A solution to the fundamental linear fractional order differential equation

Area of Science:

• Electrical Engineering
• Applied Mathematics
• Fractional Calculus

Background:

• Fractional order differential equations are increasingly used to model complex systems.
• Standard methods often lack generalized solutions for these equations.
• Understanding the behavior of fractional systems is crucial for advanced engineering applications.

Purpose of the Study:

• To provide a general solution for the linear fractional order differential equation cdtqx(t) + ax(t) = bu(t).
• To introduce and define the F-function as a generalized impulse response.
• To explore the implications of the F-function for fractional system analysis.

Main Methods:

• Solving the fractional differential equation using series expansion.
• Defining the F-function as the resulting impulse response.
• Analyzing the complex plane behavior of the F-function and its relation to fractional poles.
• Applying the theory to a specific example of a lossy transmission line.

Main Results:

• The impulse response is presented as a series, termed the F-function.
• The F-function is shown to be a generalization of the standard exponential function.
• The F-function establishes a basis for defining a qth order fractional pole.
• Complex plane analysis reveals key characteristics of fractional system dynamics.

Conclusions:

• The F-function offers a unified approach to solving linear fractional order differential equations.
• This work extends the concept of poles to fractional order systems.
• The presented theory and example demonstrate the utility of the F-function in analyzing fractional systems like lossy transmission lines.