Solving Fractional Black-Scholes and Navier-Stokes Equations via a New $\frac{t^{\varrho}}{\varrho}$-Integral Transform and Residual Power Series

Document Type : Research Article

Authors

1 Department of Mathematics‎, ‎Daykondi University‎, ‎Nili‎, ‎Afghanistan

2 ‎Department of Mathematical Sciences‎, ‎Yazd University‎, ‎Yazd‎, ‎Iran

Abstract

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.

Keywords

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Volume 11, Issue 1
January 2026
Pages 1-31
  • Receive Date: 05 October 2025
  • Revise Date: 29 December 2025
  • Accept Date: 02 February 2026
  • Publish Date: 23 February 2026