Nonhomogeneous Problem Research Articles

English

The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as 0.5.

Numerical solution of Benjamin-Bona-Mahony-Burger (BBMB) and regularized long-wave (RLW) equations using time-space pseudo-spectral method

In this research article, an improved numerical scheme is developed for solving the non-linear Benjamin-Bona-Mahony-Burger (BBMB) and regularized long-wave (RLW) equations. The time-space pseudo-spectral method is developed using Chebyshev polynomials at the CGL points. The algebraic mapping is used to transform the initial-boundary value non-homogeneous problem into the homogeneous problem and further, the BBMB and RLW equations are reduced to a system of nonlinear equations. To demonstrate the performance, the method applied to two model problems and approximate results have been compared with the exact solution and other existing methods. The proposed method succeeds to obtain high order accuracy.

Elliptic problems in long cylinders revisited

The asymptotic study of partial differential equations in cylinders becoming unbounded in one or several directions has known important developments in the last years, especially thanks to the works of Michel Chipot and his collaborators, see for example Chipot ( $$\ell $$ goes to plus infinity, Birkauser Verlag, Basel, 2002, Asymptotic issues for some partial differential equations, Imperial College Press, London, 2016), Chipot and Mardare (J Math Pures Appl 104:921–941, 2015), Chipot and Rougirel (Discrete Contin Dyn Syst Ser B 1(3):319–338, 2001), Chipot et al. (Asympt Anal 85:199–227, 2013), Chipot and Yeressian (C R Acad Sci Paris Ser I 346:21–26, 2008), Guesmia (Nonlinear Anal 70(9):3320–3331, 2009) and Xie (in: Recent advances on elliptic and parabolic issues, Proceedings of the 2004 Swiss-Japanese Seminar, World Scientific, 2006). In this paper, we prove the convergence to the solution of a linear elliptic problem on an infinite cylinder of the solutions of the same problem taken on larger and larger truncations of the cylinder. Following the methods introduced in Chipot and Yeressian (2008) and Chipot and Mardare (2015), we generalize the result of convergence found in Chipot and Yeressian (2008) for the case where the data is not necessarily independent of the coordinate along the axis of the cylinder. We also consider the non-homogenous Dirichlet problem instead of the homogenous one.

Green’s formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators

This article treats boundary value problems for the fractional Laplacian , a > 0, and more generally for classical pseudodifferential operators (ψdo’s) P of order 2a with even symbol, applied to functions on a smooth subset Ω of . There are several meaningful local boundary conditions, such as the Dirichlet and Neumann conditions , k = 0, 1, where . We show a new Green’s formula where B is a first-order ψdo on depending on the first two terms in the symbol of P. Moreover, we show in the elliptic case how the Poisson-like solution operator KD for the nonhomogeneous Dirichlet problem is constructed from P+ in the factorization obtained in earlier work. The Dirichlet-to-Neumann operator is derived from this as a first-order ψdo on , with an explicit formula for the symbol. This leads to a characterization of those operators P for which the Neumann problem is Fredholm solvable.

Finite element approximations of the nonhomogeneous fractional Dirichlet problem

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter.

Positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolution–supersolution, together with truncation and comparison techniques.

DEA for nonhomogeneous mixed networks

Practically, systems contain a set of subunits each of which has a set of inputs and produces a set of outputs. Conventional data envelopment analysis (DEA) ignores the operations of individual subunits in a system and considers the system as a black-box. Recent DEA studies developed to investigate the internal structures of a system and examine the interrelationship of the components within a system either with serial or parallel or mixed structure. Nevertheless, in many practical settings the assumption of identical inputs and outputs may not hold among the components. As an example, the case of a set of plants, each has various product lines manufactured by different inputs and outputs captures the idea. This paper proposes a network DEA-based methodology to address the problem of nonhomogeneity in settings where subunits operate in the mixed network structure. The layers and the overall system efficiencies can be evaluated through our proposed methodology. To show the applicability of our models, we then use the models and examine the efficiencies of a set of hypothetical data set.

A weighted anisotropic variant of the Caffarelli–Kohn–Nirenberg inequality and applications

We present a weighted version of the Caffarelli–Kohn–Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.

The global system‐ranking efficiency model and calculating examples with consideration of the nonhomogeneity of decision‐making units

AbstractData envelopment analyses have been widely used to evaluate the relative efficiency of decision‐making units (DMUs). However, the traditional data envelopment analysis model has not considered the problem of DMUs' nonhomogeneity. If nonhomogeneous DMUs are evaluated under the same production frontier, conclusions may not be precise. For example, some DMUs' input redundancy and output deficit cannot be adjusted as per planning results, which may lead to mistakes in management. This paper loosens the assumption of DMU homogeneity and builds a global system‐ranking efficiency model based on existing literature, which divides the problem of DMUs' nonhomogeneity into external nonhomogeneity and internal homogeneity. Data have been collected from 114 listed enterprises in China's solar power industry, and the analysis results indicate that this paper's model is stable and reliable and can be used as a reference for production managers.

Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction

We consider nonlinear Dirichlet and Neumann problems driven by a nonlinear nonhomogeneous differential operator and with a (p−1)-superlinear Carathéodory reaction which does not satisfy the Ambrosetti–Rabinowitz condition. We prove two multiplicity theorems producing two nodal solutions, and answer two open questions posed by Aizicovici et al. (2013) and by Barletta and Papageorgiou (2014). Our approach uses variational methods together with suitable truncation techniques and flow invariance arguments.

Approximation of an Inverse Initial Problem for a Biparabolic Equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous and nonlinear biparabolic equation. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we also give a regularized solution and consider the convergence between the regularized solution and the sought solution. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Hilbert space and Hilbert scale space.

Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.

Constant sign and nodal solutions for nonhomogeneous Robin boundary value problems with asymmetric reactions

We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques.

Properties of Consistent Grid Operators for Grid Functions Defined Inside Grid Cells and on Grid Faces

Using grids (meshes) formed from polyhedra (polygons in the two-dimensional case), we consider differential and boundary grid operators that are consistent in the sense of satisfying the grid analog of the integral identity – a corollary of the formula for the divergence of the product or a scalar by a vector. These operators are constructed and applied in the Mimetic Finite Difference (MFD) method, in which grid scalars are defined inside the grid cells and grid vectors are defined by their local normal coordinates on the planar faces of the grid cells. We show that the basic grid summation identity is a limit of an integral identity written for piecewise-smooth approximations of the grid functions. We also show that the MFD formula for the reconstruction of a grid vector field is obtained by approximation analysis of the summation identity. Grid embedding theorems are proved, analogous to well-known finite-difference embedding theorems that are used in finite-difference scheme theory to derive prior bounds for convergence analysis of the solutions of finite-difference nonhomogeneous boundary-value problems.

Weighted eigenvalue problems involving a fourth-order elliptic equation with variable exponent

In this paper, we consider weighted fourth order nonhomogeneous elliptic problem with variable exponent. Under appropriate conditions, interval of a numerical parameter sufficiently small is derived for which the existence results are obtained. The proof relies on simple variational arguments based on Ekeland's variational principle.

Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By strengthening the regularity of \begin{document}$f(z,\cdot)$\end{document} , we are able to generate a second nodal solution for a total of four nontrivial smooth solutions.

Effect of low velocity non-Darcy flow on pressure response in shale and tight oil reservoirs

Low velocity non-Darcy flow in shale and tight oil reservoirs is described by nonlinear or nonhomogeneous models. These models, especially for well shut-in period, are usually solved by numerical method, since the traditional pressure superposition principle is no longer applicable. The current paper presents a modified pressure superposition principle, accounting for the pseudo Threshold Pressure Gradient (TPG), and its mathematical proof. The proposed principle indicates that the total change of bottom hole pressure (BHP) in shut-in period is equal to the superposition of BHP change in a real well with pseudo TPG and that in a virtual well without pseudo TPG. The new principle is applied to the derivation of an analytical solution to the nonhomogeneous problem during the well shut-in period. Type curves calculated from the analytical solution show that the pseudo TPG leads to curve up-warping in switch-on period but down-warping in shut-in period, which agree with previous numerical results, and can be explained by the moving-boundary theory. Throughout the switch-on period, a closed moving-boundary is generated when the pressure gradient is less than the pseudo TPG. The boundary is closer to the well with higher pseudo TPG. However, during the shut-in period, a supply moving-boundary, which was generated during previous production or injection period, is earlier to be reached for virtual well with higher pseudo TPG. The flow is steady state afterwards. Matching of field data by the analytical solution results in the pseudo TPG in the investigation zone. The interpretation of the field case shows that pseudo TPG equals 0.104 MPa/m, generating a pressure drop as high as 6.35 MPa across the investigation zone during the well testing period.

Una forma alternativa para el método de series de Potencias

This article consider the classical problem of linear non-homogeneous second order Initial Value Problems with analytic coefficients. It classifies the possible kinds of analytic solutions, giving criteria for the nonexistence of analytical solutions and for the existence of multiple analytic solutions. An alternative proof for the convergence of the power series method is given and it applies for some singular irregular points.

Large solutions for a nonhomogeneous quasilinear elliptic problem

This paper focuses on a nonhomogeneous quasilinear elliptic boundary blow up problem △pu=a(x)f(u)+h(x)inΩ,u|∂Ω=∞,where p>1, Ω is a smooth bounded domain in Rn, and h∈C(Ω) may be sign-changing in Ω and singular on ∂Ω. We mainly analyze the influences caused by h on the existence of large solutions. Furthermore, it can be verified that any large solution is nonnegative under an additional assumption.

Existence and uniqueness of solution for a nonhomogeneous nonlocal problem

In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and uniqueness of solution.