Bastola Damped Infinite Derivative Equation: Convergent Operator Formulation and Applications

We introduce the Bastola Damped Infinite Derivative Equation (BDIDE): y(x) = sum from n=1 to infinity of [y^(n)(x) / (n!)^α], α > 0 and reformulate it as a linear operator eigenvalue problem: B_α y = y, where B_α = sum from n=1 to infinity of [D^n / (n!)^α], D = d/dx Factorial damping ensures convergence even for smooth non-analytic functions. Series solutions, recurrence relations, and nonlocal shifts are discussed. Applications to viscoelastic materials and memory systems demonstrate practical relevance. This provides a rigorous, original, and convergent formulation of infinite order differential equations. To the best of the author’s knowledge, no prior publication has formulated an infinite-order derivative operator with factorial weighting that ensures absolute convergence for smooth, non-analytic functions, nor introduced a dedicated transform (the Bastola Transform) to solve non-homogeneous equations systematically. All derivations, recurrence relations, and applications to viscoelastic stress-strain modeling are original.