应用与计算数学杂志

In the current mathematical model, examined the oscillatory flow of a Casson fluid in a thin walled elastic tube with a varying cross-section by assuming suction or injection at the tube wall and also permeability. The problem is developed that the elastic tube connects against longitudinal displacements. The perturbation approach is used to linearize the nonlinear governing equations to solve the flow characteristics. The differential equation for the pressure is solved numerically along with the corresponding initial conditions using Gill’s fourth order. The effects of Womersley parameter, velocity at the wall(suction/injection), elasticity parameter and Casson parameter on the modulus of wall shear stress and mean pressure drop are discussed through the graphs. Further noted that in a case of a locally constricted tube, the diverging part’s modulus of wall shear stress drops while the convergent part’s rises as Casson parameter, Womersley parameter increases.

Traditional methods for solving HJB equations face challenges, especially when dealing with high-dimensional spaces. However, deep learning offers a promising approach to overcome these limitations. The HJB equation, named after William Rowan Hamilton, Carl Gustav Jacob Jacobi, and Richard Bellman, provides a necessary condition for optimality. It is a partial differential equation (PDE) that characterizes the value function of the control problem, essentially describing the evolution of the optimal cost as a function of time and state. Solving the HJB equation is crucial for determining the optimal policy or strategy in various applications. However, as the dimensionality of the problem increases, traditional numerical methods like finite difference methods or finite element methods become computationally infeasible due to the curse of dimensionality. This is where deep learning techniques, particularly neural networks, come into play.

Rare event estimation is crucial in many fields, such as finance, engineering, and environmental science. These events, although infrequent, can have significant consequences, making their accurate prediction and understanding vital. Traditional methods often fall short due to the immense computational power required or lack of accuracy. The Adaptive Multilevel Splitting (AMS) method offers a robust alternative, providing a practical approach to estimating the probability of rare events in complex systems. The core of the AMS method is its splitting mechanism, where simulations that reach a certain intermediate threshold are duplicated. This process enhances the sampling of rare events, increasing the likelihood of observing them without requiring an excessive number of initial simulations. Define the rare event of interest and the corresponding threshold. Initialize a large number of independent simulations. Run the simulations until they reach the predefined threshold or fail. The successful paths are then analyzed to determine the next threshold. Simulations that reach the threshold are split, creating multiple copies that are slightly perturbed. This step increases the sample size for subsequent levels.

In the evolving landscape of financial derivatives, the quest for precise option pricing mechanisms remains paramount. The Black-Scholes model, despite its historical significance, falls short in addressing the complexities of modern financial markets, such as stochastic volatility and interest rates. Recognizing these limitations, financial theorists and practitioners have developed advanced models that incorporate more realistic elements. One such development is the integration of stochastic interest rates and pure jump Levy processes into option pricing models. This article explores this innovative approach and its implications for the financial industry.

Gauss quadrature rules are essential for numerical integration, especially in high-dimensional spaces. Traditional methods for computing these rules become computationally expensive and inefficient as the dimensionality increases. This article presents a novel fast algorithm for computing high-dimensional Gauss quadrature rules, significantly reducing computational complexity and improving efficiency. The proposed method leverages sparse grids, tensor decompositions, and adaptive strategies to handle the curse of dimensionality effectively.