The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients

Zhou, Fengying; Xu, Xiaoyong

doi:10.1016/j.amc.2016.01.029

Cited by 51 publications

(52 citation statements)

References 43 publications

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“…The fractional differential equations model areal problem in life that needs a solution. Therefore, there are many different numerical methods that solve these equations, such as the predictor‐corrector method, Legendre wavelets, Legendre spectral method, Legendre collocation method, pseudo‐spectral scheme, Haar wavelet collocation method, Chebyshev spectral methods,() other techniques,() and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.

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Section: Introductionmentioning

confidence: 99%

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Agarwal

2019

Math Methods in App Sciences

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“…Figure displays approximate solution and absolute error between the exact solution and the approximate solution

(| u (x, t) - u_{\hat{m}, \tilde{m}} (x, t) |)

for

k = k' = 2

and

M = M' = 3

. Table demonstrates absolute error for some different points of

[0, 1] \times [0, 1] .

In Table , we compare our method for some different values of

α, ν, β

and

\hat{m} = \tilde{m} = 6

together with the results obtained by using Legendre wavelet method for

\hat{m} = 12.

Also, The graph of approximate solutions for

k = k' = 1, x = 0.2

and different values of

M, M'

with exact solution is shown in Figure Example Consider the following time‐fractional convection diffusion equation of order

normalν

()

\frac{\partial^{2} u (x, t)}{\partial x^{2}} + \frac{\partial^{normalν} u (x, t)}{\partial t^{normalν}} + x \frac{\partial u (x, t)}{\partial x} = f (x, t), 0 \leq x,

…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Table gives the absolute errors of the approximate solutions of the present method and Ref. for

ν = 0.7

and

ν = 0.9

at different points. In Table we compare the absolute error of our method together with the results obtained by using the Haar wavelet method , Sinc‐Legendre collocation method and third kind Chebyshev wavelet method .…”

Section: Numerical Examplesmentioning

confidence: 99%

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

Rahimkhani

Ordokhani

2018

Numerical Methods Partial

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In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.

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“…In this paper, the Grunwald-Letnikov representation was used for Riemann-Liouville derivative, and the stability of the scheme (based on the finite difference method and cubic trigonometric B-spline) was discussed. Zhu and Nie [23] obtained a scheme based on exponential B-spline and wavelet operational matrix method for the time fractional convection-diffusion problem with variable coefficients. Yaseen et al [24] constructed a finite difference method for solving time fractional diffusion problem via trigonometric B-spline.…”

Section: Introductionmentioning

confidence: 99%

A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

et al. 2018

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In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection-diffusion equation. The time fractional derivative is estimated using Caputo's formulation, and the spatial derivatives are discretized using extended cubic B-spline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.

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The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients

Cited by 51 publications

References 43 publications

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

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