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

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.

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

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.

A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

Solutions to a multi-phase model of sea-ice growth

The multi-phase systems has found its applications in many fields. We shall apply this approach to investigate the multi-phase dynamics of sea-ice growth. In this paper, the weak solutions existence and uniqueness of parabolic differential equations are proved. Then large-time behavior of solutions are studied, also the existence of global attractor is proved. The key tool in this article is the energy method. Our existence proof is only in one dimension.

Second order optimality conditions for periodic optimal control problems governed by semilinear parabolic differential equations

In this paper, we study second-order optimality conditions for some optimal control problems governed by some semi-linear parabolic equations with periodic state constraint in time. We obtain a necessary condition and a sufficient condition in terms of the second order derivative of the associated Lagrangian. These two conditions correspond to the positive definite and the nonnegativity of the second order derivative of the Lagrangian on the same cone, respectively.

On Regional Boundary Gradient Strategic Sensors In Diffusion Systems

This paper is aimed at investigating and introducing the main results regarding the concept of Regional Boundary Gradient Strategic Sensors (RBGS-sensors the in Diffusion Distributed Parameter Systems (DDP-Systems . Hence, such a method is characterized by Parabolic Differential Equations (PDEs in which the behavior of the dynamic is created by a Semigroup ( of Strongly Continuous type (SCSG in a Hilbert Space (HS) . Additionally , the grantee conditions which ensure the description for such sensors are given respectively to together with the Regional Boundary Gradient Observability (RBG-Observability can be studied and achieved . Finally , the results gotten are applied to different situations with altered sensors positions are undertaken and examined.

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

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 .

Solutions to a three phase-field model for solidification

In this paper we present the phase-field models to describe nonisothermal solidification of ideal multicomponent and multiphase alloy systems. Governing equations are developed for the temporal and spatial variation of three phase-field functions, as well as the temperature field. The global existence of weak solutions to parabolic differential equations in three dimension was proved by the Galerkin method. The existence of a maximum theorem are also extensively studied.