A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term

Abbaszadeh, Mostafa; Mohebbi, Akbar

doi:10.1016/j.camwa.2013.08.010

Cited by 58 publications

(47 citation statements)

References 35 publications

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“…Consider the following linear TFMASDE :

\begin{array}{l} \frac{∂u (x, y, t)}{∂t} = ((), \frac{\partial^{1 - α}}{\partial t^{1 - α}} + \frac{\partial^{1 - β}}{\partial t^{1 - β}}) ([], \frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial x^{2}}) + g (u, x, y, t), 0 < α, β < 1, \end{array}

where

\begin{array}{l} g (u, x, y, t) = \sin (x + y) ([], (1 + α + β) t^{α + β} + \frac{2 normalΓ (2 + α + β)}{normalΓ (1 + 2 α + β)} t^{2 α + β} + \frac{2 normalΓ (2 + α + β)}{normalΓ (1 + α + 2 β)} t^{α + 2 β}), \end{array}

with the initial condition

u (x, y, 0) = 0, (x, y) \in Ω,

where Ω is the computational domain as shown in Figure (right plan). The exact solution is

u (x, y, t) = t^{1 + α + β} \sin (...

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In this paper, we apply the DRBEM for the numerical solution of TFPDEs. We consider two classes of TFPDEs: The time‐fractional modified anomalous subdiffusion equation (TFMASDE)

\begin{matrix} \frac{∂u (x, y, t)}{∂t} = (A \frac{\partial^{1 - α}}{\partial t^{1 - α}} + ℬ \frac{\partial^{1 - β}}{\partial t^{1 - β}}) [\frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial y^{2}}] + g (u, x, y, t), \\ (x, y) \in Ω, 0 < α, β < 1, t > 0, \end{matrix}

along with initial and boundary conditions

u (x, y, 0) = ϕ (x, y), (x, y) \in normalΩ,

u (x, y, t) = φ (x, y), (x, y) \in normalΓ, t > 0,

where

scriptA

and

scriptℬ

are positive constants and the nonlinear source term has the first‐order continuous partial derivative

\frac{∂g (u, x, y, t)}{∂t}

(Γ is the boundary enclosing Ω). The symbols

<...>

…”

Section: Introductionmentioning

confidence: 99%

“…They proved the stability and convergence of proposed methods. The authors of proposed a fourth‐order compact finite difference scheme for the solution of TFMASDE. They proved the stability and convergence of proposed scheme using the Fourier analysis. The time‐fractional convection–diffusion equation (TFCDE) with constant coefficient :

\begin{matrix} \frac{\partial^{α} u (x, y, t)}{\partial t^{α}} + p_{1} \frac{∂u (x, y, t)}{∂x} + p_{2} \frac{∂u (x, y, t)}{∂y} = \frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial y^{2}} + g (x, y, t), \\ (x, y) \in Ω, 1 < α \leq 2, t > 0, \end{matrix}

along with initial and boundary conditions

u (x, y, 0) = ϕ (x, y), ψ (x, y) = \frac{∂u (x, y, t)}{∂t} 1.19em |_{t = 0},

u (x, y, t) = φ (x, y), (x, y) \in normalΓ, t > 0,

where the constant p 1 and p 2 are the velocities of convection, and

D_{t}^{α} u (x, y, t) =

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

Dehghan

Safarpoor

2016

Math Methods in App Sciences

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In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two-dimensional linear and nonlinear time-fractional modified anomalous subdiffusion equations and time-fractional convection-diffusion equation. The fractional derivative of problems is described in the Riemann-Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique.

