A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay

A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A nu...
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Autor*in:	Wei Gu [verfasserIn] Yanli Zhou [verfasserIn] Xiangyu Ge [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2017

Übergeordnetes Werk:	In: Journal of Function Spaces - Hindawi Limited, 2016, (2017)
Übergeordnetes Werk:	year:2017

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc Journal toc

DOI / URN:	10.1155/2017/3679526

Katalog-ID:	DOAJ002187078

Internformat


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520			\|a A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
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951			\|a AR
952			\|j 2017

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allfields	10.1155/2017/3679526 doi (DE-627)DOAJ002187078 (DE-599)DOAJc002b313ba8d4d9b9025ee760eb275f2 DE-627 ger DE-627 rakwb eng QA1-939 Wei Gu verfasserin aut A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm. Mathematics Yanli Zhou verfasserin aut Xiangyu Ge verfasserin aut In Journal of Function Spaces Hindawi Limited, 2016 (2017) (DE-627)862936489 (DE-600)2861541-4 23148888 nnns year:2017 https://doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 kostenfrei http://dx.doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/toc/2314-8896 Journal toc kostenfrei https://doaj.org/toc/2314-8888 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2108 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2017
spelling	10.1155/2017/3679526 doi (DE-627)DOAJ002187078 (DE-599)DOAJc002b313ba8d4d9b9025ee760eb275f2 DE-627 ger DE-627 rakwb eng QA1-939 Wei Gu verfasserin aut A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm. Mathematics Yanli Zhou verfasserin aut Xiangyu Ge verfasserin aut In Journal of Function Spaces Hindawi Limited, 2016 (2017) (DE-627)862936489 (DE-600)2861541-4 23148888 nnns year:2017 https://doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 kostenfrei http://dx.doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/toc/2314-8896 Journal toc kostenfrei https://doaj.org/toc/2314-8888 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2108 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2017
allfields_unstemmed	10.1155/2017/3679526 doi (DE-627)DOAJ002187078 (DE-599)DOAJc002b313ba8d4d9b9025ee760eb275f2 DE-627 ger DE-627 rakwb eng QA1-939 Wei Gu verfasserin aut A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm. Mathematics Yanli Zhou verfasserin aut Xiangyu Ge verfasserin aut In Journal of Function Spaces Hindawi Limited, 2016 (2017) (DE-627)862936489 (DE-600)2861541-4 23148888 nnns year:2017 https://doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 kostenfrei http://dx.doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/toc/2314-8896 Journal toc kostenfrei https://doaj.org/toc/2314-8888 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2108 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2017
allfieldsGer	10.1155/2017/3679526 doi (DE-627)DOAJ002187078 (DE-599)DOAJc002b313ba8d4d9b9025ee760eb275f2 DE-627 ger DE-627 rakwb eng QA1-939 Wei Gu verfasserin aut A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm. Mathematics Yanli Zhou verfasserin aut Xiangyu Ge verfasserin aut In Journal of Function Spaces Hindawi Limited, 2016 (2017) (DE-627)862936489 (DE-600)2861541-4 23148888 nnns year:2017 https://doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 kostenfrei http://dx.doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/toc/2314-8896 Journal toc kostenfrei https://doaj.org/toc/2314-8888 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2108 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2017
allfieldsSound	10.1155/2017/3679526 doi (DE-627)DOAJ002187078 (DE-599)DOAJc002b313ba8d4d9b9025ee760eb275f2 DE-627 ger DE-627 rakwb eng QA1-939 Wei Gu verfasserin aut A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay 2017 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm. Mathematics Yanli Zhou verfasserin aut Xiangyu Ge verfasserin aut In Journal of Function Spaces Hindawi Limited, 2016 (2017) (DE-627)862936489 (DE-600)2861541-4 23148888 nnns year:2017 https://doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 kostenfrei http://dx.doi.org/10.1155/2017/3679526 kostenfrei https://doaj.org/toc/2314-8896 Journal toc kostenfrei https://doaj.org/toc/2314-8888 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2108 GBV_ILN_2111 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2017
language	English
source	In Journal of Function Spaces (2017) year:2017
sourceStr	In Journal of Function Spaces (2017) year:2017
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Mathematics
isfreeaccess_bool	true
container_title	Journal of Function Spaces
authorswithroles_txt_mv	Wei Gu @@aut@@ Yanli Zhou @@aut@@ Xiangyu Ge @@aut@@
publishDateDaySort_date	2017-01-01T00:00:00Z
hierarchy_top_id	862936489
id	DOAJ002187078
language_de	englisch
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callnumber-first	Q - Science
author	Wei Gu
spellingShingle	Wei Gu misc QA1-939 misc Mathematics A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
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topic_title	QA1-939 A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
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title	A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
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title_full	A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
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title_sort	compact difference scheme for solving fractional neutral parabolic differential equation with proportional delay
callnumber	QA1-939
title_auth	A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
abstract	A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
abstractGer	A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
abstract_unstemmed	A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
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title_short	A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay
url	https://doi.org/10.1155/2017/3679526 https://doaj.org/article/c002b313ba8d4d9b9025ee760eb275f2 http://dx.doi.org/10.1155/2017/3679526 https://doaj.org/toc/2314-8896 https://doaj.org/toc/2314-8888
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up_date	2024-07-04T00:13:44.619Z
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A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay

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