Boundary Effect in Accuracy Estimate of the Grid Method for Solving Fractional Differential Equations

2019 ◽  
Vol 55 (1) ◽  
pp. 65-80 ◽  
Author(s):  
V. L. Makarov ◽  
N. V. Mayko
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Boundary Effect ◽  
Grid Method ◽  
Accuracy Estimate ◽  
Fractional Differential

The Boundary Effect in the Accuracy Estimate for the Grid Solution of the Fractional Differential Equation

2019 ◽  
Vol 19 (2) ◽  
pp. 379-394 ◽  
Author(s):  
Volodymyr Makarov ◽  
Nataliya Mayko
Keyword(s):  
Dirichlet Boundary ◽  
Boundary Effect ◽  
Grid Method ◽  
Interpolation Polynomial ◽  
Accuracy Estimate ◽  
Accuracy Order ◽  
Liouville Fractional Derivative ◽  
Grid Solution ◽  
Fractional Differential ◽  
The Impact

AbstractA grid method for solving the first boundary value problem for ordinary and partial differential equations with the Riemann–Liouville fractional derivative is justified. The algorithm is based on using Green’s function, the Fredholm integral equation, and the Lagrange interpolation polynomial. The impact of the Dirichlet boundary condition on the accuracy of the approximate solution is revealed and quantitatively described through the weight assessment. All the estimates provide clear evidence that the accuracy order of the grid method is higher near the boundary of the domain than it is in the inner nodes of the mesh set.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

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