Exact solutions of nonlinear delay reaction–diffusion equations with variable coefficients

Aibinu, Mathew O.; Thakur, Surendra; Moyo, Sibusiso

doi:10.1016/j.padiff.2021.100170

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…Delay PDEs (7) admit traveling-wave solutions, u = u(z), where z = kx + λt (e.g., see [131,[141][142][143]), but do not have self-similar solutions, u = t β ϕ(xt λ ), which non-delay PDEs often have. Reductions and exact solutions with additive, multiplicative, and generalized separation of variables and more complex solutions for delay PDEs are obtained in [144][145][146][147][148][149][150][151][152][153][154][155][156] (see also [157] for a brief review, which, in addition to exact solutions, describes the main numerical methods for solving such equations).…”

Section: Delay Reaction-diffusion Pdesmentioning

confidence: 99%

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Polyanin

Сорокин

2023

Mathematics

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The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays.

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Section: Delay Reaction-diffusion Pdesmentioning

confidence: 99%

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Polyanin

Сорокин

2023

Mathematics

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“…Numerous papers contain more complex exact solutions expressed in elementary functions with a generalized or functional separation of variables for various classes of nonlinear delay PDEs. Solutions are constructed by the method of functional constraints [126] or its modifications [127,131,150]. The method of functional constraints is to seek for solutions with a generalized…”

Section: More Complex Exact Solutionsmentioning

confidence: 99%

“…it also describes a method for constructing such solutions based on grouping the coefficients of the equation in order to reduce it to a delay ODE of a second order. Exact solutions of such and similar equations are also considered in [150][151][152].…”

Section: More Complex Exact Solutionsmentioning

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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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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“…Naturally, occurrences of processes are not instantaneous. Delays are constituted in the dynamical systems (see, e.g., [2,15]). Behavioral responses of organisms to environmental changes takes a unit of time before it is feasible.…”

Section: Introductionmentioning

confidence: 99%

Analyzing population dynamics models via Sumudu transform

Aibinu¹,

Moyo²,

Moyo³

2022

J. Math. Computer Sci.

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This study demonstrates how to construct the solutions of a more general form of population dynamics models via a blend of variational iterative method with Sumudu transform. Evolution of population growth models are presented and new models which are more general, are proposed in form of delay differential equations of pantograph type. This study presents suitable reformulation and reconstruction for some existing population growth models in terms of delay differential equations of pantograph type. Also, presentation is given on innovative ways to obtain the solutions of population growth models where other analytic methods fail. Stimulating procedures for finding patterns and regularities in seemingly chaotic processes are elucidated in this paper. Some single and interacting species population models are illustrated graphically and analyzed. How, when and why the changes in population sizes occur can be deduced through this study.

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Exact solutions of nonlinear delay reaction–diffusion equations with variable coefficients

Cited by 10 publications

References 17 publications

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

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

Analyzing population dynamics models via Sumudu transform

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