A numerical approach for solving a class of variable-order fractional functional integral equations

Keshi, Farzad Khane; Moghaddam, Behrouz Parsa; Aghili, Arman

doi:10.1007/s40314-018-0604-8

Cited by 42 publications

(23 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More recently, Dabiri et al (2018) proposed a technique for the design of variable-order fractional proportional-integral-derivative (VFPID) controllers for linear dynamical systems. The time-fractional models are recently studied in the following works: Zaky (2018), Keshi et al (2018) and Moghaddam et al (2018). The semi-implicit time-scheme is considered to be low-cost and very attractive for their programming simplicity and the purpose of this paper was to develop a stable and efficient scheme, that, from the theoretical and the numerical standpoint, is found to be stable, reliable and feasible for the presented time-fractional nonlinear model.…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence analysis of a semi-implicit fractional FEM-scheme for non-Newtonian fluid flows of polymer aqueous solutions with fractional time-derivative

et al. 2020

View full text Add to dashboard Cite

In this paper, a fully discrete fractional semi-implicit FEM-scheme is developed to study the non-Newtonian fluid flow of polymer aqueous solutions in a two-dimensional domain using Caputo's fractional time-derivative. The fractional-time scheme is coupled with the spatial discretization using the Galerkin finite element method. The existence and uniqueness results were obtained for the weak discrete solution and with the help of a newly introduced trilinear form. The convergence and stability of the elaborated numerical scheme are demonstrated for certain criteria and temporal step bounds are obtained. Numerical simulations are elaborated and the effects of various parameters of the discrete system are investigated. The obtained results are analyzed and application to the lid-driven cavity problem in complex geometry is presented. Keywords Semi-implicit fractional scheme • Non-Newtonian fluid • Temporal step bounds • Polymer aqueous solutions • Fractional time-derivative Mathematics Subject Classification 65N12 Communicated by José Tenreiro Machado.

show abstract

Section: Introductionmentioning

confidence: 99%

Stability and convergence analysis of a semi-implicit fractional FEM-scheme for non-Newtonian fluid flows of polymer aqueous solutions with fractional time-derivative

et al. 2020

View full text Add to dashboard Cite

show abstract

“…A finite difference scheme to find the solution of VOFDE and convergence analysis is used in article (Lin et al 2009). Some latest and updated methods are discovered to find the numerical solution of VOFDE, viz., finite difference method (Moghaddam and Machado 2017b;Moghaddam and Mostaghim 2017), B-linear spline method (Machado and Moghaddam 2018), cubic spline method (Moghaddam and Machado 2017a, c), integro quadratic spline interpolations Keshi et al 2018) and spectral method (Moghaddam et al 2019). Analytic solution of variable-order differential equation is given in article Malesza et al (2019).…”

Section: Introductionmentioning

confidence: 99%

Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations

2019

View full text Add to dashboard Cite

In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space-time fractional variable-order non-linear reaction-diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variableorder differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction-diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.

show abstract

“…Due to the variable-order fractional derivatives, a few papers deal with the existence of exact solutions to such equations (Zhang 2018). Recently, several numerical methods are proposed for solving variable-order fractional differential and integral equations based on the finite difference (Moghaddam and Machado 2016), B-linear spline (Machado and Moghaddam 2018), Cubic spline (Moghaddam and Machado 2017a, b;Yaghoobi et al 2017), and integro quadratic spline interpolations (Moghaddam et al 2017;Keshi et al 2018). Moghaddam and Machado developed finite-difference approach schemes for variableorder fractional operators, and applied the method to approximate variable-order fractional Ricatti and variable-order fractional Emden-Fowler equations (Moghaddam and Machado 2016).…”

Section: Introductionmentioning

confidence: 99%

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

Yan

Han

et al. 2019

Comp. Appl. Math.

View full text Add to dashboard Cite

In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach's fixed-point theorem and the Gronwall-Bellman lemma. Third, we employ the Legendre-Gauss and Jacobi-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces L 2 [0, 1] and L ∞ [0, 1]. Finally, numerical results are given to support the theoretical conclusions. Keywords Spectral collocation method • Variable fractional order • Initial value problem • Convergence analysis Mathematics Subject Classification 65L60 • 41A05 • 41A10 • 41A25 Communicated by José Tenreiro Machado.

show abstract

A numerical approach for solving a class of variable-order fractional functional integral equations

Cited by 42 publications

References 49 publications

Stability and convergence analysis of a semi-implicit fractional FEM-scheme for non-Newtonian fluid flows of polymer aqueous solutions with fractional time-derivative

Stability and convergence analysis of a semi-implicit fractional FEM-scheme for non-Newtonian fluid flows of polymer aqueous solutions with fractional time-derivative

Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

Contact Info

Product

Resources

About