An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided.

Solution of Multi-order Fractional Differential Equation Based on Conformable Derivative by Shifted Legendre Polynomial

The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of the linear and non-linear equation, the background idea behind this method is finding system of algebra after achieving messing variable is that mean obtain approximate solution, a few examples illustrates for presented how much our method is capable.

Optimizing Traffic Engineering for Resilient Services in NFV-Based Connected Autonomous Vehicles

The massive amount of data generated daily by various sensors equipped with connected autonomous vehicles (CAVs) can lead to a significant performance issue of data processing and transfer. Network Function Virtualization (NFV) is a promising approach to improving the performance of a CAV system. In an NFV framework, Virtual Network Function (VNF) instances can be placed in edge and cloud servers and connected together to enable a flexible CAV service with low latency. However, protecting a service function chain composed of several VNFs from a failure is challenging in an NFV-based CAV system (VCAV). We propose an integer linear programming (ILP) model and two approximation algorithms for resilient services to minimize the service disruption cost in a VCAV system when a failure occurs. The ILP model, referred to as TERO, allows us to obtain the optimal solution for traffic engineering, including the VNF placement and routing for resilient services with regard to dynamic routing. Our proposed algorithms based on heuristics (i.e., TERH) and reinforcement learning (i.e., TERA) provide an approximation solution for resilient services in a large-scale VCAV system. Evaluation results with real datasets and generated network topologies show that TERH and TERA can provide a solution close to the optimal result. It also suggests that TERA should be used in a highly dynamic VCAV system.

Analytical Study of Nonlinear Vibration in a Rub-Impact Jeffcott Rotor

The purpose of this work is to explore the nonlinear vibration of a rub-impact Jeffcott rotor. In the first stage, the motion is not affected by the friction force, but in the second stage, the motion is influenced by the normal force and the friction force. The governing equations of the rotor of this model are derived in this paper. In consequence, there appears a difference between the two stages. We establish an approximate analytical solution for nonlinear vibrations corresponding to two stages with the mention of the location of jumps. The obtained results are compared with the numerical integration results. The steady-state response and the stability of the solutions are analytically determined for the two stages. The stability of a full annular rub solution is studied with the help of the Routh–Hurwitz criterion. Effects of different parameters of the system, the saddle-node bifurcation (turning points) and the Hopf bifurcation are presented. The main contribution lies in the analytical approximation solution based on the Optimal Auxiliary Functions Method.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

The Quasi-Optimal Radial Basis Function Collocation Method: A Technical Note

The traditional radial basis function parameter controls the flatness of these functions and influences the precision and stability of approximation solution. The coupled radial basis function, which is based on the infinitely smooth radial basis functions and the conical spline, achieves an accurate and stable numerical solution, while the shape parameter values are almost independent. In this paper, we give a quasi-optimal conical spline which can improve the numerical results. Besides, we consider the collocation points in the Chebyshev-type which improves solution accuracy of the method with no additional computational cost.

Approximation Solution for the Zener Impact Theory

Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener collision theory model equations in detail in this study and establish the Zener inelastic collision theory, which is a simple and fast calculation of the approximate solution to the nonlinear differential equations of motion. We propose an approximate formula to obtain the Zener nonlinear differential equation of motion in a simple manner. The approximate solution determines the relevant values of the collision force, material displacement, velocity, and contact time.

State-Aware Value Function Approximation with Attention Mechanism for Restless Multi-armed Bandits

The restless multi-armed bandit (RMAB) problem is a generalization of the multi-armed bandit with non-stationary rewards. Its optimal solution is intractable due to exponentially large state and action spaces with respect to the number of arms. Existing approximation approaches, e.g., Whittle's index policy, have difficulty in capturing either temporal or spatial factors such as impacts from other arms. We propose considering both factors using the attention mechanism, which has achieved great success in deep learning. Our state-aware value function approximation solution comprises an attention-based value function approximator and a Bellman equation solver. The attention-based coordination module capture both spatial and temporal factors for arm coordination. The Bellman equation solver utilizes the decoupling structure of RMABs to acquire solutions with significantly reduced computation overheads. In particular, the time complexity of our approximation is linear in the number of arms. Finally, we illustrate the effectiveness and investigate the properties of our proposed method with numerical experiments.