Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynom...

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Bibliographic Details
Published in:Advances in mathematical physics 2017-01, Vol.2017 (2017), p.1-15
Main Authors: Zogheib, Bashar, Shateyi, Stanford, Tohidi, Emran
Format: Article
Language:English
Subjects:
Accuracy
Algebra
Applied mathematics
Boundary conditions
Colleges & universities
Diffusion
Finite volume method
Mathematics
Numerical analysis
Partial differential equations
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Summary:A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.
ISSN:1687-9120
1687-9139
DOI:10.1155/2017/5691452