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Sylvester Equations and the numerical solution of partial fractional differential equations
- Source :
- Journal of Computational Physics. 293:370-384
- Publication Year :
- 2015
- Publisher :
- Elsevier BV, 2015.
-
Abstract
- We develop a new matrix-based approach to the numerical solution of partial differential equations (PDE) and apply it to the numerical solution of partial fractional differential equations (PFDE). The proposed method is to discretize a given PFDE as a Sylvester Equation, and parameterize the integral surface using matrix algebra. The combination of these two notions results in an algorithm which can solve a general class of PFDE efficiently and accurately by means of an O ( n 3 ) algorithm for solving the Sylvester Matrix Equation (over an m i? n grid with m ~ n ). The proposed parametrization of the integral surface allows for the solution with the more general Robin boundary conditions, and allows for high-order approximations to derivative boundary conditions. To achieve our ends, we also develop a new matrix-based approximation to fractional order derivatives. The proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions.
- Subjects :
- Numerical Analysis
Partial differential equation
Physics and Astronomy (miscellaneous)
Differential equation
Applied Mathematics
Mathematical analysis
MathematicsofComputing_NUMERICALANALYSIS
First-order partial differential equation
Exponential integrator
Computer Science Applications
Fractional calculus
Computational Mathematics
Modeling and Simulation
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
Sylvester equation
Numerical stability
Numerical partial differential equations
Mathematics
Subjects
Details
- ISSN :
- 00219991
- Volume :
- 293
- Database :
- OpenAIRE
- Journal :
- Journal of Computational Physics
- Accession number :
- edsair.doi...........6353db8b5acc996025819eb2959f0df6