Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions

The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on Cⁿ[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cⁿ[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.     

Bifurcation of the positive solutions to Henon equation

Three algorithms based on the bifurcation method is applied to solving the D₄ symmetric positive solutions to the boundary value problem of = Henon equation. Taking r in Henon equation as a bifurcation parameter, the D₄ – ∑_d (D₄ – ∑₁, D₄ – ∑₂) symmetry-breaking bifurcation point on the branch of the D₄ symmetric positive solutions is found via the extended systems. Finally, ∑_d (∑₁, ∑₂) symmetric positive solutions = are computed by the branch switching method based on the Liapunov-Schmidt reduction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow Up in a Nonlinearly Damped Wave Equation

In this paper we consider the nonlinearly damped semilinear wave equation u_tt – Δu + au_t |u_t|^{m – 2} = bu|u|^{p – 2} associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2].     

A strongly degenerate diffusion equation with strong absorption

It is shown that the Cauchy problem in ? for the strongly degenerate parabolic equation $$ u_t = u^p u_{xx} - u^{-q} \chi _{ \{u>o \} }\, , \quad p \ge 1 \, , \quad q > -1 \, , $$ has a nonnegative weak solution for any nonnegative bounded continuous initial datum, provided that q ≤ p – 1, while there is no (continuous) weak solution for q > p – 1. The evolution of the spatial positivity set {u(t) > 0}, continuity of the free boundary and the extinction rate are also investigated. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study of the Free Streaming Operator in Slab Geometry in Dependence of the Boundary Conditions

We study the free streaming operator T_∧ in a slab domain with boundary conditions described by a linear matrix operator Λ acting between the ‘incoming’ and ‘outgoing’ particle fluxes. Under suitable assumptions on the entries of Λ, it is proved that the resolvent operator of T_∧ is positive. It is proved also that T_∧ is the generator of a positive strongly continuous semigroup, whose type depends on the norm of the entries of Λ. Some examples are given. In particular the case of Maxwell type boundary conditions is studied and the location of the spectrum of T_∧ is improved. © 1997 by B. G. Teubner Stuttgart – John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 717–736 (1997).     

An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A? ?_b,cl (χ,^bΩ^1/2) (cf. [22], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C* – algebra. On the other hand, the symbolic structure of A can be extended continuously to the C* – algebra B generated by A as a subalgebra of ζ(σ^bL²(χ, ^bΩ^1/2)) by a generalization of a method of Gohberg and Krupnik. Furthermore, A is densely embedded in a Fréchet algebra A ? B which is a ?* – algebra in the sense of Gramsch [9, Definition 5.1], reflecting also smooth properties of the original algebra A.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Projection from One Co – Invariant Subspace onto Another in Character – Automorphic Hardy Space on a Multiply Connected Domain

In a case of a theory in a unit disk the solution of a problem on the invertibility of an orthogonal projection from one co–invariant subspace of the shift operator onto another turned out to be essential for the solution of the problem on the Riesz basis property of the reproducing kernels and in particular for the solution of the problem on the basis of exponentials in L₂ space on a segment. In the present paper we are dealing with the similar problems in harmonic analysis on a finitely connected domain. Namely we obtain necessary and sufficient conditions for the invertibility of an orthogonal projection from one co – invariant subspace of character – automorphic Hardy space in the domain onto another. The given condition has a form of a Muckenhoupt condition for a certain weight on the boundary of the domain, but essentially depends on a character. Namely, for two fixed character – automorphic inner functions, which define the co – invariant subspaces, the projection may be invertible for one character and not invertible for another.     

Finite element approximation of electromagnetic fields in three dimensional space

A finite element approximation scheme for electromagnetic fields in three dimensional domains is considered. The trial sub-spaces consist of piecewise linear continuous fields satisfying the boundary condition at the boundary nodes of the subdivision. Optimal (L²)³ – and (H¹)³ – convergence are proved.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

Some inverse problems raised from a mathematical model of ductal carcinoma in situ

The growth of avascular tumors has been studied by many authors. In this paper we study a special kind of cancer, ductal carcinoma in situ (DCIS) in order to investigate possible procedures to connect free boundary model of DCIS with clinical data. This paper is to present some results of our research on mathematical modeling, analysis, and numerical simulations of ductal carcinoma in situ. In particular, we formulate a number of inverse problems for the well-posed free boundary valued problem related to clinical diagnose of cancer.     

Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities

The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (u_t – Δu – f)(v – u) ? 0 for all regular v ? 0, where u ? 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ? Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {u_m}. The sequence {u_m} may be interpolated linearly in t to obtain an approximation U_Δt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for u_m – u_m-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of U_Δt and ?U_Δt, to u and ?u, respectively, on D ; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence x_Δt, where x_Δt interpolates x _Δt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.     

Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol

Let M be a compact smoot Riemannian manifold without boundary and let D = ad₀δ₀ + bδ₁d₁ – E on the space of smooth sections of the cotangent bundle where a and b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-differential operator calculus to compute the quadratic term a₄(D) in the asymptotic expansion of the heat equation trace.     

Zur Lösung einer Klasse von Gleichungen mit einem monotonen Potentialoperator

This paper is concerned with terminable and interminable paths and trails in infinite graphs. It is shown that

• The only connected graphs which contain no 2 – ∞ way and in which no finite path is terminable are precisely all the 1 – ∞ multiways.
• The only connected graphs which have no 2 – ∞ trail and in which no finite trail is terminable are precisely all the 1 – ∞ multiways all of whose multiplicities are odd numbers and which have infinitely many bridges.
• In addition the strucuture of those connected graphs is determined which have a 1 – ∞ trail and in which no 1 – ∞ trail but every finite trail is terminable.

In this paper the terminology and notation of a previous paper of the writer [1] and of F. HARARY 's book [6] will be used. Furthermore, a graph consisting of the distinct nodes n₁,…,n_δ (where δ≧1) and of one or more (n_i, n_i+1)-edges for i = 1,…, δ – 1 will be called a multiway, and analogously for 1 – ∞ and 2 – ∞ multiways. The number of edges joining n_i and n_i+1 will be called the (n_i,+1)-multiplicity. Thus a multiway in which each multiplicity is 1 is a way. Multiplicities are allowed to be infinite.     

A note on certain matrices with h(x) – Fibonacci quaternion polynomials

In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences.     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)