Double Ara–Shehu Transform for Partial Integro-Differential Equations

Abstract

We study a mixed two-variable integral transform obtained by applying the ARA transform in one variable and the Shehu transform in the other. We identify the transform explicitly as a normalized and reparameterized double Laplace transform, establish boundedness, uniqueness, and an inversion formula on weighted function spaces, and derive concise rules for partial derivatives and double convolution. The formulation keeps ARA normalization and the Shehu scale parameter visible, which permits direct use of one-dimensional transform data for mixed initial-boundary conditions. A general algebraic representation is obtained for a class of second-order linear partial differential equations. Representative applications to a partial differential equation, a nonlinear integral equation, and a Volterra integro-differential equation illustrate the scope and limitations of the method.