scholarly journals Error Analysis for a Fractional-Derivative Parabolic Problem on Quasi-Graded Meshes using Barrier Functions

2020 ◽  
Vol 58 (2) ◽  
pp. 1217-1238 ◽  
Author(s):  
Natalia Kopteva ◽  
Xiangyun Meng
Keyword(s):  
Error Analysis ◽  
Fractional Derivative ◽  
Parabolic Problem ◽  
Barrier Functions ◽  
Graded Meshes

A Numerical Scheme and an Error Analysis for a Class of Fractional Optimal Control Problems

2009 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Performance Index ◽  
Numerical Solutions ◽  
Fractional Power ◽  
The State ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Linear Algebraic Equations

There has been a growing interest in recent years in the area of Fractional Optimal Control (FOC). In this paper, we present a formulation for a class of FOC problems, in which a performance index is defined as an integral of a quadratic function of the state and the control variables, and a dynamic constraint is defined as a Fractional Differential Equation (FDE) linear in both the state and the control variables. The fractional derivative is defined in the Caputo sense. In this formulation, the FOC problem is reduced to a Fractional Variational Problem (FVP), and the necessary differential equations for the problems are obtained using the recently developed theories for FVPs. For the numerical solutions of the problems, a direct approach is taken in which the solutions are approximated using a truncated Fractional Power Series (FPS). An error analysis is also performed. It is demonstrated that the solution converges from above in the sense that the value of the approximate performance index is always higher than the optimum performance index. An expression for the error in the performance index is also given. The application of a FPS and an optimality criterion reduces the FOC to a set of linear algebraic equations which are solved using a linear solver. It is demonstrated numerically that the solution converges as the number of terms in the series increases, and the approximate solution approaches to the analytical solution as the order of the fractional derivative approaches to an integer order derivative. Numerical results are presented to demonstrate the performance of the Formulation.

scholarly journals Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Yin Yang ◽  
Yunqing Huang
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Integrodifferential Equations ◽  
Collocation Methods ◽  
Volterra Type ◽  
Numerical Examples ◽  
Rigorous Error ◽  
Spectral Collocation Methods ◽  
Theoretical Results

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially inL∞norm and weightedL2-norm. The numerical examples are given to illustrate the theoretical results.

scholarly journals Numerical Solution of Fuzzy Fractional Pharmacokinetics Model Arising from Drug Assimilation into the Bloodstream

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Ali Ahmadian ◽  
Norazak Senu ◽  
Farhad Larki ◽  
Soheil Salahshour ◽  
Mohamed Suleiman ◽  
...  
Keyword(s):  
Error Analysis ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Error Estimation ◽  
Upper Bound ◽  
Tau Method ◽  
Algebraic Equations ◽  
Special Cases ◽  
The Absolute ◽  
Fractional Pharmacokinetics

We propose a Jacobi tau method for solving a fuzzy fractional pharmacokinetics. This problem can model the concentration of the drug in the blood as time increases. The proposed approach is based on the Jacobi tau (JT) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The method reduces the solution of the problem to the solution of a system of algebraic equations. Error analysis included the fractional derivative error estimation, and the upper bound of the absolute errors is introduced for this method.

Sign in / Sign up

Export Citation Format

Share Document