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Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries,...
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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: closed (31 March 2022) | Viewed by 26529

Special Issue Editors

Prof. Dr. Andrei D. Polyanin

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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
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alexander V. Aksenov

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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 Issues, Collections and Topics in MDPI journals

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

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Keywords

  • 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

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Published Papers (12 papers)

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Research

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39 pages, 1027 KiB  
Article
Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration
by Vsevolod G. Sorokin and Andrei V. Vyazmin
Mathematics 2022, 10(11), 1886; https://doi.org/10.3390/math10111886 - 31 May 2022
Cited by 8 | Viewed by 3427
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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17 pages, 332 KiB  
Article
New Reductions of the Unsteady Axisymmetric Boundary Layer Equation to ODEs and Simpler PDEs
by Alexander V. Aksenov and Anatoly A. Kozyrev
Mathematics 2022, 10(10), 1673; https://doi.org/10.3390/math10101673 - 13 May 2022
Cited by 2 | Viewed by 2081
Abstract
Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, [...] Read more.
Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
28 pages, 480 KiB  
Article
Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions
by Andrei D. Polyanin and Alexei I. Zhurov
Mathematics 2022, 10(9), 1529; https://doi.org/10.3390/math10091529 - 3 May 2022
Cited by 4 | Viewed by 2117
Abstract
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction [...] Read more.
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
10 pages, 406 KiB  
Article
Accurate Solutions to Non-Linear PDEs Underlying a Propulsion of Catalytic Microswimmers
by Evgeny S. Asmolov, Tatiana V. Nizkaya and Olga I. Vinogradova
Mathematics 2022, 10(9), 1503; https://doi.org/10.3390/math10091503 - 1 May 2022
Cited by 4 | Viewed by 1993
Abstract
Catalytic swimmers self-propel in electrolyte solutions thanks to an inhomogeneous ion release from their surface. Here, we consider the experimentally relevant limit of thin electrostatic diffuse layers, where the method of matched asymptotic expansions can be employed. While the analytical solution for ion [...] Read more.
Catalytic swimmers self-propel in electrolyte solutions thanks to an inhomogeneous ion release from their surface. Here, we consider the experimentally relevant limit of thin electrostatic diffuse layers, where the method of matched asymptotic expansions can be employed. While the analytical solution for ion concentration and electric potential in the inner region is known, the electrostatic problem in the outer region was previously solved but only for a linear case. Additionally, only main geometries such as a sphere or cylinder have been favoured. Here, we derive a non-linear outer solution for the electric field and concentrations for swimmers of any shape with given ion surface fluxes that then allow us to find the velocity of particle self-propulsion. The power of our formalism is to include the complicated effects of the anisotropy and inhomogeneity of surface ion fluxes under relevant boundary conditions. This is demonstrated by exact solutions for electric potential profiles in some particular cases with the consequent calculations of self-propulsion velocities. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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24 pages, 1684 KiB  
Article
Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws
by Vladimir Dorodnitsyn and Evgeniy Kaptsov
Mathematics 2022, 10(8), 1250; https://doi.org/10.3390/math10081250 - 11 Apr 2022
Cited by 3 | Viewed by 1517
Abstract
Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. The construction of these schemes makes use of results of the group classification of MHD equations previously obtained by the authors. [...] Read more.
Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. The construction of these schemes makes use of results of the group classification of MHD equations previously obtained by the authors. On the basis of the classical Samarskiy–Popov scheme, new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. New, previously unknown, conservation laws are found using symmetries and direct calculations. In the case of infinite conductivity, conservative invariant schemes are constructed as well. For isentropic flows of a polytropic gas the proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. In addition, invariant difference schemes with additional conservation laws are proposed. A new scheme for the case of finite conductivity is tested numerically for various boundary conditions, which shows accurate preservation of difference conservation laws. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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9 pages, 242 KiB  
Article
First-Order Approximate Mei Symmetries and Invariants of the Lagrangian
by Umara Kausar and Tooba Feroze
Mathematics 2022, 10(4), 649; https://doi.org/10.3390/math10040649 - 19 Feb 2022
Cited by 5 | Viewed by 1302
Abstract
In this article, the formulation of first-order approximate Mei symmetries and Mei invariants of the corresponding Lagrangian is presented. Theorems and determining equations are given to evaluate approximate Mei symmetries, as well as approximate first integrals corresponding to each symmetry of the associated [...] Read more.
In this article, the formulation of first-order approximate Mei symmetries and Mei invariants of the corresponding Lagrangian is presented. Theorems and determining equations are given to evaluate approximate Mei symmetries, as well as approximate first integrals corresponding to each symmetry of the associated Lagrangian. The formulated procedure is explained with the help of the linear equation of motion of a damped harmonic oscillator (DHO). The Mei symmetries corresponding to the Lagrangian and Hamiltonian of DHO are compared. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
20 pages, 1079 KiB  
Article
New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations
by Renata Nikonorova, Dilara Siraeva and Yulia Yulmukhametova
Mathematics 2022, 10(1), 123; https://doi.org/10.3390/math10010123 - 1 Jan 2022
Cited by 3 | Viewed by 1725
Abstract
In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the [...] Read more.
In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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8 pages, 233 KiB  
Article
Approximate Mei Symmetries and Invariants of the Hamiltonian
by Umara Kausar and Tooba Feroze
Mathematics 2021, 9(22), 2910; https://doi.org/10.3390/math9222910 - 15 Nov 2021
Cited by 5 | Viewed by 1555
Abstract
It is known that corresponding to each Noether symmetry there is a conserved quantity. Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries. However, the two sets of symmetries may give different conserved quantities. In this paper, [...] Read more.
It is known that corresponding to each Noether symmetry there is a conserved quantity. Another class of symmetries that corresponds to conserved quantities is the class of Mei symmetries. However, the two sets of symmetries may give different conserved quantities. In this paper, a procedure of finding approximate Mei symmetries and invariants of the perturbed/approximate Hamiltonian is presented that can be used in different fields of study where approximate Hamiltonians are under consideration. The results are presented in the form of theorems along with their proofs. A simple example of mechanics is considered to elaborate the method of finding these symmetries and the related Mei invariants. At the end, a comparison of approximate Mei symmetries and approximate Noether symmetries is also given. The comparison shows that there is only one common symmetry in both sets of symmetries. Hence, rest of the symmetries in the two sets correspond to two different sets of conserved quantities. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
15 pages, 337 KiB  
Article
Group Analysis of the Plane Steady Vortex Submodel of Ideal Gas with Varying Entropy
by Salavat Khabirov
Mathematics 2021, 9(16), 2006; https://doi.org/10.3390/math9162006 - 21 Aug 2021
Cited by 3 | Viewed by 1908
Abstract
The submodel of ideal gas motion being invariant with respect to the time translation and the space translation by one direct has 4 integrals in the case of vortex flows with the varying entropy. The system of nonlinear differential equations of the third [...] Read more.
The submodel of ideal gas motion being invariant with respect to the time translation and the space translation by one direct has 4 integrals in the case of vortex flows with the varying entropy. The system of nonlinear differential equations of the third order with one arbitrary element was obtained for a stream function and a specific volume. This element contains from the state equation and arbitrary functions of the integrals. The equivalent transformations were found for arbitrary element. The problem of the group classification was solved when admitted algebra was expanded for 8 cases of arbitrary element. The optimal systems of dissimilar subalgebras were obtained for the Lie algebras from the group classification. The example of the invariant vortex motion from the point source or sink was done. The regular partial invariant submodel was considered for the 2-dimensional subalgebra. It describes the turn of a vortex flow in the strip and on the plane with asymptotes for the stream line. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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17 pages, 760 KiB  
Article
New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System
by Roman Cherniha and Vasyl’ Davydovych
Mathematics 2021, 9(16), 1984; https://doi.org/10.3390/math9161984 - 19 Aug 2021
Cited by 9 | Viewed by 1702
Abstract
The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of [...] Read more.
The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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22 pages, 352 KiB  
Article
Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy
by Andrei D. Polyanin and Vsevolod G. Sorokin
Mathematics 2021, 9(5), 511; https://doi.org/10.3390/math9050511 - 2 Mar 2021
Cited by 19 | Viewed by 3159
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) [...] Read more.
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
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)

