Dissipativity of θ-methods for a class of advection–reaction–diffusion equations with both fixed and distributed delays

ABSTRACT

This paper deals with dissipativity of the theoretical solutions and numerical solutions of a general class of advection–reaction–diffusion equations with both fixed and distributed delays. Some sufficient conditions for the dissipativity of the mentioned problems are given. It is shown that adaptations of linear θ-methods and one-leg θ-methods discussed in the present paper have the ability to preserve dissipativity of the underlying systems when $1/2\le \theta \le 1$. Finally, numerical experiment is presented to confirm the theoretical results.

KEYWORDS:

• Delay-integro-differential equations
• advection–reaction–diffusion equation
• dissipativity
• linear θ-methods
• one-leg θ-methods

2000 AMS SUBJECT CLASSIFICATIONS:

• 35K57
• 45D05
• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are grateful for the support of the National Natural Science Foundation of China (grant numbers 11271101 and 11301115).