An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation
Abstract In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for F...
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doaj-c97728a846c14639940017363c80faa82020-11-25T01:57:44ZengSpringerOpenAdvances in Difference Equations1687-18472019-01-012019111510.1186/s13662-018-1939-6An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equationYanjie Zhou0Zhendong Luo1School of Science, Beijing Technology and Business UniversitySchool of Mathematics and Physics, North China Electric Power UniversityAbstract In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs.http://link.springer.com/article/10.1186/s13662-018-1939-6Proper orthogonal decompositionClassical Crank–Nicolson finite difference modelFractional-order parabolic-type sine-Gordon equationOptimized Crank–Nicolson finite difference extrapolating modelExistence, stabilization, and convergence |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yanjie Zhou Zhendong Luo |
spellingShingle |
Yanjie Zhou Zhendong Luo An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation Advances in Difference Equations Proper orthogonal decomposition Classical Crank–Nicolson finite difference model Fractional-order parabolic-type sine-Gordon equation Optimized Crank–Nicolson finite difference extrapolating model Existence, stabilization, and convergence |
author_facet |
Yanjie Zhou Zhendong Luo |
author_sort |
Yanjie Zhou |
title |
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation |
title_short |
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation |
title_full |
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation |
title_fullStr |
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation |
title_full_unstemmed |
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation |
title_sort |
optimized crank–nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-gordon equation |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1847 |
publishDate |
2019-01-01 |
description |
Abstract In this paper, by means of a proper orthogonal decomposition (POD) we mainly reduce the order of the classical Crank–Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs). Toward this end, we will first review the CCNFD model for FOPTSGEs and the theoretical results (such as existence, stabilization, and convergence) of the CCNFD solutions. Then we establish an optimized Crank–Nicolson finite difference extrapolating (OCNFDE) model, including very few unknowns but holding the fully second-order accuracy for FOPTSGEs via POD. Next, by a matrix analysis we will discuss the existence, stabilization, and convergence of the OCNFDE solutions. Finally, we will use a numerical example to validate the validity of theoretical conclusions. Moreover, we show that the OCNFDE model is very valid for settling FOPTSGEs. |
topic |
Proper orthogonal decomposition Classical Crank–Nicolson finite difference model Fractional-order parabolic-type sine-Gordon equation Optimized Crank–Nicolson finite difference extrapolating model Existence, stabilization, and convergence |
url |
http://link.springer.com/article/10.1186/s13662-018-1939-6 |
work_keys_str_mv |
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