Variable-order fractional derivatives with singular kernels

2020 ◽  
pp. 311-348
Author(s):  
Xiao-Jun Yang ◽  
Feng Gao ◽  
Yang Ju
Keyword(s):  
Fractional Derivatives ◽  
Variable Order ◽  
Singular Kernels ◽  
Variable Order Fractional Derivatives

A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Behrouz Parsa Moghaddam ◽  
José António Tenreiro Machado
Keyword(s):  
Differential Equation ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Spline Interpolation ◽  
Computational Approach ◽  
Variable Order ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Variable Order Fractional Derivatives

AbstractA new computational approach for approximating of variable-order fractional derivatives is proposed. The technique is based on piecewise cubic spline interpolation. The method is extended to a class of nonlinear variable-order fractional integro-differential equation with weakly singular kernels. Illustrative examples are discussed, demonstrating the performance of the numerical scheme.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

scholarly journals Hamilton’s principle with variable order fractional derivatives

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Teodor Atanackovic ◽  
Stevan Pilipovic
Keyword(s):  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Variable Order ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Special Cases ◽  
Admissible Functions ◽  
Variable Order Fractional Derivatives

AbstractWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler-Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation. Necessary conditions for the existence of the minimizer are obtained. They imply various known results in a special cases.

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