Inhomogeneous Whittaker Equation with Initial and Boundary Conditions

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions Zi κ , μ f ( x ) , along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of Zi κ , μ f ( x ) are rigorously investigated, and explicit closed-form expressions for Zi κ , μ f ( x ) and its derivatives are derived in terms of Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) , confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling.