Fitted Numerical Scheme for Singularly Perturbed Convection‐Diffusion Equation with Small Time Delay

This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection–diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter-uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

For batch processes with small time delays and actuator partial fault, the existing methods based on iterative learning control still have some limitations, including the conservative and computationally burdensome of stability conditions and the limited fault-tolerant control capabilities. For this background, an iterative learning robust predictive fault-tolerant control method is developed, which integrates the Lyapunov–Razumikhin function method and derives stability conditions based on robust positive definite invariant set and terminal constraint set. With small time delays, the stability conditions of the system deduced using the Lyapunov–Razumikhin function are solved at a lower computational cost, which is due to the fact that the dimensionality of the stabilization condition is directly related to the size of the time delay, and thus the small time delay implies a lower dimensionality. Especially, the computational effort for solving the stability conditions online is reduced, allowing real-time control law gains to be obtained and combined with historical batches of control inputs, reducing the learning cycles of the system, and realizing stable tracking of the setpoints within shorter operating batches. Moreover, the robust positive invariant set and the set of terminal constraints are able to constrain the state of the system within a safe range for all possible uncertainties, bounded disturbances, and faults. This makes the proposed methods based on them more robust and fault tolerant. Finally, a nonlinear batch reactor is used as an example to demonstrate the effectiveness and feasibility of the developed method.

In this brief, the stability of neural networks with switching between small and large time delays is studied by developing an improved exponential stability criterion. Firstly, the delayed neural network (DNN) with small delay (SD) and large delay (LD) is modeled as a switched DNN. Then, based on an augmented piecewise Lyapunov-Krasovskii functional with LD-based terms considering relaxed switching constraints, and Wiritinger-based inequality, a stability criterion with less conservatism is developed. Finally, a numerical example is provided to demonstrate the superiority and effectiveness of the proposed method.

ABSTRACTA new and efficient numerical scheme for approximating one-dimensional nonlinear modified Burgers equation has been illustrated in this study. The method makes use of collocation of quintic splines and Crank Nicolson scheme for spatial and temporal discretization, respectively. Linearization involves a quasilinearization process. Since splines are smooth functions which do not exhibit oscillations associated with polynomials and possess easily computable derivatives, working with splines has a computational edge over other existing methods. The method is unconditionally stable, which is justified by von Neumann stability analysis. Convergence analysis of the method is performed and it is proved that the method has fourth-order convergence in space and second order in time. Performance of the method is assessed by computing the CPU time, and small values of CPU time ascertains that the method is highly time efficient. Proposed method yields good numerical results for moderate values of viscosity. Efficiency of the scheme is established from the fact that the and error norms are minute. Results achieved using the proposed numerical scheme are more precise and reliable as compared to many existing works on modified Burgers equation. This method is robust, fast, flexible and easy to implement.

The use of lower order approximations in the neighborhood of boundaries coupled with higher order interior approximations is examined for the mixed initial boundary-value problem for hyperbolic partial differential equations.Uniform error can be maintained using smaller grid intervals with the lower order approximations near the boundaries.Stability results are presented for approximations to the initial boundary-value problem for the model equation uf + cux = 0 which are fourth order in space and second order in time in the interior and second order in both space and time near the boundaries.These results are generalized to a class of methods of this type for hyperbolic systems.Computational results are presented and comparisons are made with other methods.

A linear time-invariant singularly perturbed system with point-wise and distributed small time delays is considered. A direct approach to exact slow-fast decomposition of this system is proposed. Unlike the existing method, this approach does not require the asymptotic stability of the fast subsystem. Moreover, not requiring a complicated mathematical technique, this approach is much simpler for using in various engineering applications. An illustrative example is presented.

A linear time-invariant singularly perturbed system with multiple pointwise and distributed small time delays is considered. A novel (direct) approach to exact slow-fast decomposition of this system is proposed. In contrast with the existing method, this approach uses neither a preliminary transformation of the original differential system to an integral one nor rather complicated integral manifold and operator techniques. Moreover, the approach of the present paper does not assume the exponential stability of the fast subsystem. Based on this decomposition, an exact slow-fast decomposition of the spectrum of a singularly perturbed system with a single pointwise small delay is carried out. Using the theoretical results, the stability of a multilink single-sink optical network is analyzed.

This paper deals with the numerical treatment of time-dependent singularly perturbed convection–diffusion–reaction equations with delay. In the considered equations, the highest order derivative term is multiplied by a perturbation parameter, ε taking arbitrary value in the interval . For small ε, the solution of the equations exhibits a boundary layer on the right side of the spatial domain. The considered equations contain a small delay on the convection and reaction terms of the spatial variable. The terms involving the delay are approximated using the Taylor series approximation. The resulting singularly perturbed parabolic convection–diffusion–reaction equations are treated using the Crank Nicolson method in time derivative discretization and the mid-point upwind finite difference method on piecewise uniform Shishkin mesh for the space variable derivative discretization. The stability and the uniform convergence of the scheme are investigated well. The proposed scheme gives almost first-order convergence in the spatial direction. The Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme in the spatial direction to order almost two. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter and delay parameter.

A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model

We show that certain infinite-dimensional damped second-order systems of linear differential equations become unstable when arbitrarily small time delays occur in the damping.

Time Delay Control(TDC) is a robust nonlinear control scheme using Time Delay Estimation(TDE) and also has a simple structure. To apply TDC to a real system, we must design Time Delay Controller to guarantee stability. The earlier research stated sufficient stability condition of TDC for general plants. In that research, it was assumed that time delay is infinitely small. But, it is impossible to implement infinitely small time delay in a real system. So, in this research we propose a new sufficient stabil ity condition of TDC for general plants with finite time delay. And the simulation results indicate that the previous sufficient stability condition does not work even for small time delay, while our proposed condition works well.

Fitted mesh B-spline collocation method for singularly perturbed differential–difference equations with small delay

Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

Computation of steady and oscillatory convection in saturated porous media

The problem of finding the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -norm of systems with a finite number of discrete delays and distributed delay is considered. Sufficient conditions for the system to possess an H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -norm which is less or equal to a prescribed bound are obtained in terms of the Riccati partial differential equations (RPDE's). We show that the existence of the solution to the RPDE's is equivalent to the existence of the stable manifold of the associated Hamiltonian system. The main result of the paper is a derivation of algebraic finite-dimensional criterion for the solvability of RPDE's for systems with small time-delays. The result is based on slow-fast decomposition of the Hamiltonian system.