The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for \(F^{\alpha }\) -integrable functions, introduces local fractal integrable functions and fractal distribution functionals, defines the dual space of a fractal function space, proves completeness for \(F^{\alpha }\) -differentiable function spaces, defines Fractal Sobolev spaces, and introduces fractal gradian and fractal Laplace operators on fractal Hilbert spaces. It also presents criteria for the existence of unique solutions to fractal differential equations.
