Controllability of Nonlinear Deterministic Chaotic Systems with Control-Induced Delay

This paper presents the analysis of a class of retarded nonlinear chaotic systems with control-induced delay. The existence and uniqueness of the mild solution are obtained by using the local Lipschitz condition on nonlinearity and Banach contraction principle. The approximate controllability for linear and nonlinear control delay systems has been established by sequence method and using the Nemytskii operator. The application of results is explained through an example of a parabolic partial differential equation.

Solution of Parabolic Partial Differential Equations by Non-Polynomial Cubic Spline Technique

Parabolic partial differential equation having a great impact on our scientific, engineering and technology. Enormous research have been conducted for the solution of parabolic PDEs. . In this research work, we introduced a novel technique for the numerical solution of fourth order PDEs. The novel technique is based upon the polynomial cubic cutting method (PCSM) was used with Adomian breakdown technique (ADM).The constraint for the alternative variables was decomposed by Edomian decomposition for the successive approximation. A numerical test problem of parabolic PDEs solved by purposed technique

Non-standard numerical scheme for singularly perturbed parabolic partial differential equation with long time lag arising in control theory

For the numerical solution of singularly perturbed second-order parabolic partial differential equation of one dimensional convection-diffusion type with long time delays arising in control theory, a novel class of fitted operator finite difference methods is constructed using non-standard finite difference methods. Since the two parameters; time lag and perturbation parameters are sources for the simultaneous occurrence of time-consuming and high speed phenomena of the physical systems that depends on the present and past history, our study here is to capture the effect of the two parameters on the boundary layer. The spatial derivative is suitably replaced by a difference operator followed by the time derivative is replaced by the Crank-Nicolson based scheme. A second-order parameter-uniform error bounds are established to provide numerical results.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

An Inverse Problem Approach for the Estimation of the Haverkamp and van Genuchten Retention Curves Parameters with the Luus-Jaakola Method

In academia, the study of the movement of water in the ground has been widespread since the last century. The same can be determined using the Richards equation. This is a nonlinear parabolic partial differential equation that requires parameters to generate the results of the problem. Some authors have proposed equations that represent the relationship between volumetric moisture and soil water potential, such as Haverkamp and van Genuchten. As main objective of this work, the inverse modeling was implemented to obtain the Richards equation parameters, applying the Luus-Jaakola method. To verify the mathematical model, a sensitivity analysis was performed, which allowed the observation of the effect that each parameter has on the output data, implying a linear dependence. The results produced proved to be satisfactory for the problem analyzed in our research.

New Approximations to Bond Prices in the Cox–Ingersoll–Ross Convergence Model with Dynamic Correlation

We study a particular case of a convergence model of interest rates. The bond prices are given as solutions of a parabolic partial differential equation and we consider different possibilities of approximating them, using approximate analytical solutions. We consider an approximation already suggested in the literature and compare it with a newly suggested one for which we derive the order of accuracy. Since the two formulae use different approaches and the resulting leading terms of the error depend on different parameter sets of the model, we propose their combination, which has a higher order of accuracy. Finally, we propose one more approach, which leads to higher accuracy of the resulting approximation formula.

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

Evolved auxiliary controller with applications to aerospace

Purpose To guarantee the asymptotic stability of discrete-time nonlinear systems, this paper aims to propose an evolved bat algorithm fuzzy neural network (NN) controller algorithm. Design/methodology/approach In evolved fuzzy NN modeling, the NN model and linear differential inclusion representation are established for the arbitrary nonlinear dynamics. The control problems of the Fisher equation and a temperature cooling fin for high-speed aerospace vehicles will be described and demonstrated. The signal auxiliary controlled system is represented for the nonlinear parabolic partial differential equation (PDE) systems and the criterion of stability is derived via the Lyapunov function in terms of linear matrix inequalities. Findings This representation is constructed by sector nonlinearity, which converts the nonlinear model to a multiple rule base for the linear model and a new sufficient condition to guarantee the asymptotic stability. Originality/value This study also injects high frequency as an auxiliary and the control performance to stabilize the nonlinear high-speed aerospace vehicle system.

Fitted Numerical Scheme for Solving Singularly Perturbed Parabolic Delay Partial Differential Equations

In this paper, exponentially fitted finite difference scheme is developed for solving singularly perturbed parabolic delay partial differential equations having small delay on the spatial variable. The term with the delay is approximated using Taylor series approximation. The resulting singularly perturbed parabolic partial differential equation is treated using im- plicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The parameter uniform convergence analysis has been carried out with the order of convergence one. Test examples and numerical results are considered to validate the theoretical analysis of the scheme.

Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1. Solutions near an axisymmetric surface

While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys., vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function $\eta$ such the field strength satisfies $B = B(\eta )$ . Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.