Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

Background:

• Linear delay integral equations (LDIEs) are prevalent in various scientific and engineering fields.
• Solving LDIEs analytically can be challenging due to the presence of delays.
• Existing numerical methods may have limitations in terms of accuracy or computational efficiency.

Purpose of the Study:

• To develop and present a novel computational Haar wavelet collocation technique for solving linear delay integral equations.
• To demonstrate the applicability of the proposed method to delay Fredholm, Volterra, and Volterra-Fredholm integral equations.
• To validate the accuracy and convergence of the developed technique through numerical examples.

Main Methods:

• A computational Haar wavelet collocation technique is employed.
• The technique transforms the delay integral equations into a system of algebraic equations.
• The resulting algebraic system is solved using the Gauss elimination technique.
• MATLAB software is utilized for the implementation of the algorithms.

Main Results:

• The proposed Haar wavelet collocation method effectively solves linear delay integral equations.
• Numerical examples show good agreement between the computed and exact solutions.
• The maximum absolute and root mean square errors are analyzed.
• A convergence rate of approximately 2 is achieved with varying numbers of collocation points.

Conclusions:

• The computational Haar wavelet collocation technique provides an accurate and efficient approach for solving linear delay integral equations.
• The method is versatile, applicable to different types of delay integral equations.
• The demonstrated convergence rate supports the reliability of the proposed numerical technique.