Regularization techniques are critical in the development of machine learning models. Complex models, such as neural networks, are particularly prone to overfitting and to performing poorly on the training data. $L_1$ regularization is the most extreme way to enforce sparsity, but, regrettably, it does not result in an NP-hard problem due to the non-differentiability of the 1-norm. However, the $L_1$ regularization term achieved convergence speed and efficiency optimization solution through a proximal method. In this paper, we propose a batch gradient learning algorithm with smoothing $L_1$ regularization (BGS$L_1$) for learning and pruning a feedforward neural network with hidden nodes. To achieve our study purpose, we propose a smoothing (differentiable) function in order to address the non-differentiability of $L_1$ regularization at the origin, make the convergence speed faster, improve the network structure ability, and build stronger mapping. Under this condition, the strong and weak convergence theorems are provided. We used N-dimensional parity problems and function approximation problems in our experiments. Preliminary findings indicate that the BGS$L_1$ has convergence faster and good generalization abilities when compared with BG$L_{1/2}$, BG$L_1$, BG$L_2$, and BGS$L_{1/2}$. As a result, we demonstrate that the error function decreases monotonically and that the norm of the gradient of the error function approaches zero, thereby validating the theoretical finding and the supremacy of the suggested technique.

In this study, the \(\phi ^{6}\)-model expansion method is showed to be useful for finding solitary wave solutions to the Klein–Gordon (KG) equation. We develop a variety of solutions, including Jacobi elliptic functions, hyperbolic forms, and trigonometric forms, so greatly enhancing the range of exact solutions attainable. The 2D, 3D, and contour plots clearly show different types of solitary waves, like bright, dark, singular, and periodic solitons. This gives us a lot of information about how the KG equation doesn’t work in a straight line. Our findings highlight the \(\phi ^{6}\) model as a powerful tool to study nonlinear wave equations, improve our understanding of their complex dynamics, and increase the scope for theoretical exploration. The \(\phi ^{6}\) model expansion technique is exceptionally adaptable and may be utilised for a wide array of nonlinear partial differential equations. Despite its versatility, the technique may not be applicable to all nonlinear PDEs, especially those that do not meet the specified requirements or structures manageable by this technique. In theoretical physics, particularly in field theory and quantum mechanics, the Klein–Gordon equation is a classical model. By studying this model, we can illustrate the waves and particles movements at relativistic speeds. Among other areas, its significance in cosmology, quantum field theory, and the study of nonlinear optics are widely considered. Additionally, it provides exact solutions and nonlinear dynamics have various applications in applied mathematics and physics. The study is novel because it provides a new understanding of the complex behaviours and various waveforms of the controlling model by means of detailed evaluation. Future research could focus on further exploring the stability and physical implications of these solutions under different conditions, thereby advancing our knowledge of nonlinear wave phenomena and their applications in physics and beyond.

This article introduces the theory of three-variable q-truncated exponential Gould–Hopper-based Appell polynomials by employing a generating function approach that incorporates q-calculus functions. This study further explores these polynomials by using a computational algebraic approach. The determinant form, recurrences, and differential equations are proven. Relationships with the monomiality principle are given. Finally, graphical representations are presented to illustrate the behavior and potential applications of the three-variable q-truncated exponential Gould–Hopper-based Appell polynomials.

In this research, the results of radon concentration, surface and mass exhalation rates, radium concentration, effective dose rate and the alpha index have been investigated in a number of 198 soil samples that have been collected from various residential locations of Halfa Aljadida area, Sudan. The can technique, containing CR-39 have been used. From our results, the average value of soil gas radon concentration was found to be 1.96±0.22 kBq·m⁻³. The average values of surface and mass exhalation rates were 1.73±0.19 Bq·m⁻²·h⁻¹ and 34.79±3.87 mBq·kg⁻¹·h⁻¹, respectively. The radium concentration average value was 8.06±0.90 Bq·kg⁻¹. While the average value of the effective dose rate was recorded to be 54.69±6.11 mSv·y⁻¹. The average value of alpha index of studied samples was (4.03±0.45)×10⁻². From the study, a good positive and linear correlation between radium concentration, surface and mass exhalation rates of soil samples were present. In addition to that, a positive and linear correlation between radium and radon concentrations was found. Finally, a comparison between the results and other findings was conducted and the results imply the fact that the area under consideration is safe as if the health hazard are mentioned.