High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

Chen, S.; Jiang, Xiaoyun; Liu, F.; Turner, Ian

doi:10.1016/j.cam.2014.09.028

Cited by 40 publications

(24 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, there have been attempts to present analytical methods which considerably approximate the exact solutions of these equations, such as the Adomian decomposition [17], Variational iteration [18] and Homotopy perturbation [19] methods. We refer also to recent numerical methods for solving FDEs [20,21,22,23,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

Bhrawy

Zaky

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

A shifted fractional-order Jacobi orthogonal functions (SFJFs) are proposed based on the definition of the classical Jacobi polynomials. We derive a new formula expressing explicitly any Caputo fractional-order derivative of SFJFs in terms of SFJFs themselves. We also propose a shifted fractional-order Jacobi tau technique based on the derived fractional-order derivative formula of SFJFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). A shifted fractional-order Jacobi pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order Jacobi pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on SFJFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

show abstract

Section: Introductionmentioning

confidence: 99%

Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

Bhrawy

Zaky

2016

Applied Mathematical Modelling

View full text Add to dashboard Cite

show abstract

“…In the past few decades, high and rapid growing attention related to partial differential equations (PDEs) which contain fractional derivatives and integrals occurred due to their important application in modeling of many anomalous diffusion processes. Fractional partial differential equations (FPDEs) are excellent instrument, bringing into a broader paradigm concepts of science and engineering, such as fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Consequently, the solution of FPDEs represents nowadays a vigorous research area for scientists and finding approximate and exact solutions to FPDEs is an important task.…”

Section: Introductionmentioning

confidence: 99%

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

et al. 2017

View full text Add to dashboard Cite

The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.

show abstract

“…Letting the skewness parameter θ = 0, one gets C − (α, θ = 0) = C + (α, θ = 0) = 1 2 cos π 2 α , α = 1, which is just the Riesz derivative (1). For most fractional differential equations, to obtain the analytical solutions are not easy even impossible, so many researchers have to solve fractional differential equations by using various kinds of numerical methods [1,6,7,10,11,28,29,30,31,32,33,34]. In particular, as for Riesz spatial fractional differential equations, the key issue is how to approximate the Riesz derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that (7) holds for homogeneous initial conditions. The coefficients ̟ (α) p,ℓ satisfy the following equation,…”

Section: Introductionmentioning

confidence: 99%

High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions

Ding

2016

J Sci Comput

View full text Add to dashboard Cite

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new highorder formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order O(τ 2 + h 2 ), where τ and h are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

show abstract

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

Cited by 40 publications

References 34 publications

Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions

Contact Info

Product

Resources

About