Weber's Inhomogeneous Differential Equation with Initial and Boundary Conditions

Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions

“…Furthermore, we shall extend the definition of N W (a, x) and NW (a, x) to complex argument x = z (say). We shall demonstrate that NW (a, z) → 0 as z → ∞ for | arg(z) ≤ 1 2 π − δ (δ > 0), and also as a → ∞ uniformly in part of the right half plane that contains {z : (z) ≥ 2 √ a}. In contrast, all other solutions of (1.2) are exponentially large as z → ∞, and also as a → ∞, in at least three of the four quadrants.…”

“…In [1] series solutions for N W (a, x) for small x were derived, and various other representations were derived in the above references.…”

“…In [1] the authors state "The current work involves real arguments of the Weber functions. In the general analysis of Weber equation and associated functions with complex arguments, the introduction of a Nield-Kuznetsov type function is still challenging and requires further consideration."…”

http://www.theijes.com/papers/vol6-issue3/A0603010108.pdf

“…Computations of solution to (1) when the forcing function ) ( y f is a constant function of the independent variable have been obtained by Abu Zaytoon et al [6], who used computational procedures outlined in Gil et al, [5] to obtain solutions to initial and boundary value problems involving (1) with a constant forcing function f(y). In the work of Alzahrani et al [1], general solutions to the inhomogeneous equation (1) were expressed in terms of the parametric Nield-Kuznetsov integral functions of the first and second kinds, valid for constant and variable forcing functions, respectively.…”

Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

“…See also [2] for further properties in connection with physical applications. Solutions of the inhomogeneous Weber equation (1.2) play a role in the study of flow through porous layers [1], [14].…”