Nonhomogeneous Term Research Articles

Lower Bounds on the Proportion of Leaders Needed for Expected Consensus of 3-D Flocks.

This paper considers the consensus behavior of a spatially distributed 3-D dynamical network composed of heterogeneous agents: leaders and followers, in which the leaders have the preferred information about the destination, while the followers do not have. All followers move in a 3-D Euclidean space with a given speed and with their headings updated according to the average velocity of the corresponding neighbors. Compared with the 2-D model, a key point lies in how to analyze the dynamical behavior of a "linear" nonhomogeneous equation where the nonhomogeneous term strongly nonlinearly depends on the states of all agents. Using the network structure and the estimation of some characteristics for the initial states, we present a proper decaying rate for the nonhomogeneous term and then establish lower bounds on the ratio of the number of leaders to the number of followers that is needed for the expected consensus by considering two cases: 1) fixed speed and neighborhood radius and 2) variable speed and neighborhood radius with respect to the population size. Some simulation examples are given to justify the theoretical results.This paper considers the consensus behavior of a spatially distributed 3-D dynamical network composed of heterogeneous agents: leaders and followers, in which the leaders have the preferred information about the destination, while the followers do not have. All followers move in a 3-D Euclidean space with a given speed and with their headings updated according to the average velocity of the corresponding neighbors. Compared with the 2-D model, a key point lies in how to analyze the dynamical behavior of a "linear" nonhomogeneous equation where the nonhomogeneous term strongly nonlinearly depends on the states of all agents. Using the network structure and the estimation of some characteristics for the initial states, we present a proper decaying rate for the nonhomogeneous term and then establish lower bounds on the ratio of the number of leaders to the number of followers that is needed for the expected consensus by considering two cases: 1) fixed speed and neighborhood radius and 2) variable speed and neighborhood radius with respect to the population size. Some simulation examples are given to justify the theoretical results.

A modified homogenized highly precise direct integration method for time-varying nonhomogeneous systems

This article investigates a novel method for time-varying systems with an impact term; we call it a modified homogenized highly precise direct integration method. Modified homogenized highly precise direct integration method can deal with the time-varying nonhomogeneous systems effectively by direct integration with high precision. Even though it is often difficult to select the time step size of integration properly for stiff problems, modified homogenized highly precise direct integration method can effectively deal with the this problem with a large time step size. By introducing new variants twice, modified homogenized highly precise direct integration method can easily deal with the nonhomogeneous term by a novel way, inherit the advantages of highly precise direct integration method and avoid the calculation of inverse matrix. The convergency and efficiency analyses are given in this article. Several numerical simulations and an application example are presented to demonstrate the high accuracy, eff...

Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations

ABSTRACTThis paper proposes a numerical approach to approximate the unknown solution of some high-order fractional partial differential equations. The main idea of this approach is to transform the original problem into an equivalent integral equation that depends only on the boundary values. The linear radial basis functions are used as the main tool for approximating the non-homogeneous terms and time derivative. Also the Caputo's sense is applied to approximate time derivatives. Numerical results demonstrate the order of time steps is and when and , respectively. Finally to overcome the nonlinear terms, predictor–corrector scheme is employed. The efficiency and usefulness of proposed method are demonstrated by some numerical examples.

About a composite fractional relaxation equation via regularized families

This work deals with asymptotic periodicity and compactness for a class of composite fractional relaxationequation. Some difficulties arises when the effect of different kinds of nonhomogeneous terms are taken intoconsideration. To overcome these we use methods coming from regularized families and fixed point techniques,which are an important tool to study of nonlinear phenomena. We can cover a large class of nonlinearities.

Entire Solution in Cylinder-Like Domains for a Bistable Reaction–Diffusion Equation

We construct nontrivial entire solutions for a bistable reaction–diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki et al. (Calc Var Partial Differ Equ 55(3):1–32, 2016) concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the “cylinder-like” assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We need a different approach based on the long time stability of the bistable wave in heterogeneous media. We also prove the existence of an entire solution for a one-dimensional problem with a non-homogeneous linear term.

