Vishnu Narayan Mishra

This paper investigates the numerical solution of fractional differential equations involving the Erdelyi-Kober fractional operator, which play a pivotal role in modeling memory-dependent and anomalous dynamic processes in applied mathematics, physics, and engineering, including anomalous diffusion, viscoelasticity, and heterogeneous heat conduction. We introduce a novel neural network-based framework that integrates Physics-Informed Neural Networks with the Theory of Functional Connections and employs an Extreme Learning Machine for efficient training. A new loss function is constructed using the L1 finite difference scheme in combination with Volterra integral equations, enabling accurate approximation of Erdelyi-Kober fractional derivatives while ensuring real-valued computations and minimal error. The proposed approach presents three main contributions over conventional method: (i) seamless integration of Theory of Functional Connections with Physics-Informed Neural Networks specifically for Erdelyi-Kober fractional derivatives, (ii) a novel loss function formulation tailored to the Erdelyi-Kober operator, and (iii) enhanced accuracy and computational efficiency. A neural network approach based on L1 is compared with the conventional L1 numerical scheme in order to solve fractional differential equations. The L1 discretization approach is used in a deep learning model in the proposed neural architecture to increase efficiency and accuracy. Using numerical analyses, we demonstrate that the L1 neural network technique achieves reliable convergence and exceeds the accuracy of the traditional L1 method. Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and coefficient of determination (R2) are performance metrics that further support the model's effectiveness and demonstrate its resilience while handling fractional-order problems. The results show that the combination of neural networks and the L1 scheme provides a strong and accurate solution for solving complex fractional differential equations. Several illustrative examples are provided to demonstrate the efficacy, reliability, and practical relevance of the methodology. The results highlight the capability of the proposed framework to solve complex fractional differential equations effectively, offering a powerful tool for researchers and practitioners in applied mathematics, engineering, and related scientific disciplines.

In this sequel, we employ an approach that combines the modified version of the Lagrange polynomial approximation technique with the biconjugate gradient stabilized method (BiCGSTAB) to analyze numerical solutions of fractional Volterra integral equations (FVIEs). These equations emerge as indispensable tools in modern physics, enabling the modeling and analysis of intricate dynamical systems with memory effects and nonlocal behaviors. From fractal dynamics to quantum mechanics, electromagnetism, and biophysics, these equations provide a unified framework to capture the rich complexity inherent in diverse physical phenomena. The scheme for establishing the uniquely existence of a solution to the integral equation is elucidated in relation to the Banach contraction principle and the Bielecki norm. Furthermore, we rigorously prove several theorems concerning the method′s convergence and error estimation. Notably, our proposed approach effectively handles potential singularities in the solution, a facet that, to the best of our knowledge, has remained unaddressed in existing literature. To showcase the reliability and efficacy of our method, we present illustrative examples along with comparative analyses. All numerical computations are conducted using MATLAB 23.2.0.2410171 (R2023b).

This paper presents a novel investigation into the dynamics of the Cheeger problem for rectangular domains. The study introduces a new framework by defining the dynamical system associated with the classical Cheeger problem specifically for rectangular shapes. A key finding is that the conditions m < 0 and q > m are sufficient for chaotic behavior in the dynamical system defined for the Cheeger problem on rectangles. In this context, b(t) = e^mt represents one of the time-varying dimensions of the rectangle, and εb(t) = e^qt represents the perturbation in the same dimension. The maximal deterministic Lyapunov exponent is rigorously derived, showing that chaos emerges as one rectangle dimension decays while perturbations grow. Numerical illustrations confirm chaos onset at finite times t = 0.2383 and t = 0.2494, demonstrating exponential divergence of trajectories and sensitive dependence on initial conditions. These results establishing fundamental limits on the predictability horizon and enabling precise timing for interventions, thereby bridging theoretical insights with practical foresight and control. Sensitivity analysis further reveals that increasing q accelerates chaos onset, while increasing m stabilizes the system. Altogether, the work advances both theoretical understanding and practical insights into geometric dynamics for complex systems.

This study aimed to design a linear feedback control approach for a parametrically excited energy harvesting system utilizing a piezoelectric material as the transduction element. The purpose was to significantly increase the amount of energy produced compared to that produced by the original system. To do so, firstly, it is necessary to analyze the stability of the system and perform a global sensitivity analysis to determine the physical parameters of the system that most contribute to energy production. The sensitivity analysis is done by calculating the Sobol indices, which are statistical indices that measure the relative contribution of each input variable (in this case, the physical parameters of the system) to the contribution of all input variables. In the stability analysis, the state transition matrix approximation techniques created by Sinha and Butcher and the results of the Floquet Theory for periodic systems were used. Stability analysis and global sensitivity analysis are methodologically complementary techniques for a better understanding of the dynamics of a system. In the case of this work, they are applied to an energy-harvesting system based on mechanical vibrations, providing important information to design a more efficient controller. The control technique used was proposed by Sinha and Butcher (1997), and is known as Linear Feedback Controller Design via the Lyapunov-Floquet Transform.

This study provides the first comprehensive demonstration of how to utilise ResNets to estimate a family of generalized Caputo-type fractional differential equations and their solutions, and how to limit the quantity of parameters present in these ResNets. The basis of our evidence is the variational iteration method. It determines the differential equation's exact solution with the use of the variational iteration method. Then it shows how to estimate these equations using residual neural networks, using the structure produced by the variational iteration method.