Abstract
We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).
Highlights
The presence of delay significantly complicates the analysis of equations of the form (1). Such equations admit traveling-wave solutions u = u(z), where z = x + λt, but do not admit self-similar solutions u = t β φ( xtλ )
It is important to note that delay differential equations have a number of specific qualitative features [4,21,29] that are not inherent in equations without delay
Papers [90,91,92] are devoted to numerical methods for solving pantograph-type PDEs with proportional delay [90,91] and more complex varying delay [92]
Summary
An important role is played by the study of the hereditary properties of nonlinear systems of different nature, when the rate of change of the unknown depends on the state at a given time, and on the previous evolution of the process. The state of the system is determined not by its entire history, but by the current moment and some moment in the past The differential equations modelling such systems, in addition to the unknown function u(t), contain the function u(t − τ ), where t is the time and τ > 0 is the delay [1,2,3]. The presence of delay significantly complicates the analysis of equations of the form (1). Such equations admit traveling-wave solutions u = u(z), where z = x + λt (e.g, see [4,5,6,7]), but do not admit self-similar solutions u = t β φ( xtλ ) (recall that many PDEs without delay have self-similar solutions). It is important to note that delay differential equations have a number of specific qualitative features [4,21,29] that are not inherent in equations without delay
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