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He, Qi-qiang; Kang, Lishan; Evans, David

doi:10.1023/a:1019130928606

Cited by 14 publications

(2 citation statements)

References 6 publications

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“…Performing the calculations on successive intervals, we obtain a solution for the en-time interval. In [160], the conditions for the uniqueness of the solution of scheme (65) were established. In [197], the convergence of this scheme with order h 2 + s was proved and its stability was studied.…”

Section: Finite-difference Methodsmentioning

confidence: 99%

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

Сорокин

Vyazmin

2022

Mathematics

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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.

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Section: Finite-difference Methodsmentioning

confidence: 99%

“…Remark 16. Scheme (65) is a nonlinear system of difference equations, which can be solved by iterative methods, for example, the Picard-Schwarz method [160] or the method of upper and lower solutions [198][199][200][201].…”

Section: Finite-difference Methodsmentioning

confidence: 99%

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

Сорокин

Vyazmin

2022

Mathematics

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A class of one‐parameter alternating direction implicit methods for two‐dimensional wave equations with discrete and distributed time‐variable delays

Tang

Zhang

Sun

2022

Numerical Methods Partial

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For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods. KEYWORDSdiscrete and distributed time-variable delays, error analysis, numerical stability, one-parameter ADI methods, two-dimensional wave equations

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Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

Polyanin

Сорокин

2015

Applied Mathematics Letters

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Untitled

Cited by 14 publications

References 6 publications

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

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

A class of one‐parameter alternating direction implicit methods for two‐dimensional wave equations with discrete and distributed time‐variable delays

Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

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