A robust higher order compact scheme for solving general second order partial differential equation with derivative source terms on nonuniform curvilinear meshes

A fourth order compact finite difference scheme is proposed for solving general second order steady partial differential equation (PDE) in two-dimension (2D) on geometries having nonuniform curvilinear grids. In this work, the main efforts are focused not only on nonorthogonal curvilinear grids but also on the presence of mixed derivative term and nonhomogeneous derivative source terms in the governing equation. This is in turn suitable for solving fluid flow and heat transfer problems governed by Navier–Stokes(N–S) equations on geometries having nonuniform and nonorthogonal curvilinear grids. The newly proposed scheme has been applied to solve general second order partial differential equation having analytical solution and some pertinent fluid flow problems, namely, viscous flows in a lid driven cavity such as trapezoidal cavity using nonorthogonal grid, square cavity using distorted grid, complicated enclosures using curvilinear grid, and mixed convection flow in a bottom wavy wall cavity. It is seen to efficiently capture steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. Detailed comparison data produced by the proposed scheme for all the test cases are provided and compared with existing analytical as well as established numerical results available in the literature. Excellent comparison is obtained in all the cases.

Sharp profiles in models of collective movements

We consider a parabolic partial differential equation that can be understood as a simple model for crowds flows. Our main assumption is that the diffusivity and the source/sink term vanish at the same point; the nonhomogeneous term is different from zero at any other point and so the equation is not monostable. We investigate the existence, regularity and monotone properties of semi-wavefront solutions as well as their convergence to wavefront solutions.

On a nonhomogeneous and singular quasilinear equation involving critical growth in [formula omitted

This paper establishes sufficient conditions for the existence and multiplicity of solutions for nonhomogeneous and singular quasilinear equations of the form −Δu+V(x)u−Δ(u2)u=g(x,u)|x|a+h(x)inR2,where a∈[0,2), V(x) is a continuous positive potential bounded away from zero and which can be “large” at infinity, the nonlinearity g(x,s) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and the nonhomogeneous term h belongs to Lq(R2) for some q∈(1,2]. By combining variational arguments in a nonstandard Orlicz space context with a singular version of the Trudinger–Moser inequality, we obtain the existence of two distinct solutions when ‖h‖q is sufficiently small. Schrödinger equations of this type have been studied as models of several physical phenomena.

Numerical Simulation of the Laminar Forced Convective Heat Transfer between Two Concentric Cylinders

The dual reciprocity method (DRM) is a highly efficient numerical method of transforming domain integrals arising from the non-homogeneous term of the Poisson equation into equivalent boundary integrals. In this paper, the velocity and temperature fields of laminar forced heat convection in a concentric annular tube, with constant heat flux boundary conditions, have been studied using numerical simulations. The DRM has been used to solve the governing equation, which is expressed in the form of a Poisson equation. A test problem is employed to verify the DRM solutions with different boundary element discretizations and numbers of internal points. The results of the numerical simulations are discussed and compared with exact analytical solutions. Good agreement between the numerical results and exact solutions is evident, as the maximum relative errors are less than 5% to 6%, and the R2-values are greater than 0.999 in all cases. These results confirm the effectiveness and accuracy of the proposed numerical model, which is based on the DRM.

Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrodinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2∗) with respect to non-homogeneous term g(x) ∈ L2n/2(B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H2,01,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.

Weighted Orlicz estimates for general nonlinear parabolic equations over nonsmooth domains

We introduce a parabolic analogue of Muckenhoupt weights to study optimal weighted regularity in Orlicz spaces for a general nonlinear parabolic problem of p-Laplacian-type in divergence form over a nonsmooth domain. Assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation (BMO) with respect to the spatial variables, that the lateral boundary of the parabolic cylinder is δ-Reifenberg flat and that the associated weight belongs to a suitable parabolic Muckenhoupt class, we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted Orlicz space. Our results extend the existing regularity estimates in Lebesgue spaces to weighted Orlicz spaces.

Free vibration analysis of embedded single-layered nanoplates and graphene sheets by using the multiple time scale method

An asymptotic approach using the Eringen nonlocal elasticity theory and multiple time scale method is developed for the three-dimensional (3D) free vibration analysis of simply-supported, single-layered nanoplates and graphene sheets (GSs) embedded in an elastic medium. In the formulation, the small length scale effect is first introduced to the nonlocal constitutive equations by using a nonlocal parameter, then the mathematical processes of nondimensionalization, asymptotic expansion and successive integration are performed, and finally recurrent sets of motion equations for various order problems are obtained. The interactions between the nanoplates (or GSs) and their surrounding medium are modeled as a two-parameter Pasternak foundation. Nonlocal classical plate theory (CPT) is derived as a first-order approximation of the 3D nonlocal elasticity theory, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal CPT, although with different nonhomogeneous terms. Some 3D nonlocal elasticity solutions of the natural frequency parameters of nanoplates (or GSs) with and without being embedded in the elastic medium and their corresponding through-thickness distributions of modal field variables are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory.

