Related Experiment Video
Updated: Jun 21, 2025
11:18
A Guide to Concentration Alternating Frequency Response Analysis of Fuel Cells
Published on: December 11, 2019
6.7K
Existence and uniqueness of neutral functional differential equations with sequential fractional operators
Rabah Debbar1, Hamid Boulares2, Abdelkader Moumen3
1University of 8 May 1945 Guelma, Guelma, Algeria.
Plos One
|July 16, 2024
Summary
This study proves solutions exist and are unique for neutral functional differential equations with sequential fractional orders using fixed-point theorems. Examples illustrate findings for these complex mathematical models.
Area of Science:
- Mathematics
- Fractional Calculus
- Differential Equations
Background:
- Neutral functional differential equations (NFDEs) are crucial in modeling complex systems.
- Sequential fractional orders introduce advanced dynamics.
- The [Formula: see text]-Caputo operator offers a novel approach to fractional differentiation.
Purpose of the Study:
- To investigate the existence and uniqueness of solutions for NFDEs with sequential fractional orders.
- To apply established fixed-point theorems to these specific types of equations.
- To explore the impact of an initial value integral condition.
Main Methods:
- Banach Fixed Point Theorem (BFPT)
- Nonlinear Leray-Schauder Fixed Point Theorem (SFPT)
- Krasnoselski Fixed Point Theorem (KFPT)
- Analysis of [Formula: see text]-Caputo fractional derivatives
Main Results:
- Existence and uniqueness of solutions are established for the considered NFDEs.
- The application of BFPT, SFPT, and KFPT successfully guarantees the desired results.
- Illustrative examples confirm the theoretical findings.
Conclusions:
- The study successfully demonstrates the existence and uniqueness of solutions for a class of fractional differential equations.
- Fixed-point theorems provide a robust framework for analyzing such equations.
- The findings contribute to the theoretical understanding of fractional calculus and differential equations.
Related Concept Videos
01:26Linear Approximation in Frequency Domain
88
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
88
01:26Transmission-Line Differential Equations
266
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
266
01:31Classification of Systems-II
138
Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
138
01:18Second Order systems II
96
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
96
01:24Difference Equation Solution using z-Transform
282
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
282
01:11Types of Responses of Series RLC Circuits
870
A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.
870