A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation
A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legen...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/636191 |
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| author | A. H. Bhrawy |
| author_facet | A. H. Bhrawy |
| author_sort | A. H. Bhrawy |
| collection | DOAJ |
| description | A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is
developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion
equations with variable coefficients. The method depends basically on the fact that an expansion
in a series of shifted Legendre polynomials PL,n(x)PL,m(y), for the function and its space-fractional
derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion
coefficients are then determined by reducing the PFDE with its boundary and initial conditions
to a system of ordinary differential equations (SODEs) for these coefficients. This system may be
solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method,
in contrast to common finite-difference and finite-element methods, has the exponential rate of
convergence for the two spatial discretizations. Numerical examples are presented in the form of
tables and graphs to make comparisons with the results obtained by other methods and with the
exact solutions more easier. |
| format | Article |
| id | doaj-art-fe3f8aa32d8749c5a0df8aad5b27a435 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-fe3f8aa32d8749c5a0df8aad5b27a4352025-08-20T03:25:47ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/636191636191A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion EquationA. H. Bhrawy0Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaA new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x)PL,m(y), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.http://dx.doi.org/10.1155/2014/636191 |
| spellingShingle | A. H. Bhrawy A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation Abstract and Applied Analysis |
| title | A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation |
| title_full | A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation |
| title_fullStr | A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation |
| title_full_unstemmed | A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation |
| title_short | A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation |
| title_sort | new legendre collocation method for solving a two dimensional fractional diffusion equation |
| url | http://dx.doi.org/10.1155/2014/636191 |
| work_keys_str_mv | AT ahbhrawy anewlegendrecollocationmethodforsolvingatwodimensionalfractionaldiffusionequation AT ahbhrawy newlegendrecollocationmethodforsolvingatwodimensionalfractionaldiffusionequation |