α-Analytic solutions of some linear fractional differential equations with variable coefficients

2007 ◽  
Vol 187 (1) ◽  
pp. 239-249 ◽  
Author(s):  
A.A. Kilbas ◽  
M. Rivero ◽  
L. Rodríguez-Germá ◽  
J.J. Trujillo
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytic Solutions ◽  
Variable Coefficients ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay

2020 ◽  
Vol 21 (6) ◽  
pp. 539-547
Author(s):  
Ali Konuralp ◽  
Sercan Öner
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Euler Polynomials ◽  
The Matrix ◽  
Fractional Differential ◽  
Equations With Delay ◽  
Linear Fractional Differential Equations

AbstractIn this study, a method combined with both Euler polynomials and the collocation method is proposed for solving linear fractional differential equations with delay. The proposed method yields an approximate series solution expressed in the truncated series form in which terms are constituted of unknown coefficients that are to be determined according to Euler polynomials. The matrix method developed for the linear fractional differential equations is improved to the case of having delay terms. Furthermore, while putting the effect of conditions into the algebraic system written in the augmented form in which the coefficients of Euler polynomials are unknowns, the condition matrix scans the rows one by one. Thus, by using our program written in Mathematica there can be obtained more than one semi-analytic solutions that approach to exact solutions. Some numerical examples are given to demonstrate the efficiency of the proposed method.

Systems of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Unique Solution ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Right ◽  
Linear Fractional Differential Equations

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

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