Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 1 2026 02 23 Solving Fractional Black-Scholes and Navier-Stokes Equations via a New $\frac{t^{\varrho}}{\varrho}$-Integral Transform and Residual Power Series 1 31 2069 10.22128/ansne.2026.3084.1159 EN Abbas Poya Department of Mathematics‎, ‎Daykondi University‎, ‎Nili‎, ‎Afghanistan Mohammad Ali Zirak Department of Mathematics‎, ‎Daykondi University‎, ‎Nili‎, ‎Afghanistan Mohammad Hossein Akrami ‎Department of Mathematical Sciences‎, ‎Yazd University‎, ‎Yazd‎, ‎Iran Journal Article 2025 10 05 This paper introduces a novel approach for solving two-dimensional time-fractional Navier-Stokes and Black-Scholes equations. The method integrates a new integral transform--based on a generalized power function of the form $\frac{t^{\varrho}}{\varrho}$-- with the residual power series method. This combined approach, termed the ``generalized integral transform residual power series method,'' utilizes the Katugampola fractional derivative in the Caputo sense. The convergence of the method is rigorously established, and its efficacy, accuracy, and precision are demonstrated through illustrative examples. The results highlight the method's potential for efficiently solving complex fractional partial differential equations across various scientific and engineering disciplines. https://ansne.du.ac.ir/article_2069_413a6fda792abd1e1ad14818da4ff2f2.pdf