Nonhomogeneous Problem Research Articles

Nonhomogeneous backward heat conduction problem: Compact filter regularization and error estimates

In this paper, compact filter regularization method is introduced for the numerical solution of nonhomogeneous backward heat conduction problem. Compact filter regularization is a new, simple and convenient regularization method. The error estimate for this method is provided. Meanwhile, the numerical implementation is discussed. Numerical results conclude that the proposed algorithm is efficient and effective.

Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.

Some remarks on a nonhomogeneous eigenvalue problem related to generalized trigonometric functions

In this paper, a nonhomogeneous eigenvalue problem with p-Laplacian is studied. We first show the Sturm–Liouville property and derive the asymptotic expansion of eigenvalues and eigenfunctions. Employing the derivation of the eigenfunctions, we discuss the basis property related to the eigenfunctions. Finally, we also solve two inverse problems, the inverse nodal problem and Ambarzumyan problem.

Existence and uniqueness of solution for a nonhomogeneous discrete fractional initial value problem

This paper is devoted to the study of a nonhomogeneous discrete fractional initial value problem. Using the Laplace transform, we present the existence and uniqueness of the solution. We illustrate the applicability of results by an example.

Gibbs–non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: Closing the Ising gap

We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors. As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes. The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.

Analysis of recent fractional evolution equations and applications

It is believed that Mittag-Leffler function and its variants govern the solutions of most fractional evolution equations. Is it always true? We establish some unknown and useful properties that characterize some aspects of fractional calculus above power-law. These include, the correspondence relations, the equality of mixed partials, the boundedness, the continuity and the eigen-property for the derivative with non singular kernel. Some related evolution equations are studied using those properties and they unexpectedly show that solutions are not governed by Mittag-Leffler function nor its variants. Rather, solutions are governed by unusual functions of type exponential (37), easier to handle and more friendly than power series like Mittag-Leffler functions. Numerical approximations of a second order nonhomogeneous fractional Cauchy problem are performed and show regularity in the dynamics. An application to a geoscience model, the Rössler system of equations is performed where existence and uniqueness results are provided. The system appears to be chaotic and the properties of the derivative with no singular kernel do not interrupt this chaos. We conclude by a similar application to heat model where we solve the model numerically and graphically reveal solutions comparable with expected exact ones.

Multiple Solutions with Sign Information for a Class of Parametric Superlinear (p,\xa02)-Equations

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian (with p>2) and a Laplacian (a two phase equation). The reaction consists of a parametric (p-1)-superlinear term and a (p-1)-sublinear perturbation. We show that for all lambda >0 big, the problem has at least three nontrivial smooth solutions, all with sign information. Also we determine their asymptotic behaviour as the parameter lambda rightarrow infty . When we strengthen the regularity of the perturbation term, we produce a second nodal solution, for a total of four solutions, all with sign information.

(omega ,c)-Periodic solutions for time varying impulsive differential equations

In this paper, we study a class of (omega ,c)-periodic time varying impulsive differential equations and establish the existence and uniqueness results for (omega ,c)-periodic solutions of homogeneous problem as well as nonhomogeneous problem.

Solution of 3D linearly anisotropic scattering, fixed-source multigroup xyz reactor problems by the [formula omitted] Boundary Element – Response Matrix method

This paper aims at extending the work performed by means of the AN Boundary Element – Response Matrix method in a previous paper, devoted to criticality problems, to non-homogeneous problems (i.e. the problems that involve a fixed external source, as in case of the accelerator driven systems). As in the preceding paper, the interest is given to the 3D, xyz multigroup systems in which linearly anisotropic scattering is allowed. Standard interface conditions have been adopted and no external intervention in order to improve the numerical results, such as some kind of discontinuity factors, has been introduced. The latter choice is motivated by the need of establishing a clean test of the performances of the AN method in itself. Results of 2D and 3D multigroup fixed source problems obtained with the method described in this paper are compared with those obtained by well assessed reference codes such as the MCNP Monte Carlo code and the PARTISN discrete ordinates code. Even in the absence of discontinuity factors and despite of the intrinsically approximate character of the AN method, the accuracy of the numerical results is more than satisfactory. Finally the computational time is in most cases much shorter than that required by the chosen reference code.

