Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Subal Ranjan Sahu ◽  
Jugal Mohapatra
Keyword(s):  
Time Delay ◽  
Numerical Study ◽  
Singularly Perturbed ◽  
Parabolic Differential Equations ◽  
Small Space ◽  
First Order ◽  
Taylor Series Approximation ◽  
Backward Euler ◽  
Differential Difference Equation ◽  
Negative Space

Abstract A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

scholarly journals Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

2021 ◽  
Vol 13 (2) ◽  
pp. 57-71
Author(s):  
Wakjira Tolassa Gobena ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Condition ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Parabolic Differential Equations ◽  
Parabolic Boundary ◽  
Delay Term ◽  
Local Boundary ◽  
Backward Euler ◽  
Local Boundary Condition ◽  
Non Local

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

scholarly journals Robust numerical method for singularly perturbed parabolic differential equations with negative shifts

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2383-2401
Author(s):  
Mesfin Woldaregay ◽  
Gemechis Duressa
Keyword(s):  
Differential Equations ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Spatial Discretization ◽  
Parabolic Differential Equations ◽  
First Order ◽  
Temporal Discretization ◽  
Solution Bound ◽  
The Stability

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having delay on the zeroth and first order derivative terms. The solution of the considered problem exhibits boundary layer behaviour as the perturbation parameter tends to zero. The equation is solved using ?-method in temporal discretization and exponentially fitted finite difference method in spatial discretization. The stability of the scheme is proved by using solution bound technique by constructing barrier functions. The parameter uniform convergence analysis of the scheme is carried out and it is shown to be accurate of order O(N-2/N-1+c?+(?t)2) for the case ?= 1/2, where N is the number of mesh points in spatial discretization and ?t is the mesh size in temporal discretization. Numerical examples are considered for validating the theoretical analysis of the scheme.

scholarly journals Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Mesfin Mekuria Woldaregay ◽  
Worku Tilahun Aniley ◽  
Gemechis File Duressa
Keyword(s):  
Time Delay ◽  
Numerical Test ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Parabolic Differential Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Derivative Term ◽  
The Stability ◽  
The Right

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

Optimal tuning of PID controller in time delay system: a review on various optimization techniques

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas George ◽  
V. Ganesan
Keyword(s):  
Time Delay ◽  
Pid Control ◽  
Pid Controller ◽  
Optimization Techniques ◽  
Operating Conditions ◽  
Delay Systems ◽  
Pid Controllers ◽  
Time Delay Systems ◽  
Process Industries ◽  
First Order

AbstractThe processes which contain at least one pole at the origin are known as integrating systems. The process output varies continuously with time at certain speed when they are disturbed from the equilibrium operating point by any environment disturbance/change in input conditions and thus they are considered as non-self-regulating. In most occasions this phenomenon is very disadvantageous and dangerous. Therefore it is always a challenging task to efficient control such kind of processes. Depending upon the number of poles present at the origin and also on the location of other poles in transfer function different types of integrating systems exist. Stable first order plus time delay systems with an integrator (FOPTDI), unstable first order plus time delay systems with an integrator (UFOPTDI), pure integrating plus time delay (PIPTD) systems and double integrating plus time delay (DIPTD) systems are the classifications of integrating systems. By using a well-controlled positioning stage the advances in micro and nano metrology are inevitable in order satisfy the need to maintain the product quality of miniaturized components. As proportional-integral-derivative (PID) controllers are very simple to tune, easy to understand and robust in control they are widely implemented in many of the chemical process industries. In industries this PID control is the most common control algorithm used and also this has been universally accepted in industrial control. In a wide range of operating conditions the popularity of PID controllers can be attributed partly to their robust performance and partly to their functional simplicity which allows engineers to operate them in a simple, straight forward manner. One of the accepted control algorithms by the process industries is the PID control. However, in order to accomplish high precision positioning performance and to build a robust controller tuning of the key parameters in a PID controller is most inevitable. Therefore, for PID controllers many tuning methods are proposed. the main factors that lead to lifetime reduction in gain loss of PID parameters are described in This paper and also the main methods used for gain tuning based on optimization approach analysis is reviewed. The advantages and disadvantages of each one are outlined and some future directions for research are analyzed.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

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