In this paper, we investigate the exact solutions of several fractional-order Helmholtz equations using the homotopy perturbation transform method. We specify sufficient requirements for its convergence and provide error estimations. The homotopy perturbation transform method yields a quickly converging succession of solutions. Solutions for various fractional space derivatives are compared to present approaches and explained using figures. Appropriate parameter selection produces approximations identical to the exact answer. Test examples are provided to demonstrate the proposed approach's precision and competence. The results demonstrate that our system is appealing, user-friendly, dependable, and highly effective.

The current study employs a transformation-based analytical technique, namely Riccati modified extended simple equation method (RMESEM) to construct and examine soliton phenomena in a prominent $(2+1)$-dimensional mathematical model namely Nizhnik-Novikov-Veselov system (NNVS), which has potential applications in exponentially localized structure interactions. The suggested RMESEM uses a variable transformation to turn the desired NNVS into a nonlinear ordinary differential equation (NODE). The resulting NODE is then assumed to have a closed-form solution, converting it into an algebraic system of equations. When the resulting algebraic system is dealt with RMESEM's strategy using Maple, a range of dark and bright soliton solutions in the form of rational, exponential, periodic, hyperbolic and rational-hyperbolic functions are revealed. Some 3D, contour and 2D graphs are plotted for visual representations of these soliton solutions that demonstrate their versatility. The findings deepen our understanding of the NNVS's dynamics, shedding light on its behavior and potential uses.