Further characteristics for certain newly formed solutions for two significant mathematical models by utilization of an efficient semi-analytic method

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Abstract This study analyses the details of two mathematical models: the (2+1)-dimensional nonlinear time-fractional HirotaMaccari (HM) model, which includes a novel fractional operator called the β-derivative, and the (3+1)-dimensional Korteweg-de Vries-type equation. Two different forms of the wave transformations are utilized to convert the studied models into nonlinear differential equations. The improved version of the generalized Kudryashov method (IGKhM), known for its efficiency, is applied for the first time to investigate studied models. Various computational programs provide essential visual representations, making them invaluable in academic and practical applications. The construction of traveling waves and singular solitons is achieved using exponential and hyperbolic trigonometric functions, which exhibit characteristics of darkness, brightness, and mixed states. In two- and three-dimensional representations, numerical simulations of the outcomes are incorporated to better express and understand the physical significance of the results. Every solution obtained is new and distinctive compared to previous studies. Additionally, the results are individually substituted into the corresponding equations and satisfy the conditions.