Nonlinear Nonlocal Boundary Condition Research Articles

On a Porous Medium Equation with Weighted Inner Source Terms and a Nonlinear Nonlocal Boundary Condition

On a Porous Medium Equation with Weighted Inner Source Terms and a Nonlinear Nonlocal Boundary Condition

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions

In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone.

A MODIFIED BACKWARD EULER SCHEME FOR THE DIFFUSION EQUATION SUBJECT TO NONLINEAR NONLOCAL BOUNDARY CONDITIONS

In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differ- entiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

Analytical modeling of the linear and nonlinear dynamic characteristics of the non-uniform axially functionally graded cylindrical beam based on the strain gradient theory

In this research, the nonlinear vibration behavior of axially functionally graded (AFG) nano-scale truncated conical tubes (with non-uniform cross-section), including the porosity, is presented for the first time based on the Euler-Bernoulli beam theory and the von-Kármán’s nonlinear strain-displacement and nonlinear nonlocal boundary conditions. The effect of being at the tapered nanosize model is supposedly based on the nonlocal strain gradient theory using the Hamilton principle to extract the equation of motions and nonlinear time-dependent nonlocal boundary conditions. The material properties and thickness of nano-tube are varying along the length of tubes. The generalized differential quadrature method (GDQM) is utilized with the iteration method to solve the complicated nonlinear equations. The numerical results in tabular and graphical styles present for various nonlinear amplitudes, the AFG power indexes, the rate of the cross-section change, the porosity parameter, and the nonlocal gradient strain parameter for both simply supported, fully clamped, and simply supported-clamped boundary conditions in detail.

Explicit and Implicit Crandall’s Scheme for the Heat Equation with Nonlocal Nonlinear Conditions

In this paper, explicit and implicit Crandall's formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. The integrals in the boundary equations are approximated by the composite Simpson quadrature rule. Here nonlinear terms are approximated by Richtmyer's linearization method. Finally, some numerical examples are given to show the effectiveness of the proposed method.

Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения

Studying heat transfer processes in periodic media containing vacuum interlayers or cavities, heat through which is transferred by radiation, is of significant interest for applications. Direct numerical solution of such problems involves considerable computational efforts and becomes almost impossible for systems containing a large number of heat conducting elements, especially for 2D and 3D structures. Therefore, it is of issue to develop effective approximate solution methods for such problems. This publication continues a series of studies on developing and substantiating special discrete and asymptotic approximations of radiant-and-conduction heat transfer problems in periodic systems of heat conducting elements separated by vacuum. In this study, the stationary radiant-and-conduction heat transfer problem in a system of absolutely black square rods is considered. The sought quantity is the absolute temperature, which is found from the solution of the boundary-value problem for the stationary heat conduction equation with nonlinear nonlocal boundary conditions describing radiant heat transfer between the rods through vacuum interlayers. A special discrete approximation of this problem leading to the system of linear algebraic equations with respect to the fourth power of the temperature is presented. The solution of this system as approximation of the mean temperature over the rod cross-section is described. The discrete approximation error estimate as a function of the square rod side length (the small parameter of the problem) and the thermal conductivity coefficient has been obtained. The obtained error estimate proves applicability of the discrete approximation for materials with a high thermal conductivity coefficient.

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow‐up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.

On nonlinear neutral Liouville–Caputo-type fractional differential equations with Riemann–Liouville integral boundary conditions

AbstractThis paper studies neutral Liouville–Caputo-type fractional differential equations and inclusions supplemented with nonlocal Riemann–Liouville-type integral boundary conditions. Sadovskii’s fixed point theorem is applied to establish the existence result for the single-valued case, while the multivalued case is investigated by using nonlinear alternative for contractive maps. Examples are constructed to illustrate the main results. The case of nonlinear nonlocal boundary conditions is also discussed.

具有非线性非局部边界条件的非散度型退化抛物方程的定性分析

具有非线性非局部边界条件的非散度型退化抛物方程的定性分析

Global existence and decay of solutions of a singular nonlocal viscoelastic system

In this work, we consider a singular one-dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition. By using the potential well theory we obtain the existence of a global solution. Then, we prove the general decay result, by constructing Lyapunov functional and using the perturbed energy method.

Properties of solutions for a reaction\u2013diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition

In this paper, the authors consider the reaction–diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition. They prove that the solution either exists globally or blows up in finite time depending on the initial data, the weighting function on the border, and nonlinear indexes in the equation by using the comparison principle.

Second order systems with nonlinear nonlocal boundary conditions

Second order systems with nonlinear nonlocal boundary conditions

The generalized Riemann problem and instability of delta shock to the chromatography equations

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.

Initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal boundary condition*

We consider an initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal Neumann boundary condition. We prove a comparison principle and the existence of a local solution and study the problem of uniqueness and nonuniqueness.

Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence result. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.

Positive solutions of first order boundary value problems with nonlinear nonlocal boundary conditions

Positive solutions of first order boundary value problems with nonlinear nonlocal boundary conditions

Initial-Boundary-Value Problem for a Semilinear Parabolic Equation with Nonlinear Nonlocal Boundary Conditions

We consider an initial-boundary-value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions. We prove the principle of comparison, establish the existence of local solutions, and study the problem of uniqueness and nonuniqueness.