A Method for Solving Time-Fractional Initial Boundary Value Problems of Variable Order

Abuasbeh, Kinda; Kanwal, Asia; Shafqat, Ramsha; Taufeeq, Bilal; Almulla, Muna A.; Awadalla, Muath

doi:10.3390/sym15020519

Cited by 6 publications

(3 citation statements)

References 39 publications

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“…Several techniques have been introduced in the literature to solve ordinary and partial diferential equations. Galerkin method [21], implicit fnite diference scheme [22], Adomian decomposition method [23], Crank-Nicolson scheme [24], homotopy perturbation method [25], backward Euler method [26], diferential transform method [27], and stabilized meshless technique [28] are some of them. Many of these approaches have been utilized for the numerical solution of fractional diferential equations including the Navier-Stokes equation [29,30], Schrödinger equation [31], COVID-19 model [32], Kundu-Mukherjee-Naskar equation [33], and Black-Scholes model [34].…”

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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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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“…This method involves approximating the solution of a differential equation using a series of orthogonal functions, such as Fourier series or Chebyshev polynomials and then using the corresponding operational matrices to solve for the coefficients of the series. For more detail, we refer the reader to the references [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations

et al. 2023

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In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10−14, while the L2 error is on the order of 10−13. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

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“…Indeed, FDEs have a large flood of applications in different scopes such as chemistry, physics, finance, engineering, and infectious disease. The combination of FDEs and other analytical and numerical methods can be found in many works such as impulsive FDEs [6,7], implicit hybrid FDEs [8][9][10], mathematical modelings with the help of FDEs [11][12][13][14][15], neutral FDEs [16,17], p-Laplacian FDEs [18], variable order time-fractional FDEs [19], random and fuzzy FDEs [20,21], integro-differential inclusions [22,23], and references therein.…”

Section: Introductionmentioning

confidence: 99%

A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces

et al. 2023

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In this paper, we study a coupled fully hybrid system of (k,Φ)–Hilfer fractional differential equations equipped with non-symmetric (k,Φ)–Riemann-Liouville (RL) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (GBS). Moreover, the Ulam–Hyers (UH) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results.

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A Method for Solving Time-Fractional Initial Boundary Value Problems of Variable Order

Cited by 6 publications

References 39 publications

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

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

Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations

A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces

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