Numerical Solution of Differential Equations Using the Wavelet-Based Galerkin Method with Fibonacci Wavelets

Differential equations form the foundation of scientific theories that address numerous real-world physical challenges. Numerical methods enable the resolution of complex problems through relatively simple operations. A significant advantage of numerical methods, compared to analytical methods, is their ease of implementation on modern computers, allowing for quicker solutions. Galerkin's method belongs to a broader category of numerical techniques. Additionally, wavelet analysis represents a promising domain within applied and computational research. This paper establishes a wavelet-based Galerkin method for numerically solving differential equations, utilizing Fibonacci wavelets as trial functions. The proposed method yields results comparable to existing techniques and provides solutions that closely approximate exact answers for certain problems, thereby demonstrating its effectiveness and accuracy.