New ideas for decomposing nonlinearities in differential equations

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence {c_j} satisfying 1 = c₁ ? c₂ ? ? and j ? c_j for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910-955] are derived as particular cases.     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

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).     

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

General Sobolev Orthogonal Polynomials

In this paper, we study orthogonal polynomials with respect to the inner product (f, g)_S^(N) =〈u, fg〉+∑_m=1^N λ_m〈u, f^(m)g^(m) 〉, where λ_m≥0 form=1,…,N, anduis a semiclassical, positive definite linear functional. For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated withu.     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).     

Singularity conditions for the non-existence of a common quadratic Lyapunov function for pairs of third order linear time invariant dynamic systems

The condition that a finite collection of stable matrices {A₁, … , A_M} has no common quadratic Lyapunov function (CQLF) is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters. These conditions are applied to the case of two stable 3 × 3 matrices, where they are used to derive necessary and sufficient conditions for the non-existence of a CQLF.     

On approximately simultaneously diagonalizable matrices

A collection A₁, A₂, …, A_k of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A₁, A₂, …, A_k. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A₁, …, A_k when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)     

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.     

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u_i)_t=Δu_i+f_i(u₁,…,u_m) with and u_i(x,0)=φ_i(x), i=1,…,m, in a bounded domain Ω⊂Rn. Under suitable assumptions on f_i, he proves that, if φ_i?(1+ε₀)ψ_i in , for some small ε₀>0, then the solutions blow up in a finite time, where ψ_i is a positive solution of Δψ_i+f_i(ψ₁,…,ψ_m)?0, with ψ_i|_∂Di=0 for i=1,…,m. If m=1, the initial value can be negative in a subset of Ω.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

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.     

Some trace formulae involving the split sequences of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let diag(θ₀, θ₁, … , θ_d) denote the diagonal matrix referred to in (ii) above and let denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u₀, u₁, … , u_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Au_i = θ_iu_i + u_i+1 (0 ? i ? d − 1), Au_d = θ_du_d, , . The sequence ?₁, ?₂, … , ?_d is called the first split sequence of the Leonard pair. It is known that there exists a basis v₀, v₁, … , v_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Av_i = θ_d−iv_i + v_i+1 (0 ? i ? d − 1),Av_d = θ₀v_d, , . The sequence ?₁, ?₂, … , ?_d is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function.     

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u_t = Δu + u^m(x, t)e^pv(0,t), v_t = Δv + u^q(0, t)e^nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u₀(v₀) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.     

OMC for scalar multiples of unitaries

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a₁, … , a_n and b₁, … , b_n, respectively. Then for any scalars t and s

     

Fractional differential transform method combined with the Adomian polynomials

A modification of the fractional differential transform method (FDTM) for solving nonlinear fractional differential equations (FDEs) is presented. In this technique, the nonlinear term is replaced by its Adomian polynomial of index k. Then the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index. Thus nonlinear FDEs can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the Adomian polynomials. Numerical examples with different types of nonlinearities are solved and good results are obtained.     

Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rⁿ of the quasilinear singular parabolic equations: u_t = div(|∇u|^m − 1∇u) + t^s|x|^σu^p, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that p_c ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ p_c then every non-trivial solution blows up in finite time, but for p > p_c, a small positive global solution exists.     

Adomian polynomials: A powerful tool for iterative methods of series solution of nonlinear equations

In this article, we illustrate how the Adomian polynomials can be utilized with different types of iterative series solution methods for nonlinear equations. Two methods are considered here: the differential transform method that transforms a problem into a recurrence algebraic equation and the homotopy analysis method as a generalization of the methods that use inverse integral operator. The advantage of the proposed techniques is that equations with any analytic nonlinearity can be solved with less computational work due to the properties and available algorithms of the Adomian polynomials. Numerical examples of initial and boundary value problems for differential and integro-differential equations with different types of nonlinearities show good results.