The double natural generalized Laplace transform decomposition method (DNGLTDM) is proposed as a new approach for solving singular linear and non-linear two-dimensional Boussinesq equations involving fractional partial derivatives. This method combines the decomposition technique with the double natural generalized Laplace transform to construct solutions in the form of rapidly convergent infinite series that approximate the exact solutions. The paper presents a detailed study of the fundamental properties of the transform, including the convolution theorem, the periodicity theorem, the treatment of partial derivatives with non-constant coefficients, and partial fractional derivatives. In addition, the convergence of the obtained series solutions and the associated error analysis are thoroughly investigated. Finally, two illustrative examples are provided to demonstrate the accuracy and effectiveness of the DNGLTDM, one of which considers a problem of partial fractional order.
Eltayeb et al. (Mon,) studied this question.