A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work p...

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Bibliographic Details
Published in:Fractal and fractional 2024-11, Vol.8 (11), p.647
Main Authors: Abdelfattah, Waleed Mohammed, Ragb, Ola, Salah, Mohamed, Mohamed, Mokhtar
Format: Article
Language:English
Subjects:
Accuracy
Boundary value problems
Calculus
Control theory
Convergence
Differential equations
differential quadrature technique
discrete singular convolution
Fractional calculus
fractional derivative
fractional Lorenz system
fractional Riccati
generalized Caputo
Hilbert space
Iterative methods
Lorenz system
Mathematical analysis
Methods
Nonlinear systems
Nonlinearity
Numerical analysis
Operators (mathematics)
Polynomials
Quadratures
Riccati equation
Shape functions
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Summary:The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract8110647