In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and polynomial forms are obtained. The obtained solutions are presented in the form of 2D and 3D graphics to study the behavior of the analytical solution by setting out the values of suitable parameters. The acquired results reveal that the unified method is a suitable approach for handling non-linear homogeneous evolution equations.

Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. Bio-modeling is an important tool for constructing surfaces using 2D cross sections. The aim of this manuscript was to show 3D construction using 2D CT scan contours. The fractured part of the cranial vault were constructed using a Ball curve with two shape parameters, later the 2D contours were flipped into 3D with an equidistant z component. The surface created was represented by a bi-cubic rational Ball surface with $C^2$ continuity. At the end of this article, we present two real cases, in which we had constructed the frontal and parietal bone fractures using a bi-cubic rational Ball surface. The proposed method was validated by constructing the non-fractured part.

The unidirectional propagation of long waves in certain nonlinear dispersive system is explained by the Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The purpose of this study is to investigate the BBM-Burger equation numerically using Caputo derivative and B-spline basis functions. The fractional derivative is considered in Caputo form, and formula is used for discretization of temporal derivative. The interpolation of space derivative is done with the help of B-spline functions. The effect of and time on solution profile of travelling wave for different domain of is discussed in this paper. The numerical results have been presented to show that the cubic B-spline method is effective and efficient in solving the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. Moreover, the convergence and stability of the proposed scheme are analyzed. The error norms are also calculated to check the accuracy of the proposed scheme. The numerical results reflect that the proposed scheme can be used for linear and highly nonlinear models.

This study presents a comparative analysis of tumor growth models based on logistic, exponential, and Gompertz formulations. Their response to therapeutic intervention is examined to identify which model shows better behavior with minimal decline of immune cells. The framework incorporates three main cell populations as follows: natural killer cells, cytotoxic T cells, and tumor cells, along with treatment effects. Dynamical properties such as positive invariance, existence, boundedness, and equilibrium stability are investigated. Numerical simulations indicate that the logistic model gives more favorable treatment outcomes compared to the exponential and Gompertz models. The results also show a faster decline of immune cell populations in the exponential and Gompertz models than in the logistic model under varying drug flux.

Pituitary adenoma is a clinically important brain tumor whose progression and therapeutic outcomes are influenced by intricate interactions between tumor development and the host immune response. Globally, pituitary adenomas account for approximately 15% of all intracranial tumors. This study aims to investigate clinical MRI data obtained from a patient who recovered from a pituitary adenoma. The collected data provide measurements of tumor volume (mm) at several time points throughout the treatment period. The patient received vaccine-based therapy accompanied by regular clinical assessments, and achieved recovery after nearly twenty-two months. Motivated by the clinical observations and the underlying treatment mechanism, a mathematical model describing tumor–immune–vaccine interactions is developed. The proposed ordinary differential equation (ODE) framework incorporates tumor cells, immune cells, and vaccine components to characterize the temporal evolution of tumor volume during treatment. Fundamental dynamical properties of the model, including positivity, boundedness, existence of solutions, and equilibrium stability, are established through analytical techniques. In addition, numerical simulations are performed using the fourth-order Runge–Kutta (RK4) method and validated against the available clinical measurements. The numerical results exhibit close agreement between the observed and simulated data, yielding a minimal root mean square error (RMSE). Furthermore, sensitivity analysis highlights the significant role of immune- and vaccine-related parameters in regulating tumor suppression. The findings suggest that a relatively simple mechanistic framework can effectively capture the reduction in pituitary adenoma under vaccine-based therapy.