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.     

Mathematical modelling and optimisation of a waste water treatment plant by combined oxygen electrode and biological waste water treatment model

Structured mathematical models of oxygen electrode and biological waste water treatment were combined to optimise the position of electrode in the bioreactor. Parameters sensibility testing of mathematical model has shown that the model was most sensitive to the change of these parameters (in decreasing order of significance): oxygen consumption per unit of BOD_5, specific consumption rate of adsorbed BOD₅, volumetric coefficient of oxygen transfer rate (k_La), and waste water inflow. Two positions of electrode were tested: in the bioreactor and in the bioreactors syphon outlet shaft. Results obtained for the exchange of the oxic/anoxic phase and sludge recycling plant indicated that a reduction of electrical power for the aeration could be achieved if O₂ electrode is placed in the bioreactor tank, compared to position in outlet shaft. Better positioning of electrode has resulted in shortening of the oxic/anoxic cycle by 13% and the daily working time of aerators for 0.5 h per day, respectively. In the same time, the number of oxic/anoxic cycles per day was increased by 29%. In addition, results obtained by mathematical modelling indicated no debasement of chemical composition in out coming purified water if new position will be used.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randi? index χ = χ(G) of a graph G   is defined by χ(G)=_∑(uv)(_δuδv)^-1/2$\chi (G)={\sum }_{(\mathit{uv})}({\delta }_u{\delta }_v{)}^{-1/2}$, where δ_u is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number n_k of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randi? index attained its minimum value with given k (k ? n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number n_k of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.     

Peakon, loop and solitary travelling wave solutions for the general modified CH-DP equation

Travelling wave solutions for the general modified CH-DP equation u_t − u_xxt + αu²u_x − βu_xu_xx = uu_xxx are developed. By using the dynamical system method, a peakon and a dark soliton are found to coexist for the same wave speed. Exact explicit blow-up solutions are given. By using numerical simulation, a loop solution for a special case is discussed.     

Steiner intervals, geodesic intervals, and betweenness

The concept of the k-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping S:V×?×V?2^V such that S(u₁,…,u_k) consists of all vertices in G that lie on some Steiner tree with respect to a multiset W={u₁,…,u_k} of vertices from G. In this paper we obtain, for each k, a characterization of the class of graphs in which every k-Steiner interval S has the so-called union property, which says that S(u₁,…,u_k) coincides with the union of geodesic intervals I(u_i,u_j) between all pairs from W. It turns out that, as soon as k>3, this class coincides with the class of graphs in which the k-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, S satisfies (m), if x₁,…,x_k∈S(u₁,…,u_k) implies S(x₁,…,x_k)⊆S(u₁,…,u_k), and S satisfies (b2) if x∈S(u₁,u₂,…,u_k) implies S(x,u₂,…,u_k)⊆S(u₁,…,u_k). In the case k=3, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval S satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class of graphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.     

Determination of an unknown coefficient in a nonlinear heat equation

The problem of determining an unknown term k(u) in the equation k(u)u_t=(k(u)u_x)_x is considered in this paper. Applying Tikhonov's regularization approach, we develop a procedure to find an approximate stable solution to the unknown coefficient from the overspecified data.     

Numerical similarity reductions of the (1+3)-dimensional Burgers equation

We consider the (1+3)-dimensional Burgers equation u_t = u_xx + u_yy + u_zz + uu_x which has considerable interest in mathematical physics. Lie symmetries are used to reduce it to certain ordinary differential equations. We employ numerical methods to solve a number of these ordinary differential equations.     

Estimates for k-Hessian operator and some applications

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F _k [u] = 0, where F _k [u] is the elementary symmetric function of order k, 1 ? ? 6 n, of the eigenvalues of the Hessian matrix D ² u. For example, F ₁[u] is the Laplacian Δu and F _n [u] is the real Monge-Ampère operator detD ² u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.     

Control parameter estimation in a semi-linear parabolic inverse problem using a high accurate method

Parabolic inverse problems have an important role in many branches of science and technology. The aim of this research work is to solve these classes of equations using a high order compact finite difference scheme. We consider the following inverse problem for finding u(x, t) and p(t) governed by u_t = u_xx + p(t)u + φ(x, t) with an over specified condition inside the domain. Spatial derivatives are approximated using central difference scheme. The time advancement of the simulation is performed using a “third order compact Runge-Kutta method”. The convergence orders for the approximation of both u and p are of o(k³ + h²) which improves the results obtained in the literature. An exact test case is used to evaluate the validity of our numerical analysis. We found that the accuracy of the results is better than that of previous works in the literature.     

On the proof of continued fraction expansions for irrationals

We use matrices to prove two theorems in regard to the continued fraction expansion for Σ_{k = 0}^∞u^−2k and for Σ_{k = 0}^∞u^−c(k), where c(k) is any sequence of positive integers that increase quickly.     

Applications of the Fourier—Wiener transform to differential equations on infinite dimensional spaces,II

In this paper, we investigate the existence and regularity of solutions for Cauchy problems associated with the following two equations: (1) u_t(x, t) = (?1)^{k + 1}Δ^ku(x, t), and (2) u_tt(x, t) = (?1)^{k + 1}Δ^ku on infinite dimensional spaces. The nonhomogeneous cases associated with (1) and (2) are also considered.     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

On palindromic factorization of words

Given a finite word u, we define its palindromic length  _|u|pal${|u|}_{\mathrm{pal}}$ to be the least number n   such that u=_v1v2…_vn$u=v_1{v}_2\dots v_n$ with each _vi$v_i$ a palindrome. We address the following open question: let P be a positive integer and w   an infinite word such that _|u|pal?P${|u|}_{\mathrm{pal}}?P$ for every factor u of w. Must w be ultimately periodic? We give a partial answer to this question by proving that for each positive integer k, the word w must contain a k  -power, i.e., a factor of the form u^k$u^k$. In particular, w cannot be a fixed point of a primitive morphism. We also prove more: for each pair of positive integers k and l, the word w must contain a position covered by at least l distinct k-powers. In particular, w cannot be a Sierpinski-like word.     

One-point pseudospectral collocation for the one-dimensional Bratu equation

The one-dimensional planar Bratu problem is u_xx + λ exp(u) = 0 subject to u(±1) = 0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x) ≈ u₀ (1 − x²) where u₀ is determined by collocation at a single point x = ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0) ≈ −W(−λ(1 − ξ²)/2)/(1 − ξ²) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

On uniqueness properties of solutions of the k-generalized KdV equations

In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg-de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference u₁−u₂ of two solutions u₁,u₂ of (1.1) at two different times t₀=0 and t₁=1 which guarantee that u₁≡u₂.     

Mutually independent bipanconnected property of hypercube

A graph is denoted by G with the vertex set V(G) and the edge set E(G). A path P = 〈v₀, v₁, … , v_m〉 is a sequence of adjacent vertices. Two paths with equal length P₁ = 〈 u₁, u₂, … , u_m〉 and P₂ = 〈 v₁, v₂, … , v_m〉 from a to b are independent if u₁ = v₁ = a, u_m = v_m = b, and u_i ≠ v_i for 2 ? i ? m − 1. Paths with equal length from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, d_G(u, v) ? l ? ∣V(G) − 1∣ with (l − d_G(u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with d_Qn(u,v)?n-1, is (n − 1, l)-mutually independent bipanconnected for every with (l-d_Qn(u,v)) being even. As for d_Qn(u,v)?n-2, it is also (n − 1, l)-mutually independent bipanconnected if l?d_Qn(u,v)+2, and is only (l, l)-mutually independent bipanconnected if l=d_Qn(u,v).