scholarly journals Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces

2017 ◽  
Vol 263 (7) ◽  
pp. 3989-4006 ◽  
Author(s):  
Zhenhai Liu ◽  
Stanisław Migórski ◽  
Shengda Zeng
Keyword(s):  
Boundary Conditions ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Partial Differential ◽  
Differential Variational Inequalities

An Accurate Jacobi Pseudospectral Algorithm for Parabolic Partial Differential Equations With Nonlocal Boundary Conditions

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
M. A. Abdelkawy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Jacobi Polynomials ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Exponential Rate Of Convergence ◽  
Order Four

A new spectral Jacobi–Gauss–Lobatto collocation (J–GL–C) method is developed and analyzed to solve numerically parabolic partial differential equations (PPDEs) subject to initial and nonlocal boundary conditions. The method depends basically on the fact that an expansion in a series of Jacobi polynomials Jn(θ,ϑ)(x) is assumed, for the function and its space derivatives occurring in the partial differential equation (PDE), the expansion coefficients are then determined by reducing the PDE with its boundary conditions into a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit the Runge–Kutta (IRK) method of order four. The proposed method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the spatial discretizations. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

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