An efficient collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions

• Published in: Applied mathematical modelling Vol. 39; no. 18; pp. 5616 - 5635
• Main Authors: Bhrawy, A.H., Doha, E.H., Abdelkawy, M.A., Hafez, R.M.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2015
• Subjects: Collocation method; Integral conservation condition; Legendre–Gauss–Lobatto quadrature; Neumann boundary condition; Nonclassic boundary value problems; Nonlocal boundary conditions; Integral conservation condition; Neumann boundary condition; Legendre–Gauss–Lobatto quadrature; Nonlocal boundary conditions; Collocation method; Nonclassic boundary value problems
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2015.01.029

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach.