Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

Vsevolod G. Sorokin and Andrei V. Vyazmin
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Vsevolod G. Sorokin: Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, Bldg 1, 119526 Moscow, Russia
Andrei V. Vyazmin: Chemical Engineering and Biotechnology Department, Moscow Polytechnic University, 38 Bolshaya Semenovskaya St., 107023 Moscow, Russia

Mathematics, 2022, vol. 10, issue 11, 1-39

Abstract: The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

Keywords: delay models; nonlinear delay reaction–diffusion equations; exact solutions; numerical methods; method of lines; Mathematica (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (4)

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