Special Issue "Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 March 2022.

Special Issue Editors

Prof. Dr. Andrei Dmitrievich Polyanin
E-Mail Website
Guest Editor
1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia
2. Department of Applied Mathematics, Bauman Moscow State Technical University, 5 Second Baumanskaya Street, 105005 Moscow, Russia
Interests: exact solutions, reductions, and symmetries; nonlinear partial differential equations; delay partial differential equations; mathematical physics equations; functional differential equations; methods of generalized and functional separation of variables; methods of differential and functional constraints; heat and mass transfer; hydrodynamics
Prof. Dr. Alexander V. Aksenov
E-Mail Website
Guest Editor
1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 1 Leninskiye Gory, 119991 Moscow, Russia
2. Keldysh Institute of Applied Mathematics RAS, Miusskaya Square, 125047 Moscow, Russia
Interests: exact solutions of nonlinear equations; group analysis; mathematical physics; asymptotic analysis; partial differential equations for scientists and engineers; hydrodynamics and gas dynamics

Special Issue Information

Dear Colleagues,

Partial differential equations, delay partial differential equations, and functional differential equations are indispensable in modeling various phenomena and processes in natural, engineering, and social sciences. Exact solutions represent rigorous standards (reference solutions) that help understand better the properties and qualitative features of differential equations. They allow one to test, thoroughly and accurately, various numerical and approximate analytical methods for solving these equations. Notably, exact solutions can provide a basis for examining and improving computer algebra packages for solving partial differential equations.

This Special Issue aims to collect original and significant contributions on exact solutions to various partial differential and functional differential equations. Equally welcome are relevant topics related to symmetry reductions, the development and refinement of methods for finding exact solutions, and new applications of exact solutions. The Special Issue can also serve as a platform for exchanging ideas between scientists interested in partial differential and functional differential equations.

Prof. Dr. Andrei Dmitrievich Polyanin
Prof. Dr. Alexander V. Aksenov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.


  • nonlinear partial differential equations
  • reaction-diffusion equations
  • wave type equations
  • higher-order nonlinear PDEs
  • partial differential equations with delay
  • partial functional differential equations
  • exact solutions
  • self-similar solutions
  • invariant solutions
  • generalized separable solutions
  • functional separable solutions
  • classical symmetries
  • nonclassical symmetries
  • symmetry reductions
  • weak symmetries
  • differential constraints

Published Papers (1 paper)

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Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy
Mathematics 2021, 9(5), 511; https://doi.org/10.3390/math9050511 - 02 Mar 2021
Viewed by 420
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). Full article
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