Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

On the equivalence of linear equations under nonlocal transformation

Linear nth order (n?3) ordinary differential equations have been shown to possess n+1, n+2 or n+4 Lie point symmetries. Each class contains equations which are equivalent under point transformation. By taking the example of third order equations, we show that all linear equations are equivalent if the class of transformation is broadened to include nonlocal transformations and hence the representative of this class of equations is y⁽ⁿ⁾=0.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Boundedness of solutions of difference equations and application to numerical solution of Volterra integral equations of the second kind

We study the difference equations obtained when some numerical methods for Volterra integral equations of the second kind are applied to the linear test problem y(t) = 1 + ∝₀^t (λ + μt + vs) y(s) ds, t ⩾ 0, with fixed stepsize h. The resulting difference equations are of Poincaré type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von Neumann polynomial. This result is then used in stability analysis of reducible quadrature methods for Volterra integral equations.     

On generalized fuzzy relational equations and their applications

The paper provides an idea of generalization of fuzzy relational equations where t- and s-norms are introduced. The first part contains an extensive presentation of the resolution of fuzzy relational equations; next the solutions are specified for a list of several triangular norms. Moreover the dual equations are considered. The second part deals with the applicational aspects of these equations in systems analysis, decision-making, and arithmetic of fuzzy numbers.     

Numerical studies of slow viscous rotating fluid past a sphere

Navier-Stokes equations for steady, viscous rotating fluid, rotating about the zaxis with angular velocity ω are linearized using Stokes approximation. The linearized Navier-Stokes equations governing the axisymmetric flow can be written as three coupled partial differential equations for the stream function, vorticity and rotational velocity component. Only one parameterR _eω =2ωa ²/v enters the resulting equations. Even the linearized equations are difficult to solve analytically and the method of matched asymptotic expansions is to be applied. Central differences are applied to the two-dimensional partial differential equations and are solved by the Peaceman-Rachford ADI method. The resulting algebraic equations are solved by successive over relaxation method. Streamlines are plotted for Ψ=0·01, 0·05, and 0·25 andR _eω =0·1, 0·3, 0·5.     

Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

Separation of variables and exact solutions to nonlinear diffusion equations with x-dependent convection and absorption

This paper considers nonlinear diffusion equations with x-dependent convection and source terms: ut=(Dx(u)ux)+Q(x,u)ux+P(x,u). The functional separation of variables of the equations is studied by using the generalized conditional symmetry approach. We formulate conditions for such equations which admit the functionally separable solutions. As a consequence, some exact solutions to the resulting equations are constructed. Finally, we consider a special case for the equations which admit the functionally separable solutions when the convection and source terms are independent of x.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

On Derivation and Classification of Vlasov Type Equations and Equations of Magnetohydrodynamics. The Lagrange Identity,the Godunov Form,and Critical Mass

The derivation of the Vlasov–Maxwell and the Vlasov–Poisson–Poisson equations from Lagrangians of classical electrodynamics is described. The equations of electromagnetohydrodynamics (EMHD) type and electrostatics with gravitation are obtained. We obtain and compare the Lagrange equalities and their generalizations for different types of the Vlasov and EMHD equations. The conveniences of writing the EMHD equations in twice divergent form are discussed. We analyze exact solutions to the Vlasov–Poisson–Poisson equations with the presence of gravitation where we have different types of nonlinear elliptic equations for trajectories of particles with critical mass m ² = e ²/G, which has an obvious physical sense, where G denotes the gravitation constant and e is the electron charge. As a consequence we have different behaviors of particles: divergence or collapse of their trajectories.     

Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi-infinite stretching sheet

We investigate the steady two-dimensional flow of an incompressible water based nanofluid over a linearly semi-infinite stretching sheet in the presence of magnetic field numerically. The basic boundary layer equations for momentum and heat transfer are non-linear partial differential equations. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations for this investigation are solved numerically using Nachtsheim–Swigert shooting iteration technique together with fourth order Runge–Kutta integration scheme. Effects of the nanoparticle volume fraction ϕ, magnetic parameter M, Prandtl number Pr on the velocity and the temperature profiles are presented graphically and examined for different metallic and non-metallic nanoparticles. The skin friction coefficient and the local Nusselt number are also discussed for different nanoparticles.     

Stability of one-leg θ-methods for nonlinear neutral differential equations with proportional delay

The stability properties of one-leg θ-methods for nonlinear neutral differential equations with proportional delay is investigated. In recent years, the stability of one-leg θ-methods for this class of equations on a quasi-geometric mesh is investigated. Instead, in the present paper, the focus is on stability of one-leg θ-methods for the neutral differential equations with constant delay obtained by applying the approach of transformation to the proportional delay equations. Some sufficient conditions for global stability and asymptotic stability are established. Two numerical examples are also included.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

Adjoint methods for static Hamilton�CJacobi equations

We use the adjoint methods to study the static Hamilton?CJacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions ?? ^?? of those we can get the properties of the solutions u of the Hamilton?CJacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.     

Separable anisotropic differential operators and applications

The nonlocal boundary value problems for anisotropic partial differential-operator equations with a dependent coefficients are studied. The principal parts of the appropriate generated differential operators are nonself-adjoint. Several conditions for the maximal regularity and the fredholmness in Banach-valued Lp-spaces of these problems are given. These results permit us to establish that the inverse of corresponding differential operators belongs to Schatten q-class. Some spectral properties of the operators are investigated. In applications, the nonlocal BVP's for quasielliptic partial differential equations and for systems of quasielliptic equations on cylindrical domain are studied.     

The solubility of certain p-adic equations

Estimates are given for the number of variables required to solve p-adic equations. In particular, systems of homogeneous and of inhomogeneous additive equations, as well as single homogeneous equations in general, are studied.     

Nonlinear differential equations satisfied by certain classical modular forms

A unified treatment is given of low-weight modular forms on ?? ₀(N), N = 2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under ?? ₀(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard?CFuchs equations of triangle subgroups of PSL(2, R), which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with ??(1) is treated.     

Convergent series solution of nonlinear equations

The author's decomposition method [1] provides a new, efficient computational procedure for solving large classes of nonlinear (and/or stochastic) equations. These include differential equations containing polynomial, exponential, and trigonometric terms, negative or irrational powers, and product nonlinearities [2]. Also included are partial differential equations [3], delay-differential equations [4], algebraic equations [5], and matrix equations [6] which describe physical systems. Essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance. The procedure depends on calculation of the author's A_n, a finite set of polynomials [1,13] in terms of which the nonlinearities can be expressed. This paper shows important properties of the A_n which ensure an accurate and computable convergent solution by the author's decomposition method [1]. Since the nonlinearities and/or stochasticity which can be handled are quite general, the results are potentially extremely useful for applications and make a number of common approximations such as linearization, unnecessary.     

Differential Galois theory of linear difference equations

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric equations.