Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Integral Equations

Background:

• Mixed Volterra-Fredholm integral equations present significant computational challenges.
• Existing numerical methods may lack efficiency or accuracy for these complex equations.

Purpose of the Study:

• To introduce a novel numerical approach for solving mixed Volterra-Fredholm integral equations.
• To extend the method to one and two-dimensional cases.
• To analyze the convergence properties of the proposed algorithm.

Main Methods:

• A shifted Jacobi-Gauss collocation method is employed.
• The integral equations are reduced to a solvable system of algebraic equations.
• The technique utilizes shifted Jacobi-Gauss nodes for discretization.

Main Results:

• The proposed algorithm effectively solves mixed Volterra-Fredholm integral equations.
• The method is successfully extended to one and two-dimensional problems.
• Convergence analysis confirms exponential convergence, characteristic of spectral methods.

Conclusions:

• The shifted Jacobi-Gauss collocation method offers a powerful and accurate technique for solving mixed Volterra-Fredholm integral equations.
• The spectral nature of the algorithm ensures high precision and efficiency.
• This approach provides a robust tool for computational mathematics and related fields.