show abstract

“…Consider the following linear TFMASDE :

\begin{array}{l} \frac{∂u (x, y, t)}{∂t} = ((), \frac{\partial^{1 - α}}{\partial t^{1 - α}} + \frac{\partial^{1 - β}}{\partial t^{1 - β}}) ([], \frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial x^{2}}) + g (u, x, y, t), 0 < α, β < 1, \end{array}

where

\begin{array}{l} g (u, x, y, t) = \sin (x + y) ([], (1 + α + β) t^{α + β} + \frac{2 normalΓ (2 + α + β)}{normalΓ (1 + 2 α + β)} t^{2 α + β} + \frac{2 normalΓ (2 + α + β)}{normalΓ (1 + α + 2 β)} t^{α + 2 β}), \end{array}

with the initial condition

u (x, y, 0) = 0, (x, y) \in Ω,

where Ω is the computational domain as shown in Figure (right plan). The exact solution is

u (x, y, t) = t^{1 + α + β} \sin (...

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In this paper, we apply the DRBEM for the numerical solution of TFPDEs. We consider two classes of TFPDEs: The time‐fractional modified anomalous subdiffusion equation (TFMASDE)

\begin{matrix} \frac{∂u (x, y, t)}{∂t} = (A \frac{\partial^{1 - α}}{\partial t^{1 - α}} + ℬ \frac{\partial^{1 - β}}{\partial t^{1 - β}}) [\frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial y^{2}}] + g (u, x, y, t), \\ (x, y) \in Ω, 0 < α, β < 1, t > 0, \end{matrix}

along with initial and boundary conditions

u (x, y, 0) = ϕ (x, y), (x, y) \in normalΩ,

u (x, y, t) = φ (x, y), (x, y) \in normalΓ, t > 0,

where

scriptA

and

scriptℬ

are positive constants and the nonlinear source term has the first‐order continuous partial derivative

\frac{∂g (u, x, y, t)}{∂t}

(Γ is the boundary enclosing Ω). The symbols

<...>

…”

Section: Introductionmentioning

confidence: 99%

\begin{matrix} \frac{\partial^{α} u (x, y, t)}{\partial t^{α}} + p_{1} \frac{∂u (x, y, t)}{∂x} + p_{2} \frac{∂u (x, y, t)}{∂y} = \frac{\partial^{2} u (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} u (x, y, t)}{\partial y^{2}} + g (x, y, t), \\ (x, y) \in Ω, 1 < α \leq 2, t > 0, \end{matrix}

along with initial and boundary conditions

u (x, y, 0) = ϕ (x, y), ψ (x, y) = \frac{∂u (x, y, t)}{∂t} 1.19em |_{t = 0},

u (x, y, t) = φ (x, y), (x, y) \in normalΓ, t > 0,

where the constant p 1 and p 2 are the velocities of convection, and

D_{t}^{α} u (x, y, t) =

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

Dehghan

Safarpoor

2016

Math Methods in App Sciences

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show abstract

“…In the literature, many works have been devoted to this study [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Based on the Grünwald-Letnikov approximation, Yuste and Acedo [5] constructed an explicit difference scheme for fractional diffusion equations, and investigated the stability using the von Neumann method.…”

Section: Introductionmentioning

confidence: 99%

“…Then a high order unconditionally stable scheme for this equation was derived in [12]. A modified anomalous time fractional sub-diffusion equation with a nonlinear source term was studied in [15,16], where a finite difference scheme of first order temporal accuracy and fourth order spatial accuracy was proposed. By using the Fourier transform, a compact difference scheme with O τ 2 + h 4 was then constructed in [17], the stability and convergence are shown by the energy method.…”

Section: Introductionmentioning

confidence: 99%

A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

2015

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In this paper, a compact finite difference scheme with global convergence order O τ 2−α + h 4 is derived for fractional sub-diffusion equations with the spatially variable coefficient subject to Neumann boundary conditions. The difficulty caused by the variable coefficient and the Neumann boundary conditions is overcome by subtle decomposition of the coefficient matrices. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. The theoretical results are supported by numerical experiments.

show abstract

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

Dehghan

Safarpoor

2015

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs . We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation.

show abstract

A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term

Cited by 58 publications

References 35 publications

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

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