On a Discrete Fractional Boundary Value Problem with Nonlocal Fractional Boundary Conditions

In this paper, we investigate the nonlinear fractional difference equation with nonlocal fractional boundary conditions. We derive the Green's function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnosel-skii's fixed point theorem.

Global and Blow-up Solutions to a p-Laplace Equation with Nonlocal Source and Nonlocal Boundary Condition

This paper deals with an evolution p-Laplace equation with nonlocal source subject to weighted nonlocal Dirichlet boundary conditions. We give sufficient conditions for the existence of global and non-global solutions.

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, we prove that the unique solution has global blow-up property whenever blow-up occurs.

Highly Efficient Method for Solving Parabolic PDE with Nonlocal Boundary Conditions

In this work, a highly efficient algorithm is developed for solving the parabolic partial differential equation (PDE) with the nonlocal condition. For this purpose, we employ orthogonal Chelyshkov polynomials as the basis. The convergence analysis of the proposed scheme is derived. Numerical experiments are carried out to explain the efficiency and precision of the proposed scheme. Furthermore, the reliability of the scheme is verified by comparisons with assured existing methods.

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL NONLINEAR DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM WITH NONLOCAL BOUNDARY CONDITION

Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.

Nonlocal isogeometric analysis for bidirectional functionally graded porous curved microbeams with arbitrary boundary conditions

This research aims to analyze the static bending and free vibration behaviors of bidirectional functionally graded porous(BFGP)curved microbeams with elastic boundary conditions resting on a Pasternak’s elastic foundation in a hygro-thermal environment.Isogeometric analysis based on non-uniform rational B-splines,first-order shear deformation theory,nonlocal elasticity theory combined with the modified strain gradient theories,modified Timoshenko beam theory are formulated to depict bending and shear deformations of BFGP curved microbeams.Especially,because using the modified Timoshenko beam theory,this study removes the requirement for a shear correction factor and describes shear stress by zero at the upper and lower cross-sectional sites of the BFGP curved microbeam.Different from traditional boundary conditions,where the beginning and end positions of a curved beam are connected by an elastic system of straight and torsion springs.This allows for greater flexibility in controlling the stiffness of the springs to obtain arbitrary boundaries.To assess the accuracy and convergence of the proposed approach,validation numerical examples were conducted in the various examples.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.

An isogeometric approach for nonlocal bending and free oscillation of magneto-electro-elastic functionally graded nanobeam with elastic constraints

This work uses isogeometric analysis(IGA),which is based on nonlocal hypothesis and higher-order shear beam hypothesis,to investigate the static bending and free oscillation of a magneto-electro-elastic functionally graded(MEE-FG)nanobeam subject to elastic boundary constraints(BCs).The magneto-electric boundary condition and the Maxwell equation are used to calculate the variation of electric and magnetic potentials along the thickness direction of the nanobeam.This study is innovative since it does not use the conventional boundary conditions.Rather,an elastic system of straight and torsion springs with controllable stiffness is used to support nanobeams’beginning and end positions,creating customizable BCs.The governing equations of motion of nanobeams are established by applying Hamilton’s principle and IGA is used to determine deflections and natural frequency values.Verification studies were performed to evaluate the convergence and accuracy of the proposed method.Aside from this,the impact of the input parameters on the static bending and free oscillation of the MEE-FG nanobeam is examined in detail.These findings could be valuable for analyzing and designing innovative structures constructed of functionally graded MEE materials.

A Quasilinear Parabolic System with Nonlocal Sources and Weighted Nonlocal Boundary Conditions

In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.

Numerical Computations of Nonlocal Schrodinger Equations on the Real Line

The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work,the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction.By utilizing the variational principle of virtual work,the governing equations and the associated standard boundary conditions are systematically extracted,and the thermal effect,equivalent to the induced thermal load,is explicitly assessed by using the nonlocal heat conduction law.The stressdriven constitutive integral equation is equivalently transformed into a differential form with two non-standard constitutive boundary conditions.By employing the eigenvalue method,the critical buckling loads of the beams with different boundary conditions are obtained.The numerically predicted results reveal that the growth of the nonlocal parameter leads to a consistently strengthening effect on the dimensionless critical buckling loads for all boundary cases.Additionally,the effects of the influential factors pertinent to the nonlocal heat conduction on the buckling behavior are carefully examined.

Growing Sandpile Problem with Local and Non-Local Boundaries Conditions

In this paper, we study a class of Prigozhin equation for growing sandpile problem subject to local and a non-local boundary condition. The problem is a generalized model for a growing sandpile problem with Neumann boundary condition (see [1]). By the semi-group theory, we prove the existence and uniqueness of the solution for the model and thanks to a duality method we do the numerical analysis of the problem. We finish our work by doing numerical simulations to validate our theoretical results.

A degenerate parabolic system with localized sources and nonlocal boundary condition

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

ON A SEMILINEAR PARABOLIC EQUATION SYSTEM WITH NONLOCAL AND COUPLED BOUNDARY CONDITIONS

The purpose of this paper is to investigate the existence, uniqueness and dynamics of a nonlinear system of partial differential equations with nonlocal and coupled boundary conditions which is motivated by a model problem arising from quasi-state thermoelasticity. A sufficient condition for the uniqueness of a steady-state solution is obtained. The behavior of solutions to the evolution problem and the relation between the solutions to the evolution problem and its corresponding steady-state problem are also discussed.

Iterative algorithm for parabolic and hyperbolic PDEs with nonlocal boundary conditions

In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.

Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries

This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.

Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain

This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

Nonnegative Solutions for a Riemann-Liouville Fractional Boundary Value Problem

We investigate the existence of nonnegative solutions for a Riemann-Liouville fractional differential equation with integral terms, subject to boundary conditions which contain fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval.Using an extension of the analysis of Douglas and Dupont[23]for Dirichlet boundary conditions,we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space.We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROWfor solving almost block diagonal linear systems.We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.