Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Liu, Fawang; Meerschaert, Mark M.; McGough, Robert J.; Zhuang, Pinghui; Liu, Qingxia

doi:10.2478/s13540-013-0002-2

Cited by 274 publications

(134 citation statements)

References 33 publications

(31 reference statements)

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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: Introductionsupporting

confidence: 59%

“…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: 48%

“…Consider the following FDWE [3,7]: 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; ð4:2Þ where hðx; tÞ ¼ 6t 3Àq e x Cð4 À qÞ þ 3t 2 e x À t 3 e x :…”

Section: Exampleunclassified

“…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%

“…In [2], Bhrawy et al applied a spectral tau method based on the Jacobi operational matrix to solve the problem. Liu et al [3] proposed the fractional predictor-corrector method to solve this problem. In [4], Mainardi derived the fundamental solutions for the FDWE.…”

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.

show abstract

Section: Introductionsupporting

confidence: 59%

“…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: 48%

Section: Exampleunclassified

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

El‐Sayed,

Boulaaras

2024

Int J Numerical Modelling

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This work presents a numerical approach for solving the time–space fractional‐order wave equation. The time‐fractional derivative is described in the conformal sense, whereas the space‐fractional derivative is given in the Caputo sense. The investigated technique is based on the third‐kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third‐kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non‐standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.

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Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

Shivanian

2015

Math Methods in App Sciences

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H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.

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Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Cited by 274 publications

References 33 publications

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

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

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

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