Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions

Heydari, Mohammad Hossein; Hooshmandasl, Mohammad Reza; Mohammadi, Fakhrodin

doi:10.1016/j.amc.2014.02.047

Cited by 84 publications

(48 citation statements)

References 21 publications

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“…, 2 k , k is any arbitrary non-negative integer, m is the degree of the Legendre polynomials and independent variable t is defined on [0, 1]. They are defined on the interval [0, 1] by [32]:…”

Section: The Lws and Their Propertiesmentioning

confidence: 99%

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Wavelets method for the time fractional diffusion-wave equation

et al. 2015

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Section: The Lws and Their Propertiesmentioning

confidence: 99%

“…In the last few years, several numerical methods have been proposed for solving FDWE, for instance see [12,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In recent years, the LWs have been applied for solving some fractional differential equations, for instance see [31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Wavelets method for the time fractional diffusion-wave equation

et al. 2015

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“…The technique is an amalgamation of the Legendre wavelets approximation and differential evolution algorithm. The Legendre wavelets method has been extensively used to approximate the unknown functions of integer-and fractional-order differential equations [24,25]. Besides, the differential evolution (DE) algorithm, the famous meta-heuristic scheme, has nowadays gained popularity for its global optimization attribute [26,27].…”

Section: Introductionmentioning

confidence: 99%

Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling

et al. 2018

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In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

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“…Particularly, orthogonal wavelets are widely used in approximating numerical solutions of various types of fractional order differential equations in the relevant literatures; see [18][19][20][21][22]. Among them, the second-kind Chebyshev wavelets have gained much attention due to their useful properties ( [23][24][25][26]) and can handle different types of differential problems.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Zhou

2017

Advances in Mathematical Physics

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The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method.

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Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions

Cited by 84 publications

References 21 publications

Wavelets method for the time fractional diffusion-wave equation

Wavelets method for the time fractional diffusion-wave equation

Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

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