Band 25, Heft 7-8

This article presents a computationally efficient numerical algorithm that relies on fractional-order Lagrange polynomials (FOLPs) for solving initial and boundary value problems involving nonlinear fractional-order Bratu equation. This Bratu equation is a core component of the fabrication process framework for electroscope nanofibers. We create an operational matrix of fractional integration using the FOLPs to tackle the relevant initial and boundary value problems. Additionally, when using the collocation approach, it is reduced to a set of algebraic equations that can be solved using Newton’s iterative method. The Caputo notion of the fractional derivative is utilized. The effectiveness and accuracy of the present strategy are then illustrated using a few numerical examples. It has been found that for accurate numerical computation, just a few terms of Lagrange’s polynomials are needed.

In this research, we provide certain novel requirements for the existence and uniqueness of fuzzy solutions for a type of nonlinear fuzzy Caputo fractional generalized Hukuhara Volterra–Fredholm integro-differential equations under integral boundaries. The required findings are demonstrated by employing the Banach and successive iterations convergence technique. First, we give the two analytic form solution equivalent the main equation and then we show the existence and uniqueness of solutions with the help of the successive iterations convergence technique, as well as Banach’s fixed point theorem. Moreover, we examine the generalized Ulam–Hyers (GUH) and Ulam–Hyers–Rassias stability for our main problem. An example is provided to demonstrate the reliability of our findings.

The viral transmission is a complex process which is influenced by certain factors such as infection rate, production rate of new viruses and impact of the antibody-virus complex. These factors affect the transmission process unpredictably under the influence of external noise. In order to incorporate this phenomenon, the viral transmission is manifested under random environment by introducing noise to the deterministic SIVA model which is formulated by dividing the dynamical system into four distinct parts, namely, susceptible cells ( S ), infected cells ( I ), virus ( V ) and immune cells ( A ). A deterministic viral transmission model is analyzed as a representation of real-life situations that involves certain assumptions. In order to derive the intensity fluctuations and variances of the cell and virus population of the stochastic model with the presence of Gaussian white noise, Fourier transformation method is used. The deterministic model exhibits two distinct equilibria: virus-free steady (VFS) and endemic steady (ES) state. The intensities and variances of different cells and virus populations are derived and the values are computed numerically. The results of the sensitivity analysis, shown in graphical form, indicate that the infection rate and the production rate of viruses are highly sensitive. Our findings suggest that the presence of noise causes fluctuations in the model, leading to a noticeable impact on the transmission of the virus.

Dynamical systems are mathematical models often represented by differential equations (DEs), which can be linear or nonlinear. The developments in machine learning (ML) applications have paved the way to solve dynamical systems with ease. However, the major hindrances involved in all those algorithms are their inability to predict noisy and sparse datasets as well as to incorporate the prior physical information. Most of the implementation of neural networks is to perform nonlinear transformations from input to output and also data-driven. The new scientific computing paradigms viz. physics-informed neural networks (PINNs) have revolutionized traditional utilization of ML algorithms. It represents a promising set of algorithms that integrate the training process with the known physical properties governed by DEs. Here, we have implemented PINNs, to solve the vibration equation of large membranes. Finally, the obtained results are verified with the ground truths through simulations for tracking the performance of proposed algorithm.