On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

Imposing boundary conditions in Sinc method using highest derivative approximation

Sinc approximate methods are often used to solve complex boundary value problems such as problems on unbounded domains or problems with endpoint singularities. A recent implementation of the Sinc method [Li, C. and Wu, X., Numerical solution of differential equations using Sinc method based on the interpolation of the highest derivatives, Applied Mathematical Modeling 31 (1) 2007 1–9] in which Sinc basis functions are used to approximate the highest derivative in the governing equation of the boundary value problem is evaluated for structural mechanics applications in which interlaminar stresses are desired. We suggest an alternative approach for specifying the boundary conditions, and we compare the numerical results for analysis of a laminated composite Timoshenko beam, implementing both Li and Wu’s approach and our alternative approach for applying the boundary conditions. For the Timoshenko beam problem, we obtain accurate results using both approaches, including transverse shear stress by integration of the 3D equilibrium equations of elasticity. The beam results indicate our approach is less dependent on the selection of the Sinc mesh size than Li and Wu’s SIHD. We also apply SIHD to analyze a classical laminated composite plate. For the plate example, we experience difficulty in obtaining a complete system of equations using Li and Wu’s approach. For our approach, we suggest that additional necessary information may be obtained by applying the derivatives of the boundary conditions on each edge. Using this technique, we obtain accurate results for deflection and stresses, including interlaminar stresses by integration of the 3D equilibrium equations of elasticity. Our results for both the beam and the plate problems indicate that this approach is easily implemented, has a high level of accuracy, and good convergence properties.     

Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions

We formulate and analyze a Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Crank-Nicolson method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions. For each method, optimal error estimates are derived in the maximum norm.Dedicated to Professor J. Crank on the occasion of his 80th birthdaySupported in part by the National Science Foundation grant CCR-9403461.Supported in part by project DGICYT PB95-0711.     

Global blow-up for a localized quasilinear parabolic system with nonlocal boundary conditions

In this article, we investigate the positive solution of a localized quasilinear parabolic system with nonlocal boundary conditions. Under certain conditions, the global existence and finite time blow-up criteria are established, and the global blow-up behaviour is also obtained.     

Solving parabolic Wick-stochastic boundary value problems using a finite element method

A class of linear parabolic stochastic boundary value problems of Wick-type is studied. The equations are understood in a weak sense on a suitable stochastic distribution space, and existence and uniqueness results are provided. The paper continues to discuss a numerical method for this type of problem, based on a Galerkin type of approximation. Estimates showing linear convergence in time and space are derived, and rate of convergence results for the stochastic dimension are reported.     

The Fourier problem with nonlocal boundary conditions for a class of nonlinear parabolic equations

The paper contains theorems on the uniqueness and existence of a generalized solution to the Fourier problem (the problem without initial conditions) for a class of strongly nonlinear parabolic equations, without any restrictions on the behavior of the data and of the solution ast → ∞. A priori estimates of the solution are obtained as well. Bibliography: 8 titles. Dedicated to Olga Arsenievna Oleinik Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions

We use a matrix formulated algorithm to approximate solutions of a class of nonlinear reaction-diffusion equations with nonlocal boundary conditions. Some theoretical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis and error estimation of the method are also investigated. Finally, some numerical examples are given to demonstrate accuracy and efficiency of the proposed method.     

A nonlinear parabolic equation with a nonlocal boundary term

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe’s method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.     

A parabolic equation with nonlocal conditions

For a linear parabolic equation with the principal part in divergence form, a boundary-value problem with nonlocal (irregular) conditions of integral type is considered. Sufficient conditions of the unique solvability are found for the above-mentioned problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Solving nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space

In this paper, a function space is constructed, in which an arbitrary function satisfies the nonlocal boundary conditions of a nonlinear pseudoparabolic equation. A very simple numerical algorithm for the approximations of the nonlinear pseudoparabolic equation with nonlocal boundary conditions based on the function space is provided. A numerical example is given to illustrate the applicability and efficiency of the algorithm.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

A degenerate parabolic system with nonlocal boundary condition

In this article, we investigate the blow-up properties of the positive solutions to a degenerate parabolic system with nonlocal boundary condition. We give the criteria for finite time blow-up or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate for small weighted nonlocal boundary.     

带非局部边界条件的热方程的LEGENDRE配置法

对带非局部边界条件的热方程的初边值问题提出了LEGENDRE配置法,并给出其半离散逼近和全离散逼近的稳定性和收敛性分析.数值试验验证了方法的有效性.     

Sinc collocation method with boundary treatment for two-point boundary value problems

Sinc collocation method is proven to provide an exponential convergence rate in solving linear differential equations, even in the presence of singularities. But in order to treat the derivatives on boundaries, people often relied on the finite difference method, which would be expected to limit the accuracy. The present paper develops a Sinc collocation method with boundary treatment for two-point boundary value problems. Numerical results show that the method can directly and efficiently handle the boundary derivatives.     

On the existence of global solutions of a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions

We establish conditions for the existence and nonexistence of global solutions of an initial–boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal boundary conditions. The results depend on the behavior of variable coefficients as t→∞.     

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.