Higher differentiability for solutions of nonhomogeneous elliptic obstacle problems

This paper focuses on weak solutions to the following variational inequality∫Ω〈a(x,Du),D(φ−u)〉dx≥∫Ω〈|F|p−2F,D(φ−u)〉dx for all φ∈Aψ(Ω)={w∈W1,p(Ω)|w≥ψa.e. inΩ}, where the vector field a satisfies some growth and ellipticity conditions. We derive that the higher differentiability property of the weak solution u is related to the regularity of the assigned obstacle ψ and the nonhomogeneous term F under a suitable integer or fractional differentiability assumption on a(x,ξ) with respect to x.

Hybrid-Trefftz finite elements for non-homogeneous parabolic problems using a novel dual reciprocity variant

A new hybrid-Trefftz finite element for the solution of transient, non-homogeneous parabolic problems is formulated. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases are regular, and defined independently for each finite element, using arbitrary orders. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same, regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. To enhance the reproducibility of this work, the formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community through the FreeHyTE web page.

Nonhomogeneous polyharmonic elliptic problems involving GJMS operator on Riemannian manifold

Let [Formula: see text] be a closed Riemannian manifold, under some assumptions on [Formula: see text] and [Formula: see text] by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: [Formula: see text]. When [Formula: see text] is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

Three-dimensional transient heat conduction analysis by boundary knot method

This paper makes the first attempt to apply the boundary knot method (BKM), in conjunction with dual reciprocity technique, for the solution of three-dimensional transient heat conduction problems. The BKM is a meshless, integration-free, easy-to-program boundary-only numerical technique for high-dimensional problems. The first step of our strategy is to use the finite difference method for temporal derivative to convert the transient heat conduction equation into a nonhomogeneous modified Helmholtz equation. And then the corresponding nonhomogeneous problem is solved using the proposed BKM strategy in conjunction with dual reciprocity technique. Four benchmark numerical examples are investigated in detail, and the numerical results show that the present scheme has the merits of high accuracy, wide applicability, good stability, and rapid convergence and is appealing to solve 3D transient heat conduction problems.

An efficient boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials under heat source load

This paper presents a boundary collocation scheme for transient thermal analysis in large-size-ratio functionally graded materials (FGMs) with heat source load. In the proposed scheme, Laplace transformation and the numerical inverse Laplace transformation (NILT) are implemented to avoid the troublesome time-stepping effect on numerical efficiency. The collocation Trefftz method (CTM) coupled with composite multiple reciprocity method is used to obtain the high accurate results in the solution of nonhomogeneous problems in Laplace-space domain. The extended precision arithmetic is introduced to overcome the ill-posed issues generated from the CTM simulation, the NILT process and the large-size-ratio FGM. Heuristic error analysis and numerical investigation are presented to demonstrate the effectiveness of the proposed scheme for transient thermal analysis. Several benchmark examples are considered under large-size-ratio FGMs with some specific spatial variations (quadratic, exponential and trigonometric functions). The proposed scheme is validated in comparison with known analytical solutions and COMSOL simulation.

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.

Existence and multiplicity results for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

In this paper, existence results for positive solutions to an eigenvalue non-homogeneous Neumann problem are established. Multiplicity results are also pointed out. The proofs are based on variational methods and topological arguments in Orlicz-Sobolev spaces.

A non-homogeneous boundary-value problem for the KdV–Burgers equation posed on a finite domain

We consider the initial–boundary value problem for the KdV–Burgers equation posed on a bounded interval [a,b]. This problem features non-homogeneous boundary conditions applied at x=a and x=b and is known to have unique global smooth solution.

A virtual element method for transversely isotropic elasticity

This work studies the approximation of plane problems concerning transversely isotropic elasticity, using a low-order Virtual Element Method (VEM), with a focus on near-incompressibility and near-inextensibility. Additionally, both homogeneous problems, in which the plane of isotropy is fixed; and non-homogeneous problems, in which the fibre direction defining the isotropy plane varies with position, are explored. In the latter case various options are considered for approximating the non-homogeneous fibre directions at element level. Through a range of numerical examples the VEM approximations are shown to be robust and locking-free for several element geometries and for fibre directions that correspond to mild and strong non-homogeneity.

On a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

ABSTRACTIn this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.

Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem

This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution for the forward and backward fractional heat problems is expressed in terms of eigen function expansion and Mittag–Leffler function. Due to the instability of determining initial data, a regularized truncated solution is considered. Further, the stability estimate for the exact solution and the convergence estimates for the regularized solution using an á-priori choice rule and an á-posteriori choice rule are derived.