Alexei F. Cheviakov

Applications of symmetry methods to partial differential equations

Local Transformations and Conservation Laws.- Construction of Mappings Relating Differential Equations.- Nonlocally Related PDE Systems.- Applications of Nonlocally Related PDE Systems.- Further Applications of Symmetry Methods: Miscellaneous Extensions.

AN ASYMPTOTIC ANALYSIS OF THE MEAN FIRST PASSAGE TIME FOR NARROW ESCAPE PROBLEMS: PART II: THE SPHERE ∗

The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in

Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples

\mathbb{R}^3

Framework for potential systems and nonlocal symmetries: Algorithmic approach

that contains N small nonoverlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The three-term asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling point charges on the unit sphere. Numerical results, based on globa...

Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations

Any partial differential equation (PDE) system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2n−1 independent nonlocally related systems by considering these potential systems individually (n singlets), in pairs (n(n−1)∕2couplets),…, taken all together (one n-plet). In turn, any one of these 2n−1 systems could lead to the discovery of new nonlocal symmetries and/or nonlocal conservation laws of the given PDE system. Moreover, such nonlocal conservation laws could yield further nonlocally related systems. A theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems. The planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples. Many new local...

Multidimensional partial differential equation systems: Nonlocal symmetries, nonlocal conservation laws, exact solutions

An algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations DEs. Each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree. Useful conservation laws play an essential role in the construction of an extended tree. A useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system. Nonlocal symmetries for a given system of DEs can arise from any system in its extended tree. We construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations, and in both cases obtain new nonlocal symmetries. More importantly, due to the equivalence of solution sets, any coordinate-independent method of analysis qualitative, numerical, perturbation, etc. can be applied to any system within the tree, and may yield simpler computations and/or results that cannot be obtained when the method is directly applied to the given system.

Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems

The paper illustrates the use of a symbolic software package GeM for Maple to compute local symmetries of nonlinear and linear differential equations (DEs). In the cases when a given DE system contains arbitrary functions or parameters, symbolic symmetry classification is performed. Special attention is devoted to the computation of point symmetries of linear PDE systems. Routines are available that effectively eliminate infinite obvious symmetries of linear DEs.

Conservation laws of surfactant transport equations

For systems of partial differential equations (PDEs) with n≥3 independent variables, construction of nonlocally related PDE systems is substantially more complicated than is the situation for PDE systems with two independent variables. In particular, in the multidimensional situation, nonlocally related PDE systems can arise as nonlocally related subsystems as well as potential systems that follow from divergence-type or lower-degree conservation laws. The theory and a systematic procedure for the construction of such nonlocally related PDE systems is presented in Part I [A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103521 (2010)]. This paper provides many new examples of applications of nonlocally related systems in three and more dimensions, including new nonlocal symmetries, new nonlocal conservation laws, and exact solutions for various nonlinear PDE systems of physical interest.

Exact anisotropic MHD equilibria

For many systems of partial differential equations (PDEs), including nonlinear ones, one can construct nonlocally related PDE systems. In recent years, such nonlocally related systems have proven to be useful in applications. In particular, they have yielded systematically nonlocal symmetries, nonlocal conservation laws, noninvertible linearizations, and new exact solutions for many different PDE systems of interest. However, the overwhelming majority of new results and theoretical understanding pertain only to PDE systems with two independent variables. The situation for PDE systems with more than two independent variables turns out to be much more complicated due to gauge freedom relating potential variables. The current paper, together with the companion paper [A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103522 (2010)], synthesizes and systematically extends known results for nonlocally related systems arising for multidimensional PDE systems, i.e., for PDE systems with three or more independe...

Conservation properties and potential systems of vorticity-type equations

Equations of the interfacial convection and convection-diffusion describing the transport of surfactants, and more general interfacial balance laws, in the context of a three-dimensional incompressible two-phase flow are considered. Here, the interface is represented implicitly by a zero level set of an appropriate function. All interfacial quantities and operators are extended from the interface to the three-dimensional domain. In both convection and convection-diffusion settings, infinite families of conservation laws that essentially involve surfactant concentration are derived, using the direct construction method. The obtained results are also applicable to the construction of the general balance laws for other excess surface physical quantities. The system of governing equations is subsequently rewritten in a fully conserved form in the three-dimensional domain. The latter is essential for simulations using modern numerical methods.