An approximate analytical solution of the fractional multi-dimensional Burgers equation by the homotopy perturbation method

Sripacharasakullert, Pattira; Sawangtong, Wannika; Sawangtong, Panumart

doi:10.1186/s13662-019-2197-y

Cited by 8 publications

(2 citation statements)

References 28 publications

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“…Different fractional differential operators are applied for the analytical result of the TFNB equation [26]. Analytical approaches for approximating the fractional Burger's equation are presented in [27,28].…”

Section: Introductionmentioning

confidence: 99%

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

Tamsir,

Nigam,

Dhiman

et al. 2023

Beni-Suef Univ J Basic Appl Sci

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Background This study proposes an efficient and stable technique based on new hybrid B-spline (HB-spline) functions for the numerical treatment of the Caputo time fractional nonlinear Burgers’ (TFNB) equation. The time derivative is discretized using the definition of the Caputo derivative, whereas HB-spline functions are used to discretize the spatial derivatives. The Rubin–Graves technique is used to linearize the nonlinear terms. Results The performance and efficacy of the established method are tested using three examples. The graphical results represent the smoothness between numerical and exact solutions. The absolute errors are very low as $$1{0}^{-4}$$ 1 0 - 4 and $$1{0}^{-5}$$ 1 0 - 5 . The convergence rate shows that the proposed method is second-order accurate in space. Conclusions The proposed method provides better results than the methods available in the literature. The method yields highly accurate results and can handle large-scale problems, which is the novelty of the present work.

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Section: Introductionmentioning

confidence: 99%

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

Tamsir,

Nigam,

Dhiman

et al. 2023

Beni-Suef Univ J Basic Appl Sci

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“…Currently, considerable attention has been given to approximate analytic and/or numerical solutions of fractional Black-Scholes equation resulting from its remarkable scope and applications in several disciplines. Some of the analytical methods are the variational iteration method [21], Adomian decomposition method [22], homotopy perturbation method [23], homotopy analysis method [17], Laplace transform homotopy perturbation method [15,16,24], and Green's function homotopy perturbation method [25].…”

Section: Introductionmentioning

confidence: 99%

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Ampun

Sawangtong

2021

Mathematics

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In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

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An Approximate Analytic Solution for the Multidimensional Fractional-Order Time and Space Burger Equation Based on Caputo-Katugampola Derivative

Sawangtong,

Ikot,

Sawangtong

2023

Int J Theor Phys

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An approximate analytical solution of the fractional multi-dimensional Burgers equation by the homotopy perturbation method

Cited by 8 publications

References 28 publications

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

A hybrid B-spline collocation technique for the Caputo time fractional nonlinear Burgers’ equation

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

An Approximate Analytic Solution for the Multidimensional Fractional-Order Time and Space Burger Equation Based on Caputo-Katugampola Derivative

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