This study introduces a novel approach to modeling cancer tumor dynamics within a fractional framework, emphasizing the critical role of the net killing rate in determining tumor growth or decay. We explore a generalized cancer model where the net killing rate is considered across three domains: time-dependent, position-dependent, and concentration-dependent. The primary objective is to derive an analytical solution for time-fractional cancer models using the Residual Power Series Method (RPSM), a technique not previously applied in this conformable context. Traditional methods for solving fractional-order differential models face challenges such as perturbations, complex simplifications, discretization issues, and restrictive assumptions. In contrast, the RPSM overcomes these limitations by offering a robust solution that reduces both complexity and computational effort. The method provides exact analytical solutions through a convergence series and reliable numerical approximations when needed, making it a versatile tool for simulating fractional-order cancer models. Graphical representations of the approximate solutions illustrate the method’s effectiveness and applicability. The findings highlight the RPSM’s potential to advance cancer treatment strategies by providing a more precise understanding of tumor dynamics in a fractional context. This work contributes to both theoretical and practical advancements in cancer research and lays the groundwork for more accurate and efficient modeling of cancer dynamics, ultimately aiding in the development of optimal treatment strategies.

In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.

In this study, we consider the Benjamin Bona Mahony equation which is in the form of ut + ux + uux - uxxt = 0: The sn-ns method and the tanh-coth method have been applied to this equation. And then, exact solutions have been obtained.

In this study, the Benney-Luke equation is considered. In order to derive new type of solutions, the sn-ns method is applied to this equation. Then, we introduce trigonometric and elliptic functions solutions in addition to the hyperbolic ones which are gained by tanh-coth. Three types of solutions are derived at the same time with the help of this method. Therefore, it can be said that this method is convenient to obtain more solutions to many kinds of nonlinear partial differential equations.

This article offers a comprehensive analysis of nonlinear time fractional Whitham-Broer-Kaup equations under Aboodh Decomposition Transform (ADT). The study model examines the effect of different fractional derivative operators on the solution behavior of these equations by comparing them with accuracy, calculation efficiency, and physical lecturer. By employing the Aboodh Transform, a mathematical tool that is powerful in solving fractional differential equations by implementing the Adomian decomposition method, we derive an approximate solutions for models assessed for specific values of the fractional order; the solutions obtained are shown in 2D and 3D. In addition, comparative analyses are performed to clarify the effect of various fractional derivative operators on the solutions achieved, which shows the accuracy and efficiency of ADT in the handling of these complex non-linear fractional partial differential equations. Furthermore, the exact and approximate solutions are compared to the constructed problem to identify the absolute errors.