Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects

This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.     

Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions , and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.     

Comparison theorems for symplectic systems of difference equations

In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of conjoined bases of two symplectic systems with matrices W _i and $ \hat W_i $ \hat W_i as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number of eigenvalues λ of a symplectic boundary value problem on the interval (λ ₁, λ ₂]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes the well-known reciprocity principle for discrete Hamiltonian systems.     

Boundary problems for systems of partial differential operators of mixed order

On a compact n-dimensional C^∞ manifold $\text{Ω}$ with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere.     

On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound‐soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient _n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.     

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Given a smooth projective toric variety X, we construct an A_∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.      

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.     

Generalized three-point difference schemes of high-order accuracy for systems of second-order nonlinear ordinary differential equations

For systems of second-order nonlinear ordinary differential equations with the Dirichlet boundary conditions, we develop generalized three-point difference schemes of high-order accuracy on a nonuniform grid. The construction of the suggested schemes requires solving four auxiliary Cauchy problems (two problems for systems of nonlinear ordinary differential equations and two problems for matrix linear ordinary differential equations) on the intervals [x _j−1, x _j ] (forward) and [x _j , x _j+1] (backward) at each grid point; this is done at each step by any single-step method of accuracy order $ \bar m $ \bar m = 2[(m+1)/2]. (Here m is a given positive integer, and [·] is the integer part of a number.) We prove that such three-point difference schemes have the accuracy order $ \bar m $ \bar m for the approximation to both the solution u of the boundary value problem and the flux K(x)d u/dx at the grid points.     

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.     

Isomorphism Classes of A-Hypergeometric Systems

Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.     

On the solvability of a nonlocal boundary value problem with the equality of fluxes at a part of the boundary and of the adjoint problem

In the present paper, we consider a nonlocal boundary value problem for the Laplace operator in a circular sector with the equality of fluxes on the radii and with zero value of the solution on one of the radii. We also consider the adjoint problem. We prove the uniqueness of the solution of these problems and obtain an explicit form for the solution by the spectral method. When proving the solvability of the problems, we study the completeness and the basis property of systems of root functions for problems of the type of the Samarskii-Ionkin problem in L _p , which can be of interest in itself.     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of optimal strategies for a fire confinement problem

We consider a class of variational problems for differential inclusions related to the control of forest fires. The area burned by the fire at time t > 0 is modeled as the reachable set for a differential inclusion ? ∈ F(x) starting from an initial set R₀. To block the fire, a barrier can be constructed progressively in time at a given speed. In this paper we prove the existence of an optimal strategy, which minimizes the value of the area destroyed by the fire plus the cost of constructing the barrier. © 2008 Wiley Periodicals, Inc.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.     

Regular Degenerate Separable Differential Operators and Applications

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in L _p -spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.     

On well posedness of best simultaneous approximation problems in Banach spaces

The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.