Surjectivity of partial differential operators with good fundamental solutions

We characterize the surjectivity of a linear partial differential operator with constant coefficients on E(Ω) as well as on D^′(Ω) in terms of the existence of “good” shifted fundamental solutions. This characterization complements results of Meise, Taylor, and Vogt as well as Frerick and the present author.     

Solutions of stochastic partial differential equations as extremals of convex functionals

Summary Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations.     

Existence of solutions for impulsive partial neutral functional differential equations

This paper is concerned with partial neutral functional differential equations of first and second order with impulses. We establish some results of existence of mild solutions for these classes of equations.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

On solutions of linear differential equations with entire coefficients

The first result of the paper concerns the effect of perturbation of the entire coefficients of certain linear differential equations on the oscillation of the solutions. Subsequent results involve the separation of the zeros of a Bank-Laine function.     

Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations

We investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations , t?s, us=φ, where B(t) is a family of nonlinear multivalued, α-accretive operators with D(B(t)) possibly depending on t, and the operators F(t,.) being defined—and Lipschitz continuous—possibly only on “thin” subsets of the initial history space E. The results are applied to population dynamics models. We also study the asymptotic behavior of solutions to this equation. Our analysis will be based on the evolution operator associated to the equation in the initial history space E.     

A method for constructing solutions to linear systems of partial differential equations

We obtain some conditions of solvability in Sobolev spaces for the systems of linear partial differential equations and deduce the corresponding formulas for solutions to these systems. The solutions are given as the sum of the series whose terms are the iterations of some pseudodifferential operators constructed explicitly.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.     

The study of nonlinear fractional partial differential equations via the Khalouta-Atangana-Baleanu operator

This paper studies nonlinear fractional partial differential equations via the Khalouta-Atangana-Baleanu operator. Using Banach’s fixed point theorem we obtain new results on the existence and uniqueness of solutions to the proposed problem. Furthermore, two new semi-analytical methodscalled Khalouta homotopy perturbation method (KHHPM) and Khalouta variational iteration method (KHVIM) are presented to find new approximate analytical solutions to our nonlinear fractional problem. The first of the two new proposed methods, KHHPM, is a hybrid method thatcombines homotopy perturbation method and Khalouta transform in the sense of Atangana-Baleanu-Caputo derivative. The other method, KHVIM is also a hybrid method that combines variational iteration method and Khalouta transform in the sense of Atangana-Baleanu-Caputo derivative. Convergence and absolute error analysis of KHHPM and KHVIM were also performed. A numerical example is provided to support our results. The results obtained showed that the proposed methods are very impressive, effective, reliable, and easy methods for dealing with complex problems in various fields of applied sciences and engineering.     

A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries

By using the index theory for unbounded self-adjoint operator equations and the symmetric mountain pass theorem, we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions. We prove an abstract theorem, and give some applications to first order Hamiltonian systems with Sturm–Liouville boundary conditions and delay differential equations.     

Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presenceof upper and lower solutions as well as the method of quasilinearization(QSL) for general non-linear second-order singular ordinarydifferential equations. We show the existence of solutions underthe assumption of weak continuity of the non-linear part. Ifthe non-linear part is monotone decreasing, a solution may beobtained by the QSL method as the strong limit of a quadraticallyconvergent sequence of approximate solutions. Under strongerassumptions on the linear and the non-linear parts, a solutionis quadratically bracketed between two monotone sequences ofapproximate solutions of certain related linear equations.     

Positive self-adjoint operators generated by products of differential expressions

Under the assumption of the product τ:=l⁺l of the regular differential expression l and its adjoint l⁺, being well-formed, a complete characterization of all the positive self-adjoint extensions of the minimal operator generated by τ in terms of boundary conditions is given.     

Universal perturbations of linear differential equations

Let be a fundamental solution of with and bounded on . We prove that there exist arbitrary small matrix functions with limit as such that has solutions with dense in .

     

Explicit Hamiltonian realizations of non-degenerate three-dimensional real linear differential systems

The purpose of this work is to give explicit Hamiltonian realizations for all non-degenerate real three-dimensional linear differential systems.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

A normalized sparse linear equations solver

Normalized factorization procedures for the solution of large sparse linear finite element systems have been recently introduced in [3]. In these procedures the large sparse symmetric coefficient matrix of irregular structure is factorized exactly to yield a normalized direct solution method. Additionally, approximate factorization procedures yield implicit iterative methods for the finite difference or finite element solution. The numerical implementation of these algorithms is presented here and FORTRAN subroutines for the efficient solution of the resulting large sparse symmetric linear systems of algebraic equations are given.