A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular...

Area of Science:

• Numerical analysis
• Computational mathematics
• Mathematical physics

Background:

• Volterra integro-partial differential equations with weakly singular kernels (VIPDEWSK) are crucial for modeling various physical phenomena.
• Existing methods may face challenges in computational efficiency and accuracy for these complex equations.

Purpose of the Study:

• To propose an efficient and accurate numerical method for solving VIPDEWSK.
• To utilize shifted Chebyshev polynomials of the fifth kind (SCPFK) for developing a matrix collocation approach.

Main Methods:

• A matrix collocation method is developed using SCPFK to create pseudo-operational matrices.
• This approach avoids explicit integration, enhancing computational speed.
• Error bounds are analyzed in a Chebyshev-weighted space to ensure approximation accuracy.

Main Results:

• The proposed method demonstrates effectiveness in solving VIPDEWSK.
• Computational efficiency is improved by avoiding direct integration.
• The accuracy of the approximation is validated through error bound analysis.

Conclusions:

• The matrix collocation method with SCPFK provides a robust and efficient technique for VIPDEWSK.
• The findings offer a valuable alternative to existing wavelet-based methods.
• The study contributes to the advancement of numerical solutions for complex differential equations.