Linear Differential Equations of Fractional Order with Recurrence Relationship

2021 ◽  
Vol 7 (1) ◽  
pp. 1-21
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Linear Differential Equations ◽  
Recurrence Relationship ◽  
Linear Differential

scholarly journals New Method for Solving Linear Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
S. Z. Rida ◽  
A. A. M. Arafa
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Linear Differential Equations ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

On Theory of Systems of Fractional Linear Differential Equations

2005 ◽  
Author(s):  
Blanca Bonilla ◽  
Margarita Rivero ◽  
Juan J. Trujillo
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Linear Differential Equations ◽  
Constant Matrix ◽  
Natural Connection ◽  
Fractional Differential ◽  
Linear Differential ◽  
Matrix Exponential Function ◽  
System 1

This paper is a continuation of a previous one dedicated to establishing a general theory of linear fractional differential equations. This paper deals with the study of linear systems of fractional differential equations such as the following: Y¯(α=A(x)Y¯+B¯(x)(1) where DαY ≡ Y(α is the Riemann-Liouville or Caputo fractional derivative of order α(0 < α ≤ 1), and: A(x)=a11(x)...a1n(x)…….....…….....…….....an1(x)...ann(x);B¯(x)=b1(x)…….…….…….bn(x)(2) are matrices of known real functions. We introduce a generalisation of the usual matrix exponential function and the Green function of fractional order, in connection with the Mittag-Leffler type functions. This function allows us to obtain an explicit representation of the general solution to system (1) when A is a constant matrix, in a way analogous to the usual case. Some applications of this theory are presented through the natural connection between system (1) and linear differential equations of fractional order. Some new models are presented.

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