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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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/5691452 |
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| author | Bashar Zogheib Emran Tohidi Stanford Shateyi |
| author_facet | Bashar Zogheib Emran Tohidi Stanford Shateyi |
| author_sort | Bashar Zogheib |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-68048779265247cdba87bc55a69af246 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-68048779265247cdba87bc55a69af2462025-02-03T01:01:42ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/56914525691452Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary ConditionsBashar Zogheib0Emran Tohidi1Stanford Shateyi2Department of Mathematics and Natural Sciences, American University of Kuwait, Salmiya, KuwaitDepartment of Mathematics, Kosar University of Bojnord, P.O. Box 9415615458, Bojnord, IranDepartment of Mathematics and Applied Mathematics, University of Venda, Thohoyandou 0950, South AfricaA 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.http://dx.doi.org/10.1155/2017/5691452 |
| spellingShingle | Bashar Zogheib Emran Tohidi Stanford Shateyi Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions Advances in Mathematical Physics |
| title | Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions |
| title_full | Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions |
| title_fullStr | Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions |
| title_full_unstemmed | Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions |
| title_short | Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions |
| title_sort | bernoulli collocation method for solving linear multidimensional diffusion and wave equations with dirichlet boundary conditions |
| url | http://dx.doi.org/10.1155/2017/5691452 |
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