Machine Learning for Modeling the Singular Multi-Pantograph Equations.

Area of Science:

• Computational Mathematics
• Intelligent Systems
• Numerical Analysis

Background:

• Singular multi-pantograph differential equations (SMDEs) present significant challenges in numerical analysis.
• Existing methods often struggle with accuracy, stability, and computational efficiency for SMDEs.
• Intelligent systems and machine learning offer potential for novel solution approaches.

Purpose of the Study:

• To introduce a new, accurate, and efficient method for solving singular multi-pantograph differential equations (SMDEs).
• To formulate a type-2 fuzzy logic system (T2-FLS) for approximating SMDE solutions.
• To optimize the T2-FLS using the square root cubature Kalman filter (SCKF) for improved performance.

Main Methods:

• Development of a type-2 fuzzy logic system (T2-FLS) for approximating solutions to SMDEs.
• Optimization of T2-FLS rules using the square root cubature Kalman filter (SCKF) to minimize a defined fineness function.
• Mathematical proof of estimation error stability and boundedness using Lyapunov theorem.

Main Results:

• The proposed T2-FLS optimized by SCKF provides accurate approximate solutions for SMDEs.
• The method demonstrates rapid convergence and a reduced computational cost compared to existing approaches.
• Statistical examinations confirm the algorithm's accuracy and robustness.

Conclusions:

• The novel T2-FLS approach offers an effective and efficient solution for SMDEs.
• The integration of SCKF optimization enhances the performance and reliability of fuzzy logic systems for differential equations.
• This research contributes a valuable tool for computational mathematics and intelligent systems applications.