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![The ROENBE solutions when 0⩽t⩽2000\documentclass[12pt]{minimal}...](publication/330115306/figure/fig1/AS:963428621512708@1606710691267/The-ROENBE-solutions-when-0t2000documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The OCNFDE solutions when 0⩽t⩽2.5\documentclass[12pt]{minimal}...](publication/330115306/figure/fig3/AS:963428621512732@1606710691703/The-OCNFDE-solutions-when-0t25documentclass12ptminimal-usepackageamsmath_Q320.jpg)
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, sta...
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... the records for settling the CCNFD model and the OCNFDE model in the same Laptop (iPone Mac book: Int Core i5 Processor, 16 GB RAM) we find that the CPU consumption time for set- tling the CCNFD model on 0 ≤ t ≤ 2000 is 762 min, whereas the CPU consumption time for settling the OCNFDE model is less than 6 min, that is, the CPU consumption time for settling the CCNFD model is 126 times more than that for settling the OCNFDE model. More comparisons of errors and CPU time of the CCNFD and OCNFDE solutions are listed in Table 1, where˜Vwhere˜ where˜V(t n ) -V n 2 and˜Vand˜ and˜V(t n ) -V n d 2 are approximately estimated by V n+1 -V n 2 and V n+1 d -V n d 2 , respectively. Table 1 further shows that the numerical computing conclusions accord with the theo- retical ones. ...
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... comparisons of errors and CPU time of the CCNFD and OCNFDE solutions are listed in Table 1, where˜Vwhere˜ where˜V(t n ) -V n 2 and˜Vand˜ and˜V(t n ) -V n d 2 are approximately estimated by V n+1 -V n 2 and V n+1 d -V n d 2 , respectively. Table 1 further shows that the numerical computing conclusions accord with the theo- retical ones. This implies that the OCNFDE model is effective for settling the FOPTSGEs (1)-(3) and that the OCNFDE model is far superior to the CCNFD model. ...
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... One of the techniques researchers employ to reduce the computational span of computational methods is the proper orthogonal decomposition method (POD) [1]. This has previously been used to reduce the dimensions of numerical methods such as non-standard finite difference and Crank-Nicolson to solve problems such as the Sobolev, hyperbolic, Burgers and Navier-Stokes equations [2][3][4]. In this paper, we couple the POD and HUPFD to obtain a new enhanced hybrid method that we name the enhanced higher-order unconditionally positive finite difference method (EHUPFD). ...
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... Zhou et al. [5,6] also built some FD schemes for space fractional diffusion equations. Luo and Wang [7] established a reduced-order FD scheme for the fractionalorder parabolic-type sine-Gordon equations, and Zhou and Luo [8] founded an optimized FD algorithm for the fractional-order parabolic-type sine-Gordon equations. Especially, Çelik and Duman [9] constructed the Crank-Nicolson finite element (FE) (CNFE) method with the unconditionally stable second-order time accuracy for the symmetric tempered fractional diffusion equation (STFDE), which is one of the most effective FE numerical methods. ...
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... That significantly affects their efficiency in terms of computational speed and memory requirement. The POD has been used to reduce the dimensions of numerical methods, such as the Crank-Nicolson method [13][14][15][16][17][18]. The POD is useful when it is impossible to perform numerical simulations due to large-scale computing requirements. ...
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... In future work, the proposed second-order operator splitting scheme will be extended to numerically solve the spatial fractional-order parabolic-type sine-Gordon equation. 10,27 ...
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