1 research outputs found
Exact Solutions and Reductions of Nonlinear Diffusion PDEs of Pantograph Type
We study nonlinear pantograph-type reaction-diffusion PDEs, which, in
addition to the unknown , also contain the same functions with
dilated or contracted arguments of the form , , and
, where and are the free scaling parameters (for equations
with proportional delay we have , ). A brief review of
publications on pantograph-type ODEs and PDEs and their applications is given.
Exact solutions and reductions 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.Comment: arXiv admin note: text overlap with arXiv:2102.0481