Nonhomogeneous Initial Boundary Value Problems Research Articles

Chaos of the initial and boundary value problems for the reaction-diffusion equations

This paper investigates an initial and boundary value problem for the reaction-diffusion equations, which can be considered as a linearized form of the advective Fisher-KPP equations. It is demonstrated that the initial and boundary value problem is chaotic when the three parameters of the reaction-diffusion equation vary above a specific surface. However, stable solutions are obtained both on and below this surface within a particular subset of initial values. The chaos and stability of the nonhomogeneous initial boundary value problem are further studied. Finally, some numerical examples are provided to illustrate the validity of the obtained results.

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

Non-homogeneous initial boundary value problems for the biharmonic Schrödinger equation on an interval

Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

Initial-boundary value problem for the Hirota equation posed on a finite interval

In this paper, we turn our attention to a nonhomogeneous initial boundary value problem for Hirota equation posed on a bounded interval (0,L). In particular, the explicit solution formula of linear nonhomogeneous boundary value problem is established by Laplace transform. Using space L2(0,T;H0s−1(0,1)), which is preparing for trilinear estimates and Lions-Magenes interpolation theorem, we prove the local existence, uniqueness, and Lipschitz continuous in C(0,T;Hs(0,1))∩L2(0,T;Hs+1(0,1)) corresponding to the initial and boundary data. Moreover, the local solution extends to a global one by a priori bound.

A NONHOMOGENEOUS BOUNDARY-VALUE PROBLEM FOR THE MODIFIED ANISOTROPIC HEISENBERG SPIN CHAIN POSED ON A FINITE DOMAIN

This article is concerned with the Modified Anisotropic Heisenberg Spin Chain. We prove that the associated nonhomogeneous initial-boundary value problem has a unique globally smooth solution in $H^{2k+1}(m, n)$ for $k\geq 1$. Our main new ingredient is a technique of spatial difference and crucial uniformed estimates of the step-size $h.$ Meanwhile, to prove the global existence, we overcome drawbacks which are not exist in corresponding Cauchy problem.

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. Here the well-posedness of bounded solutions for a non-homogeneous initial-boundary value problem associated with this equation is studied.

Solutions of Delayed Partial Differential Equations With Space-Time Varying Coefficients

A constructive algorithm using Chebyshev spectral collocation is proposed for computing trustworthy approximate solutions of linear and weakly nonlinear delayed partial differential equations or initial boundary value problems, with continuous and bounded coefficients. The boundary conditions are assumed to be Dirichlet. The solution of linear problems is obtained at Chebyshev grid points in space and a given interval of time. The algorithm is then extended to systems with weak nonlinearities using perturbation series, which yields nonhomogeneous initial boundary value problems without delay. The proposed methodology is illustrated using examples of linear and weakly nonlinear heat and wave equations with bounded continuous space-time varying coefficients.

Initial Boundary Value Problem and Asymptotic Stabilization of the Two‐Component Camassa‐Holm Equation

The nonhomogeneous initial boundary value problem for the two‐component Camassa‐Holm equation, which describes a generalized formulation for the shallow water wave equation, on an interval is investigated. A local in time existence theorem and a uniqueness result are achieved. Next by using the fixed‐point technique, a result on the global asymptotic stabilization problem by means of a boundary feedback law is considered.

Odd-order quasilinear evolution equations posed on a bounded interval

We study in a rectangle Q_T = (0, T)×(0, 1) global well-posedness of nonhomo-geneous initial-boundary value problems for general odd-order quasilinear partial differential equations. This class of equations includes well-known Korteweg–de Vries and Kawahara equations which model the dynamics of long small-amplitude waves in various media. Our study is motivated by physics and numerics and our main goal is to formulate a correct nonhomogeneous initial-boundary value problem under consideration in a bounded interval and to prove the existence and uniqueness of global in time weak and regular solutions in a large scale of Sobolev spaces as well as to study decay of solutions while t → ∞.

Initial boundary value problem and asymptotic stabilization of the Camassa–Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa–Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.

Optimal Results for the Nonhomogeneous Initial-Boundary Value Problem for the Two-Dimensional Navier–Stokes Equations

In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes equations in a general open space domain in R2 with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve the equations for data with optimally low regularity in both space and time.

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

Nonhomogeneous initial-boundary value problems for nonlinear diffusion–convection equations

Nonhomogeneous initial-boundary value problems for nonlinear diffusion–convection equations

Rational methods with optimal order of convergence for partial differential equations

We construct rational approximations to linear, non-homogeneous initial boundary value problems. They are based on A-acceptable rational approximations to the exponential for the time discretization. The order reduction phenomenon is avoided and the optimal order of convergence in time is achieved. The theoretical results are illustrated numerically.

Nonhomogeneous Initial-Boundary Value Problems for Coercive and Self-Controlling Models of Monotone Type

Based on the idea of partial Yosida approximation we prove global in time existence of nonhomogeneous initial-boundary value problems for the class of coercive models of monotone type from the theory of inelastic deformations of metals. Then using this result and the coercive limits idea introduced in [8], we approximate self-controlling problems with nonhomogeneous boundary data by a sequence of coercive models and prove a convergence result.

Explicit single step methods with optimal order of convergence for partial differential equations

We construct explicit single step discretizations of linear, non-homogeneous initial boundary value problems. These methods use polynomial approximations for the time discretization. The order reduction phenomenon, present when a fully discrete Runge–Kutta is used, is avoided and the maximum expected order of convergence is achieved. The results are illustrated numerically.

Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gases

Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spacesL ∞(Ω) orL 2(Ω) and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved, i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.