On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.     

Analytic methods applied to a sequence of binomial coefficients

In previous work we have shown that the binomial coefficients C_n··kr, r are strongly logarithmically concave for 0?r?[n/(k+1)] and hence have at most a double maximum. Let r_{n, k} be the least integer at which this maximum occurs. Properties of {r_{n, k}}_{n, k} are best derived by introducing the polynomial family Q_k(x,y) defined by Q_k(x,y) = ∏^k+1_j=1 [1?(k+1)x+jy]?x∏^k_j=1[1?kx+jy]. It is shown that for each k there is a unique function η_k(y) defined on [0, 1/(k+1)] which is analytic in a neighbourhood of zero and which satisfies Q_k(η_k(y), y)=0. Setting η_k(0)=δ_k, η¹_k(0)=α_k we prove that r_rn,k = [nδ_k] o[nδ_k]+1 and further, that the number of times that r_m,k = [mδ_k]+1 for k+1? m ? n is asymptotically nα_k. Several other properties of α_k are derived, including 0<α_k <1/2.     

Delay-dependent stability of symmetric schemes in Boundary Value Methods for DDEs

We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations y^′(t)=ay(t)+by(t-τ) with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with (k₁+m,k₂)-boundary conditions is called τ_k1,k2(0)-stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR₂s) kind, and the Top Order Methods (TOMs), are considered in this paper.By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be τ_ν,ν-1(0)-stable.     

Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping

In this paper, we are concerned with the oscillation of third order nonlinear delay differential equations of the form

(r₂(t)(r₁(t)y^′)′)′+p(t)y^′+q(t)f(y(g(t)))=0.     

Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

On the oscillation of a second-order delay equation

The oscillatory nature of two equations (r(t) y′(t))′ + p₁(t)y(t) = f(t), (r(t) y′(t))′ + p₂(t) y(t ? τ(t))= 0, is compared when positive functions p₁ and p₂ are not “too close” or “too far apart.” Then the main theorem states that if h(t) is eventually negative and a twice continuously differentiable function which satisfies (r(t) h′(t))′ + p₁(t) h(t) ? 0, then this inequality is necessary and sufficient for every bounded solution of (r(t) y′(t))′ + p₂(t) y(t ? τ(t)) = 0 to be nonoscillatory.     

On the growth of solutions of certain second-order nonlinear differential equations

In this paper, we determine the growth of real-valued solutions of certain second-order algebraic differential equations. Our main result, together with a result of G. Valiron, shows that if y₀ is an entire function which has only real, nonnegative coefficients in its power series around the origin, and which is a solution of a quadratic second-order algebraic differential equation, then y₀ satisfies a growth estimate of the form,y₀(x) ? exp(exp x^c), where c is a constant, for all sufficiently large x. The determination of the growth of such solutions was an open problem since the Valiron-Wiman theory fails to provide any information on growth, if the equation possesses a solution of infinite order of growth.     

Oscillation criteria for second-order nonlinear neutral delay dynamic equations

In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation

(r(t)((y(t)+p(t)y(t−τ)Δ⁾γ⁾Δ⁾+f(t,y(t−δ))=0     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y=Φx of a k-sparse signal x∈? ^N , where Φ satisfies the restricted isometry property, then the approximate recovery x ^# via ? ₁-minimization is within O(δ) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma–Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order δ(k/m)^(r?1/2)α for any 0<α<1, if m? _r,α k(logN)^1/(1?α). The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.     

Accuracy of several multidimensional refinable distributions

Compactly supported distributions f₁,..., f_r on ℝ^d are fefinable if each f_i is a finite linear combination of the rescaled and translated distributions f_j(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝^d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σ_k∈Λ c_k f(Ax−k), where Λ is a finite subset of Γ, the c_k are r×r matrices, and f=(f₁,...,f_r)^T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f₁,...,f_r along the lattice Γ. We determine the accuracy p from the matrices c_k. Moreover, we determine explicitly the coefficients y_α,i(k) such that x^α=Σ _i=1 ^r Σ_k∈Γy_α,i(k) fi(x+k). These coefficients are multivariate polynomials y_α,i(x) of degree |α| evaluated at lattice points k∈Γ.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

Functions with Nonzero Wronskian form a fundamental solution set

This self-contained note could find classroom use in a course on differential equations. It is proved that if y₁(x) and y₂(x) are C ² -functions whose Wronskian is never zero for α < x < β, then y₁ and y₂ form a fundamental solution set for a uniquely determined second-order linear homogeneous ordinary differential equation, y″ + p(x)y′ + q(x)y = 0, whose coefficients, p(x) and q(x), are continuous on (α, β).     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

On the existence of periodic solutions to p-Laplacian Rayleigh differential equation with a delay

In this paper, the authors study the existence of periodic solutions to a p-Laplacian Rayleigh differential equation with a delay as follows:

(φp(y^′(t))^)′+f(y^′(t))+g(y(t−τ(t)))=e(t),     

Oscillation criteria and growth of nonoscillatory solutions of higher order differential equations

We look for conditions under which all solutions of the nonlinear ordinary differential equation y⁽ⁿ⁾ + f(t, y) = 0, t ? 0, ?∞ < y < ∞, are oscillatory, as well as consider the asymptotic behaviour of the nonoscillatory solutions.     

Control problems for parabolic equations on controlled domains

Given the one-dimensional heat equation v_t = v_xx on the controlled domain Q(y) = {(t, x); 0 < x < y(t), 0 < t < T} subject to some initial-boundary conditions, we study the problem of optimally selecting y(·) from some admissible class so as to maximize a given payoff of fixed duration. Q(y) is thus a controlled domain. We also study the problem in which the heat equation holds in Q(y, z) = {z(t) < x < y(t), 0 < t < T}; z minimizing, y maximizing, i.e., the differential game. The principle techniques involved are (i) transforming the controlled domain to an uncontrolled domain and then (ii) using the method of lines for parabolic equations to enable us to use known results for control systems governed by ordinary differential equations. Sufficient conditions for existence in an admissible class is given and the method of lines allows numerical techniques to be applied to determine the optimal control in our class.     

The Jacobian problem as a system of ordinary differential equations

LetP=x ⁿ +P _n?1(y)x ^n?1+…+P ₀(y),Q=x ^m +Q _m?2(y)x ^m?2+…+Q ₀(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP _x Q _y ?P _y Q _x =1. Then:*

1. K[Q _m?2(y), …,Q ₀(y)]=K[y],
2. K[P, Q]=K[x, y] ifQ=x ^m +Q _k (y)x ^k +Q _r (y)x ^r

     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.