Review

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32 pages, 817 KiB  
Review
Exact Solutions for Gravity-Segregated Flows in Porous Media
by Pavel Bedrikovetsky and Sara Borazjani
Mathematics 2022, 10(14), 2455; https://doi.org/10.3390/math10142455 - 14 Jul 2022
Cited by 4 | Viewed by 1916
Abstract
The review is devoted to exact analytical solutions for quasi-2D gravity segregated flows or gravity currents in subterranean porous formations. The problems under consideration are quasi-linear. The driving forces are two components of the buoyancy—one exerting the bulk of the light fluid and [...] Read more.
The review is devoted to exact analytical solutions for quasi-2D gravity segregated flows or gravity currents in subterranean porous formations. The problems under consideration are quasi-linear. The driving forces are two components of the buoyancy—one exerting the bulk of the light fluid and one due to the curvilinearity of the interface between the fluids. In the case of homogeneous formation or where the seal slope is negligible, the transport equation is parabolic and allows for a wide set of self-similar solutions. In a large-scale approximation of the buoyancy domination, the governing equation is hyperbolic; the method of characteristics allows for a detailed analytical description of gravity current propagation with final accumulation in the geological trap. Analytical models for leakage via the caprock seal are also discussed. The work was completed by formulating some unsolved problems in segregated flows in porous media. Full article
(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)
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