scholarly journals ON NUMERICAL SOLUTIONS OF LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 45 (1) ◽  
pp. 35-47
Author(s):  
YLLDRITA SALIHI ◽  
GJORGJI MARKOSKI ◽  
ALEKSANDAR GJURCHINOVSKI
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
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.

scholarly journals Memory effects on the proliferative function in the cycle-specific of chemotherapy

2021 ◽  
Author(s):  
Najma Ahmed ◽  
Dumitru Vieru ◽  
Fiazud Din Zaman
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Classical Model ◽  
Cell Mass ◽  
Time Derivative ◽  
Fractional Model ◽  
Fractional Differential ◽  
Mathcad Software

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

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