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

Abstract

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 COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

On the Spectrum Structure for One Difference Eigenvalue Problem With Nonlocal Boundary Conditions

“…The results of the investigation of stability are described in some articles. Short overview of these investigations up to the year 2013 is provided in [1]. In the present paper we continue investigation of conditions of stability.…”

“…was investigated in many papers without the study of spectrum structure of corresponding eigenvalue problem (1.1)-(1.2) ( see, f.e. [1,6] and the references therein). The other approach oh investigation of this stability is the spectrum structure of the matrix of difference equation system [3,15,19,24,25].…”

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

“…. The operational matrices  (1) t ,  (1) x and  (2) x using (2.11) and (2.13) . The functions Λ 1 (x) and Λ j (t) for j = 2, 3 using Equation (3.3) .…”

“…Therefore, their analytical and numerical solutions have been an interesting area of research for many years. Recently, several numerical methods such as Crank-Nicolson Hermite cubic orthogonal spline collocation method [1], polynomial-based mean weighted residuals methods [2], finite difference schemes [3][4][5][6][7], Laplace transform method [8], finite element method [9], wavelets method [10], Ritz-Galerkin method [11], spectral meshless radial point interpolation method [12], -method [13], implicit Euler method [14] and operational approach of the Tau method [15] have been proposed to solve systems including nonlocal boundary conditions. It is well known that, Bernstein polynomials (BPs) play an important role as basis functions for various numerical techniques for solving different mathematical systems.…”

Polynomial-based mean weighted residuals methods for elliptic problems with nonlocal boundary conditions in the rectangle

“…Both issues (more working precision and slower convergence) could make that the computational effort clearly grows to solve these nonlinear equations. For nonlinear hyperbolic and parabolic equations, some spectral collocation methods have also been combined with Runge-Kutta methods, finite-difference or spline methods [37,38], but with neither highly oscillatory nor other PDEs with large derivatives therein. This should be deeply analyzed in another paper.…”