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

Abstract

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.

Решения Линейных Начально-Краевых Задач Для Уравнений Типа Клейна–гордона С Постоянным И Пропорциональным Запаздыванием

“…В [33] методом разделения переменных построены точные решения линейных уравнений вида (5) и (6). Многие нелинейные уравнения с пропорциональным запаздыванием вида (5) и ( 6), допускающие редукции и точные решения с аддитивным, мультипликативным, обобщенным и функциональным разделением переменных рассматриваются в [34][35][36][37]. Приближенные аналитические методы решения некоторых линейных и нелинейных уравнений в частных производных с пропорциональным запаздыванием описаны в [38,39]; численные методы -в [40,41].…”

“…Нетривиальные решения X = X n (𝑥) линейной однородной задачи на собственные значения (36), (38) Чтобы найти начальные условия для ОДУ второго порядка с пропорциональным запаздыванием (37), функции ϕ(x) и ψ(x), входящие в начальные условия (34), представим в виде разложений по собственным функциям:…”

A Systematic Approach to Delay Functions

“…Proof. It follows from Proposition 1 that dcos(x; σ) = dexp(ix; iσ) − idsin(x; σ) (39) and replacing i with −i in Equation (30), we arrive at…”

Lagrangian formalism and Noether-type theorems for second-order delay ordinary differential equations

“…Applications of Lie groups of transformations have been extended to finite-difference, discrete and differential-difference equations and integro-differential equations [11][12][13][14][15][16][17][18][19][20][21][22][23]. Exact solutions of delay PDEs were considered in [24,25]. The symmetry approach was also used to construct numerical schemes, which preserve qualitative properties of the underlying differential equations [26].…”