On additive solutions of a linear equation

We investigate the functional equation $$ \sum\limits_{i = 1}^n {\alpha _i A(\beta _i x)} = 0 $$ which holds for all x ∈ ? with an unknown additive function A: ? → ? and fixed real parameters α _i , β _i , where i = 1; …; n. Here we give sufficient and necessary conditions for the existence of non-trivial additive solutions of the equation above in some cases depending on the algebraic properties of the parameters.     

Properties of stationary solutions of a generalized Tjon-Wu equation

A generalized version of the Tjon-Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ_∗ of the equation has the following property: Either μ_∗ is supported at one point or suppμ_∗=[0,∞). Moreover we show that in the second case the distribution function of μ_∗ is continuous. Some open questions are discussed.     

On a linear iterative equation

We consider the following iterative equation $$ \sum_{i=0}^{k}a_{i}f^{i}(x)=0, $$ where a₀,…, a _k are given real numbers and ? is an unknown function. Assuming some conditions on the coefficients a₀,…, a _k we prove that this equation has exactly one solution and that the solution depends continuously on the coefficients.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

Existence and uniqueness of solutions for a non-uniformly parabolic equation

In this paper, we study a generalized Burgers equation u_t+(u²)_x=tu_xx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to .     

Safe solutions of a nonlinear car-following equation

A leader-follower pair of cars whose motion is subject to a non-linear delay differential equation are travelling with the same constant velocity u_I when the leader begins to change his velocity in a smooth way to the non-negative velocity u_F < u_I. Conditions are found for the response of the follower to be a safe one according to certain natural safety criteria.     

Solutions with exponential character to a linear functional equation and roots of its characteristic equation

We study the connection between solutions of the functional equation φ(x)=S_∫φ(x+M(s))σ(ds) which are comparable at infinity with functions of the form xeλx, and real roots of the characteristic equation S_∫eλM(s)σ(ds)=1.     

Exact solutions of a novel integrable partial differential equation

We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P_II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P_II with its shift in parameter.We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P_II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel’s equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

On a functional equation arising from comparison of utility representations

We solve the functional equation F₁(t)−F₁(t+s)=F₂[F₃(t)+F₄(s)] for real functions defined on intervals, assuming that F₂ is positive valued and strictly monotonic and that F₃ is continuous. The equation arose from the equivalence problem of utility representations under assumptions of separability, homogeneity and segregation (e-distributivity).     

Boundedness of global positive solutions of a porous medium equation with a moving localized source

In this paper a porous medium equation with a moving localized source ut=ur(Δu+af(u(x₀(t),t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, in one space dimension case, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Separation of variables of a generalized porous medium equation with nonlinear source

This paper considers a general form of the porous medium equation with nonlinear source term: u_t=(D(u)u_xⁿ)_x+F(u), n≠1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated.     

Traveling-wave solution to a nonlinear equation in semiconductors with strong spatial dispersion

The third-order nonlinear differential equation (u _xx ? u) _t + u _xxx + uu _x = 0 is analyzed and compared with the Korteweg-de Vries equation u _t + u _xxx ? 6uu _x = 0. Some integrals of motion for this equation are presented. The conditions are established under which a traveling wave is a solution to this equation.     

Spectral properties of the neutron transport equation

The general equation describing the steady-state flow through a porous column is λu ? D_xA(D_x?(u) + G(u)) = f, where λ is a nonnegative constant. In this paper existence, uniqueness and comparison results for solutions to the Dirichlet and mixed boundary value problems associated with this equation are proven. The existence of a weak solution to the evolution problems associated with the equation u_t = D_x(D_x?(u) + G(u)) are deduced.     

Hopf bifurcation in numerical approximation of the sunflower equation

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point ata =a ₀, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point ata _h =a ₀ +O(h).     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.     

Nonclassical analog of the Goursat problem for a three-dimensional equation with highest derivative

In the present paper, we study the Goursat problem for a three-dimensional equation with highest derivative of fifth order with L _p -coefficients and establish a homeomorphism between certain pairs of Banach spaces by reducing this problem to the equivalent Volterra integral equation.     

On a linear transcendence measure for the solutions of a universal differential equation at algebraic points

In this paper the author continues his work on arithmetic properties of the solutions of a universal differential equation at algebraic points. Every real continuous function on the real line can be uniformly approximated by C^∞-solutions of a universal differential equation. An algebraic universal differential equation of order five and degree 11 is explicitly given, such that every finite set of nonvanishing derivatives y^(k₁)(τ),…,y^(k_r)(τ) (1?k₁<?<k_r) at an algebraic point τ is linearly independent over the field of algebraic numbers. A linear transcendence measure for these values is effectively computed.     

On a boundary value problem for a higher-order elliptic equation

For an elliptic 2lth-order equation with constant (and only leading) real coefficients, we consider the boundary value problem in which the (k _j ? 1)st normal derivatives, j = 1,..., l, are specified, where 1 ≤ k ₁ < ... < k _l . If k _j = j, then it becomes the Dirichlet problem; and if k _j = j + 1, then it becomes the Neumann problem. We obtain a sufficient condition for this problem to be Fredholm and present a formula for the index of the problem.     

High frequency behavior of a rolling ball and simplification of the separation equation

The Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential V (z = cosθ, D, λ) that is difficult to analyze. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of z = cosθ. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency ω ₃-dependence of the width of the nutational band, the depth of motion above V(z _min,D, λ) and the ω ₃-dependence of nutational frequency $\tfrac{{2\pi }} {T} $ .