Resolvent Kernel and Haar Wavelet Techniques for Solving Coupled of Fredholm Integro-Differential Equations

This study focuses on the fredholm integro-differential equations and is frequently found in areas of applied mathematics, physics, and engineering fields. to address these systems and obtain exact of solutions under suitable conditions, we propose a novel analytical approach. we choose the haar wavelet colocation method for processing due to its simplicity, effectiveness, and ability to handle non-smooth solutions. the integral terms in these equations are determined using the trapezoidal rule, which effectively strikes a balance between accuracy and computational efficiency. these results are compared with analytical solutions. the comparisons show that the suggested strategy yields highly accurate results and offers a solid framework for solving the fredholm integro-differential equations.