Asymptotic nonlocal elasticity theory for the buckling analysis of embedded single-layered nanoplates/graphene sheets under biaxial compression

A three-dimensional (3D) asymptotic formulation is developed for the buckling analysis of simply-supported, single-layered nanoplates/graphene sheets (SLNP and SLGS) embedded in an elastic medium and under biaxial compressive loads. In the formulation, the Eringen nonlocal elasticity theory is used to capture the small length scale effect, and the interaction between the SLNP/SLGS and its surrounding medium is simulated using a Pasternak-type foundation. After performing the mathematical processes of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recursive sets of governing equations for various order problems. The nonlocal classical plate theory (CPT) is derived as a first-order approximation of the 3D nonlocal elasticity theory, and the governing equations for higher-order problems retain the same differential operators as those of nonlocal CPT, although with different nonhomogeneous terms. Some accurate nonlocal elasticity solutions of the critical load parameters of simply-supported, biaxially-loaded SLNP/SLGS with and without being embedded in the elastic medium are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory.

An Analytical Solution for Radiofrequency Ablation with a Cooled Cylindrical Electrode

We present an analytical solution to the electrothermal mathematical model of radiofrequency ablation of biological tissue using a cooled cylindrical electrode. The solution presented here makes use of the method of separation of variables to solve the problem. Green’s functions are used for the handling of nonhomogeneous terms, such as effect of electrical currents circulation and the nonhomogeneous boundary condition due to cooling at the electrode surface. The transcendental equation for determination of eigenvalues of this problem is solved using Newton’s method, and the integrals that appear in the solution of the problem are obtained by Simpson’s rule. The solution obtained here has the possibility of handling different functional dependencies of the source term and nonhomogeneous boundary condition. The solution provides a tool to understand the physics of the problem, as it shows how the solution depends on different parameters, to provide mathematical tools for the design of surgical procedures and to validate other modeling techniques, such as the numerical methods that are frequently used to solve the problem.

A non-singular method of fundamental solutions for two-dimensional steady-state isotropic thermoelasticity problems

We consider a boundary meshless numerical solution for two-dimensional linear static thermoelastic problems. The formulation of the problem is based on the approach of Marin and Karageorghis, where the Laplace equation for the temperature field is solved first, followed by a particular solution of the non-homogenous term in the Navier-Lamé system for the displacement, the solution of the homogenous equilibrium equations, and finally the application of the superposition principle. The solution of the problem is based on the method of fundamental solutions (MFS) with source points on the boundary. This is, by complying with the Dirichlet boundary conditions, achieved by the replacement of the concentrated point sources with distributed sources over the disk around the singularity, and for complying with the Neumann boundary conditions by assuming a balance of the heat fluxes and the forces. The derived non-singular MFS is assessed by a comparison with analytical solutions and the MFS for problems that can include different materials in thermal and mechanical contact. The method is easy to code, accurate, efficient and represents a pioneering attempt to solve thermoelastic problems with a MFS-type method without an artificial boundary. The procedure makes it possible to solve a broad spectra of thermomechanical problems.

A survey of the precise integration method

For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method (PIM) gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.

Global $$W^{1,p}$$ estimates for solutions to the linearized Monge–Ampère equations

In this paper, we investigate regularity for solutions to the linearized Monge–Ampere equations when the nonhomogeneous term has low integrability. We establish global \(W^{1,p}\) estimates for all \(p<\frac{nq}{n-q}\) for solutions to the equations with right-hand side in \(L^q\) where \(n/2<q\le n\). These estimates hold under natural assumptions on the domain, Monge–Ampere measures, and boundary data. Our estimates are affine invariant analogues of the global \(W^{1,p}\) estimates of N. Winter for fully nonlinear, uniformly elliptic equations.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACTThe unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhoff term

Consider the equation −M(x,‖u‖)Δpu=λf(x,u,∇u)−g(x,u,∇u) in Ω, u=0 on ∂Ω. The main aim of this paper is to prove existence results for both non-degenerate and degenerate cases of the function M. Our approach relies on the fixed point index and the cone theoretic argument.

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón–Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same Lq space for every q∈[p,∞). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction.