Nonhomogeneous Problem Research Articles

Long-time behaviour of solutions to an evolution PDE with nonstandard growth

Abstract In this paper, we prove time estimates for solutions to a general nonhomogeneous parabolic problem whose operator satisfies nonstandard growth conditions. We also study the asymptotic behaviour of solutions to an anisotropic problem.

On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two

ABSTRACT In this article, the Ekeland variational principle, mountain pass theorem and Trudinger–Moser inequality are applied to establish the existence and multiplicity of solutions for the following nonhomogeneous Kirchhoff-type elliptic problem: where Ω is a smooth bounded domain in , is a Kirchhoff function, is a continuous function, , , , and ε is a small positive parameter. The nonlinearity term behaves like when for some .

A non-homogeneous elliptic problem in low dimensions with three symmetric solutions

We investigate the existence of multiple solutions for a low-dimensional Neumann problem with non-standard growth set on a strip-like domain of Rn. Variational techniques, depending in particular on a minimax theorem and a version of the principle of symmetric criticality, are employed to establish the existence of three non-zero solutions endowed with axial symmetry. Some consequences are also derived.

A Fat boundary-type method for localized nonhomogeneous material problems

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a conforming, highly non-uniform mesh numerically unfeasible. In fact, such meshes may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. The problem becomes even more prohibitive when the region requiring a finer-level mesh changes in time, requiring the introduction of refinement and derefinement techniques. To address the aforementioned challenges, we employ a technique related to the Fat boundary method as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings with a series of two-dimensional numerical tests.

Existence and multiplicity of positive solutions for parametric nonlinear nonhomogeneous singular Robin problems

We consider nonlinear Robin problems driven by a nonhomogeneous differential operator and with a reaction that has a singular term and a parametric $$(p-1)$$-superlinear perturbation which need not satisfy the Ambrosetti–Rabinowitz condition. We are looking for positive solutions. Using variational arguments and a suitable truncation and comparison techniques, we prove a bifurcation-type theorem which describes the set of positive solutions as the parameter $$\lambda > 0$$ varies. Also we show the for every admissible value of the parameter $$\lambda >0$$, the problem has a smallest solution $${\bar{u}}_{\lambda }$$ and we determine the monotonicity and continuity properties of the map $$\lambda \rightarrow {\bar{u}}_{\lambda }$$.

A boundary collocation method for anomalous heat conduction analysis in functionally graded materials

This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

The concentrated solutions on the nonhomogeneous scalar curvature problem

We consider the following scalar curvature equation−Δu=(1+εK(x))u2⁎−1,inRN. If K(x) is nonhomogeneous at the critical points, we give the precise description on the concentrated solutions, which contains the necessary and existence on the two peak solutions. Here we generalize the results of Cao-Noussair-Yan in [6] to the nonhomogeneous case.

One-dimensional scalar wave propagation in multi-region domains by the boundary element method

Two boundary element method formulations are presented for the analysis of the one-dimensional scalar wave propagation problem in multi-region domains. One of the formulations employs the time-domain fundamental solution; the other, the fundamental solution related to the static problem. The problem domain is constituted of sub-domains with different material properties and, consequently, with different wave propagation velocities. Thus, the sub-region technique, akin to the boundary element method, is used to deal with the non-homogeneous problem. At the end of the article, some examples are presented, illustrating the characteristics of each formulation and their accuracy.

Crack-induced electrical resistivity changes in cracked CNT-reinforced composites

Carbon nanotube (CNT)-reinforced composites exhibit a piezoresistive behavior that permits their use as sensors in novel structural health monitoring (SHM) applications, by measuring the electrical resistivity change of the CNT modified laminate. However, the presence of cracks in such composite materials may not only compromise their structural integrity, but may as well alter their capability to act as reliable piezoresistive sensors. In this paper, we conduct a numerical study aimed at quantifying how the presence of cracks in reinforced polymer composites does influence their electrical conductivity and, consequently, their sensor performance. To this end, the electromechanical constitutive properties of the composite are determined by a mixed micromechanics approach that allows characterizing both the elastic properties and the strain-induced alterations in the overall electrical conductivity of the CNT-reinforced composite. The strain response of the cracked composite domain is accurately determined by means of a dual Boundary Element (BE) approach. Electrical conductivity in the cracked composite follows from its computed strain state at each point in the domain. Subsequently, the resulting non-homogeneous electrical conductivity problem is solved using a finite differences scheme that also accounts for semipermeable crack-face electrical boundary conditions. Several parametric studies are conducted to illustrate the influence of various crack geometries in the piezoresistive behavior of CNT-reinforced composites at varying CNTs concentrations.

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.

DISCONTINUOUS STURM-LIOUVILLE PROBLEMS INVOLVING AN ABSTRACT LINEAR OPERATOR

In this paper we introduce to consideration a new type boundary value problems consisting of an Sturm-Liouville equation on two disjoint intervals as $ -p(x)y^{\prime \prime }+ q(x)y+\mathfrak{B}y|_{x} = \mu y , x\in [a, c)\cup(c, b] $ together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction $x=c, $ where $\mathfrak{B}:L_{2}(a, c)\oplus L_{2}(c, b)\rightarrow L_{2}(a, c)\oplus L_{2}(c, b)$ is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define alternative inner products in Sobolev spaces, incorporating with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator $\mathfrak{B}.$

Multiple Solutions for Nonhomogeneous Kirchoff-Type Problem with Hardy-Sobolev Critical Exponent

In this work, we show the existence of multiple solutions for nonhomogeneous Kirchoff-type problem with Hardy-Sobolev critical exponent, by using Ekeland's variational principle and mountain pass theorem without Palais-Smale conditions.

On the particular solution of the nonhomogeneous Riemann problem in the weighted Smirnov classes with the general weight

On the particular solution of the nonhomogeneous Riemann problem in the weighted Smirnov classes with the general weight

Stress state of a rectangular domain with the mixed boundary conditions

The problem stated at the paper is not new, and some analytical methods and a lot of numerical methods are used to solve it. The approach proposed in the paper Popov G., Vaysfeld N. (2011) is based on the integral transform method and reduce the stated problem to a singular integral equation. This paper is continuation of paper Pozhylenkov O. V. (2019), where a rectangular domain with conditions of ideal contact at the lateral sides was considered. The novelty of the presented paper consists of a new statement of the problem when the conditions of the second main elasticity problem are given at the lateral sides. With the help of the Fourier transform, the one-dimensional vector boundary problem in the transform’s domain is derived. The solution of the homogeneous problem is found using matrix differential calculations, the fundamental solution matrix is constructed in the form of the contour integral, which is found using the residue theorem. As a result, we get the non-homogeneous one-dimensional problem, so the final solution is found with help of Green’s matrix Popov G., Abdimanapov S., Efimov V. (1999), which is constructed as a combination of the fundamental basis solution matrix. The solution is constructed as a superposition of the homogeneous and non-homogeneous solutions and contains the unknown derivatives of the displacement. The aim is to find the unknown function, which must satisfy the boundary condition on the upper edge of the domain. It leads to the singular integral equation which is solved with the help of the orthogonal method. The stress state of a domain was investigated depending on load properties and domain size.

Solving the Transport-Coagulation Problem in a Two-Dimensional Spatial Region

The article presents a numerical scheme for solving the spatially nonhomogeneous coagulation problem. The problem is solved in a two-dimensional spatial region using unstructured grids. The finite-volume method is used with monotonicity-preserving limiters. The coagulation kernel in the Smoluchowski collision integrals is approximated by a low-rank decomposition, which reduces the machine time requirement. Reflection is reduced by introducing a perfectly matched layer on the spatial boundary.

Modified quasi‐boundary value method for the multidimensional nonhomogeneous backward time fractional diffusion equation

This study deals with the inverse problem of retrieving the initial status for a fractional diffusion system from the measured final and source data. Here, the mentioned system is taken as a nonhomogeneous time fractional diffusion problem in a general bounded domain . A regularized sought solution is obtained by the regularization scheme, namely, modified quasi‐boundary value method. Further, the convergence estimates between the exact and regularized solution are derived based on the choice of strategies of the regularization parameter. Eventually, numerical examples are illustrated to show the efficiency of the proposed method.

Effect of Torso Non-Homogeneities in the quasi-static inverse problems arising in electrocardiology

Abstract In the present paper, an homogeneous and non-homogeneous inverse problem constrained by the stationary problem in electrocardiology representing the heart, lungs surfaces, and torso model is investigated. Our goal is to reconstruct the electrical potentials on the surface of the heart from the information obtained non invasively on the torso surface. The existence and uniqueness of solution for the heart-torso problem and the related inverse problem is assessed, and the primal and dual problems are discretized using a finite element method. We present some preliminary numerical experiments using an efficient implementation of the proposed scheme in homogeneous and non-homogeneous cases. Finally, we demonstrate the effect of the non-homogeneity on the reconstructed epicardial potential and show that the inverse ECG problem cannot be solved by the classical BEM (boundary element method).

A new scheme to obtain pollution-free solution for plane waves and to generate a continuous Petrov–Galerkin method for the Helmholtz problem

In this paper, we propose a new scheme to solve the Helmholtz problem. The proposed scheme has the ability to obtain a pollution-free solution and with good accuracy, when the exact solution is a plane wave or a superposition of plane waves with the same direction of propagation. This ability is valid for all the directions of plane waves. The scheme can be naturally extended to include other types of solutions of the homogeneous Helmholtz equation, as for example: cylindrical waves. The presented results confirm the ability described previously to obtain a pollution-free solution for all the directions of propagation of the plane waves. Also in this paper we propose a new continuous Petrov–Galerkin method with the goal to analyze Helmholtz problems where the solution is not a plane wave or a superposition of plane waves. This method was applied to non-homogeneous Helmholtz problems having good accuracy.

On the energy functionals derived from a non-homogeneous p-Laplacian equation: Γ-convergence, local minimizers and stable transition layers

In this paper we consider a family of singularly perturbed non-homogeneous p-Laplacian problems ϵpdiv(k(x)|∇u|p−2∇u)+k(x)g(u)=0 in Ω⊂Rn subject to Neumann boundary conditions. We establish the Γ-convergence of the energy functionals associate to this family of problems. As an application, we obtain the existence and profile asymptotic of a family of local minimizers in the one-dimensional case (i.e. Ω=(0,1)). In particular, these minimizers are stable solutions which develop inner transition layer in (0,1).

Existence of solutions for a nonhomogeneous Dirichlet problem involving p(x)-Laplacian operator and indefinite weight

We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the p(x)-Laplacian operator with Dirichlet boundary condition: \t\t\t−Δp(x)u+V(x)|u|q(x)−2u=f(x,u)in Ω,u=0 on ∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta _{p(x)}u+V(x) \\vert u \\vert ^{q(x)-2}u =f(x,u)\\quad \\text{in }\\varOmega , u=0 \\text{ on }\\partial \\varOmega , $$\\end{document} where Ω is a smooth bounded domain in mathbb{R}^{N}, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, f(x,t) is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.