Wavelets method for the time fractional diffusion-wave equation

Heydari, Mohammad Hossein; Hooshmandasl, Mohammad Reza; Ghaini, F. M. Maalek; Cattani, Carlo

doi:10.1016/j.physleta.2014.11.012

Cited by 101 publications

(41 citation statements)

References 37 publications

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“…Moreover, this is the first time the Ritz-Galerkin method in the two-dimensional BWs basis and with utilizing the satisfier function is employed to give an approximate solution of FDWE. We also compare our results with those results obtained by [3] and [7]. Comparison for the numerical examples shows the more accuracy and less computations of our scheme in comparison to other published methods.…”

Section: Introductionmentioning

confidence: 63%

“…We compare our results with obtained results in [3] and [7]. In all examples the package of Mathematica ver.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 84%

“…Notice the following FDWE [7]: o q uðx; tÞ ot q þ ouðx; tÞ ot ¼ o 2 uðx; tÞ ox 2 þ hðx; tÞ; ðx; tÞ 2 ½0; 1 Â ½0; 1 ; 1\q 2;…”

Section: Examplementioning

confidence: 99%

“…In the case q ¼ 2, this equation is named telegraph equation. Recently, considerable amount of papers have been proposed methods for solving the FDWE [2][3][4][5][6][7][8][9][10][11][12][13]. Chen et al [1] obtained the analytical solution by the method of separation of variables and proposed the numerical solution with finite difference method.…”

Section: Introductionmentioning

confidence: 99%

“…A fully discrete difference scheme is recommended for a diffusion-wave system by Wess [6]. Heydari et al [7] applied fractional operational matrix (FOM) of integration for the Legendre wavelets (LWs) to solve the problem. In [8] a compact finite-difference scheme is used for the fourth-order fractional diffusion-wave system.…”

Section: Introductionmentioning

confidence: 99%

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Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Barikbin

2017

Math Sci

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In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz-Galerkin method are applied for the numerical solution of the time fractional diffusionwave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz-Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.

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Section: Introductionmentioning

confidence: 63%

“…We compare our results with obtained results in [3] and [7]. In all examples the package of Mathematica ver.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 84%

“…Notice the following FDWE [7]: o q uðx; tÞ ot q þ ouðx; tÞ ot ¼ o 2 uðx; tÞ ox 2 þ hðx; tÞ; ðx; tÞ 2 ½0; 1 Â ½0; 1 ; 1\q 2;…”

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Barikbin

2017

Math Sci

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A new Wavelet Method for Variable‐Order Fractional Optimal Control Problems

Heydari

Avazzadeh

2017

Asian Journal of Control

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In this paper, a new computational method based on the Legendre wavelets (LWs) is proposed for solving a class of variable‐order fractional optimal control problems (V‐FOCPs). To do this, a new operational matrix of variable‐order fractional integration (OMV‐FI) in the Riemann‐Liouville sense for the LWs is derived and used to obtain an approximate solution for the problem under study. Along the way the hat functions (HFs) are introduced and employed to derive a general procedure to compute this matrix. In the proposed method, the variable‐order fractional dynamical system is transformed to an equivalent variable‐order fractional integro‐differential dynamical system, at first. Then, the highest integer order of the derivative of the state variable and the control variable are expanded by the LWs with unknown coefficients. Next, the OMV‐FI in the the Riemann‐Liouville sense together with some properties of the LWs are employed to achieve a nonlinear algebraic equation in place of the performance index and a nonlinear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency and accuracy of the proposed method are demonstrated for some concrete examples. The obtained results show that the proposed method is very efficient and accurate.

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