swift‐Hohenberg Equation through the Conformable Shehu Daftardar‐Jafari Approach with Comparative Analysis

Cited by 4 publications

References 40 publications

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Comparative Analysis of the Time-Fractional Black–Scholes Option Pricing Equations (BSOPE) by the Laplace Residual Power Series Method (LRPSM)

Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

Well-posedness and Ulam-Hyers stability results of solutions to pantograph fractional stochastic differential equations in the sense of conformable derivatives

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