Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May 1st 2018
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- Lisa Sherman
- 6 years ago
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1 ﺟﻠﺪ 2 ﺷﻤﺎره 2 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 395 )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ( 65 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﭘﺬﯾﺮش 95/4/9 درﯾﺎﻓﺖ 95//5 ﭼﮑﯿﺪه در اﯾﻦ ﻣﻘﺎﻟﻪ روﺷﯽ ﺟﺪﯾﺪ ﺑﺮ ﭘﺎﯾﮥ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻏﯿﺮﺧﻄﯽ ﺑﯿﺎن ﺷﺪه اﺳﺖ. ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال ﮐﺴﺮي ﺑﻪدﺳﺖ آﻣﺪه و ﺑﺮاي ﺗﺒﺪﯾﻞ ﻣﻌﺎدﻟﻪ اﺻﻠﯽ از دﺳﺘﮕﺎه ﻣﻌﺎدﻻت ﺟﺒﺮي اﺳﺘﻔﺎده ﺷﺪه اﺳﺖ. اﯾﻦ روش ﺟﻮاب را ﺑﻪﺻﻮرت ﯾﮏ ﺳﺮي ﺑﻪﺳﺮﻋﺖ ﻫﻢﮔﺮا ﺑﻪدﺳﺖ ﻣﯽآورد. ﻋﻼوه ﺑﺮ اﯾﻦ ﺗﺠﺰﯾﻪ و ﺗﺤﻠﯿﻞ ﺧﻄﺎي روش ﻣﻮرد ﻧﻈﺮ ﺗﺤﺖ ﭼﻨﺪ ﺷﺮط ﺳﺎده داده ﺷﺪه اﺳﺖ. ﺳﻪ ﻣﺜﺎل ﻋﺪدي ﺑﺮاي ﻧﺸﺎن دادن ﮐﺎراﯾﯽ و دﻗﺖ روش داده ﺷﺪه اﺳﺖ. ﻣﺜﺎلﻫﺎ ﮐﺎراﯾﯽ و ﮐﺎرﺑﺮد روش را ﻧﺸﺎن ﻣﯽدﻫﻨﺪ. واژهﻫﺎي ﮐﻠﯿﺪي : ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺣﺴﺎﺑﺎن ﮐﺴﺮي ﻣﺸﺘﻖ ﮐﺎﭘﻮﺗﻮ. ﻣﻘﺪﻣﻪ در دﻧﯿﺎي واﻗﻌﯽ ﺑﺮاي ﻣﺪلﺳﺎزي و ﺗﺠﺰﯾﻪ و ﺗﺤﻠﯿﻞ ﺣﺠﻢ ﻋﻈﯿﻤﯽ از ﻣﺴﺎﺋﻞ ﻧﯿﺎزﻣﻨﺪ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻫﺴﺘﯿﻢ. ﺣﺴﺎﺑﺎن ﮐﺴﺮي در ﺑﺴﯿﺎري از زﻣﯿﻨﻪﻫﺎي ﻋﻠﻮم رﯾﺎﺿﯽ و ﻣﻬﻨﺪﺳﯽ از ﺟﻤﻠﻪ ﺷﺒﮑﻪﻫﺎي ﺑﺮق ﻣﮑﺎﻧﯿﮏ ﺳﯿﺎﻻت ﻧﻈﺮﯾﮥ ﮐﻨﺘﺮل اﻟﮑﺘﺮوﻣﻐﻨﺎﻃﯿﺲ زﯾﺴﺖﺷﻨﺎﺳﯽ ﺷﯿﻤﯽ اﻧﺘﺸﺎر و وﯾﺴﮑﻮاﻻﺳﺘﯿﺴﯿﺘﻪ اﺳﺘﻔﺎده ﻣﯽﺷﻮد ].[5]-[ در ﺳﺎلﻫﺎي اﺧﯿﺮ ﺑﻪ ﺣﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﻣﻌﻤﻮﻟﯽ ﮐﺴﺮي ﻣﻌﺎدﻻت اﻧﺘﮕﺮال و ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﺑﺎ ﻣﺸﺘﻘﺎت ﺟﺰﺋﯽ ﮐﺴﺮي ﺑﺴﯿﺎر ﺗﻮﺟﻪ ﺷﺪه اﺳﺖ. از آنﺟﺎ ﮐﻪ اﮐﺜﺮ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﺟﻮابﻫﺎي ﺗﺤﻠﯿﻠﯽ دﻗﯿﻖ ﻧﺪارﻧﺪ ﺑﺮاي ﺣﻞ آنﻫﺎ روشﻫﺎي ﺗﻘﺮﯾﺒﯽ و ﻋﺪدي ﺑﻪﻃﻮر ﮔﺴﺘﺮده اﺳﺘﻔﺎده ﻣﯽﺷﻮد. از ﺟﻤﻠﻪ روشﻫﺎي ﻋﺪدي ﺑﺮاي ﺣﻞ اﯾﻦ ﻣﻌﺎدﻻت ﺗﺒﺪﯾﻼت ﻻﭘﻼس ] [6 ﺗﺒﺪﯾﻼت ﻓﻮرﯾﻪ ] [7 ﺑﺴﻂ ﺑﺮدار وﯾﮋه ] [8 روش ﺗﺠﺰﯾﮥ آدوﻣﯿﻦ ] [] [9 روش ﺗﮑﺮار ﺗﻐﯿﯿﺮات ] [2] [ روش ﺗﺒﺪﯾﻞ دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ] [4] [3 روش ﺗﻔﺎﺿﻞ ﮐﺴﺮي ] [5 روش ﺗﺠﺰﯾﻪ و ﺗﺤﻠﯿﻞ ﻫﻤﻮﺗﻮﭘﯽ ] [6 روش ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ] [7 روش ﺗﺒﺪﯾﻞ دﯾﻔﺮاﻧﺴﯿﻞ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ ] [9] [8 روش ﮔﺴﺴﺘﻪﺳﺎزي زﻣﺎن ] [2 روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ] [2 روش ﺳﺮي ﺗﻮاﻧﯽ ] [22 و ﺳﺎﯾﺮ روشﻫﺎ ] [27]-[23 اﺳﺖ. ﻫﻢﭼﻨﯿﻦ در ﭼﻨﺪ ﻣﻘﺎﻟﻪ ﺣﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﻣﺮﺗﺒﻪ ﮐﺴﺮي ﻏﯿﺮﺧﻄﯽ ] [3] [29] [28] [25 ﺑﺮرﺳﯽ ﺷﺪه اﺳﺖ. در اﯾﻦ ﻣﻘﺎﻟﻪ از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﺑﻪ ﺷﮑﻞ ) ( اﺳﺘﻔﺎده ﺷﺪه اﺳﺖ ) ( r [, T ], D f ( ) a ( )W( f ( )) ak ( ) D k f ( ) g ( ), k ﺑﺎ ﺷﺮاﯾﻂ اوﻟﯿﻪ s,,,, *ﻧﻮﯾﺴﻨﺪه ﻣﺴﺌﻮل f.mirzaee@malayeru.ac.ir f ( s ) () bs, Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﻓﺮﺷﯿﺪ ﻣﯿﺮزاﺋﯽ* اﻟﻬﺎم ﺣﺪادﯾﺎن ﻧﮋاد ﯾﻮﺳﻔﯽ داﻧﺸﮕﺎه ﻣﻼﯾﺮ داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ و آﻣﺎر ﻣﻼﯾﺮ
2 66 ﺟﻠﺪ 2 ﺷﻤﺎره 2 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 395 )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ( ﮐﻪ در آن D 2 r ﺑﯿﺎنﮔﺮ ﻣﺸﺘﻖ ﮐﺴﺮي ﮐﺎﭘﻮﺗﻮ از ﻣﺮﺗﺒﮥ. ﺗﺎﺑﻊ ﺳﻘﻒ W ﺗﺎﺑﻊ ﻏﯿﺮﺧﻄﯽ در ) ak ( ) f ( ﺑﺮاي k,,, r ﺗﻮاﺑﻌﯽ ﻣﻌﻠﻮم g ( ) ﺳﯿﮕﻨﺎل ورودي و ) f ( ﭘﺎﺳﺦ ﺧﺮوﺟﯽ اﺳﺖ. ﻣﻘﺪﻣﺎت و ﻧﻤﺎدﮔﺬاريﻫﺎ ﮐﻪ در آن ﺗﺎﺑﻊ ﮔﺎﻣﺎ و ﺑﯿﺎنﮔﺮ ﺿﺮب ﭘﯿﭽﺸﯽ اﺳﺖ. ﺑﺮاي ﻋﻤﻠﮕﺮ I اﯾﻦ رواﺑﻂ ﺑﺮﻗﺮار اﺳﺖ : ) ( ( a ), ) ( a,, I ( a) I I f ( ) I f ( ) I I f ( ). ﺳﺎﯾﺮ ﺧﻮاص ﻋﻤﻠﮕﺮ I را ﻣﯽﺗﻮان در ] [3] [6 ﻣﻼﺣﻈﻪ ﮐﺮد. ﺗﻌﺮﯾﻒ :2 ﻣﺸﺘﻖ ﮐﺴﺮي D در ﻣﻔﻬﻮم ﮐﺎﭘﻮﺗﻮ ﺑﻪ ﺻﻮرت زﯾﺮ ﺗﻌﺮﯾﻒ ﻣﯽ ﺷﻮد n n, n. f (t ) dt, n dn ( n ) d n ) ( t D f ( ) در ﺣﺎﻟﺖ ﮐﻠﯽ ﻋﻤﻠﮕﺮﻫﺎي اﻧﺘﮕﺮال ﮐﺴﺮي و ﻣﺸﺘﺘﻖ ﮐﺴﺮي ﺗﻌﻮﯾﺾﭘﺬﯾﺮ ﻧﯿﺴﺖ. و n k I D f ( ) f ( ) f ( k ) (), n n, n,! k k ﺑﺮاي ﺑﺮﻋﮑﺲ اﯾﻦ ﻋﻤﻠﮕﺮﻫﺎ دارﯾﻢ : f ( ). f ( ) D I D ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ و ﺧﻮاص آنﻫﺎ ﺗﻌﺮﯾﻒ : ﺑﺮدار m ﻋﻀﻮي ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ H m ( ) روي ﺑﺎزة ﺑﻪﺻﻮرت ) (3 ﺗﻌﺮﯾﻒ ﻣﯽ ﺷﻮد ] :[33] [32 ) (3 T H m ( ) h ( ), h ( ),, hm ( ), ﮐﻪ در آن 2h,, در ﻏــــﯿﺮ اﯾــــﻦ ﺻــــﻮرت و اﮔﺮ i ﻓﺮد و i m ﺑﺎﺷﺪ ) ( h )( 2h h ( ) 2h 2 ( (i )h )( (i )h ) (i )h (i )h, hi ( ) h 2, در ﻏــــﯿﺮ اﯾــــﻦ ﺻــــﻮرت و اﮔﺮ i زوج و 2 i m 2 ﺑﺎﺷﺪ Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﺗﻌﺮﯾﻒ : اﻧﺘﮕﺮال ﮐﺴﺮي رﯾﻤﺎن - ﻟﯿﻮوﯾﻞ از ﻣﺮﺗﺒﻪ ﺑﻪﺻﻮرت ) (2 ﺗﻌﺮﯾﻒ ﻣﯽﺷﻮد ] :[6 I f ( ) ( t ) f (t )dt f ( ),, ) (2 ) ( ) ( I f ( ) f ( ),
3 67 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي (i 2)h ih, ih (i 2)h,, در ﻏــــﯿﺮ اﯾــــﻦ ﺻــــﻮرت و T 2h T,, در ﻏــــﯿﺮ اﯾــــﻦ ﺻــــﻮرت T m 2 ﯾﮏ ﻋﺪد ﺻﺤﯿﺢ و زوج اﺳﺖ و m )) ( (T h ))( (T 2h hm ( ) 2h 2 h اﺳﺖ. ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﯾﻒ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﻪ ﺳﺎدﮔﯽ ﻣﯽﺗﻮان ﻧﺸﺎن داد : i j, hi ( jh) ) (4 i j, i, ﻋــﺪدي زوج اﺳــﺖ و i j 3 i, ﻋــﺪدي ﻓــﺮد اﺳــﺖ و i j 2 hi ( )h j ( ) و m h ( ). i i ﺗﻘﺮﯾﺐ ﺗﺎﺑﻊ ﺗﺎﺑﻊ دﻟﺨﻮاه f روي داﻣﻨﻪ را ﻣﯽﺗﻮان ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﻪ ﻓﺮم ﺑﺮداري زﯾﺮ ﺗﻘﺮﯾﺐ زد : m ) (5 f ( ) f m ( ) fi hi ( ) F T H m ( ) H mt ( ) F, i ﺑﻪﻃﻮريﮐﻪ T F f, f,, f m. ﻣﺰﯾﺖ ﺟﺎﻟﺐ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺮاي ﺗﻘﺮﯾﺐ ) f ( اﯾﻦ اﺳﺖ ﮐﻪ ﺿﺮاﯾﺐ f i در ﻣﻌﺎدﻟﮥ ) (5 از اﯾﻦ راﺑﻄﻪ ﺑﻪدﺳﺖ ﻣﯽ آﯾﻨﺪ : fi f (ih); i,,, m. ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال ﮐﺴﺮي در اﯾﻦ ﺑﺨﺶ ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ) H m ( از ﻣﺮﺗﺒﻪ ﺻﺤﯿﺢ و ﮐﺴﺮي ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﻪدﺳﺖ ﻣﯽآﯾﺪ. ﺑﺮاي ﺑﻪدﺳﺖ آوردن ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال از ﻣﺮﺗﺒﻪ ﺻﺤﯿﺢ ﺑﺎ اﺳﺘﻔﺎده از ﺑﺴﻂ ) hi ( ﺑﺮاي i,,, m دارﯾﻢ, H m (t )dt PH m ( ), Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ) 2h 2 ( (i )h )( (i 2)h ) hi ( ) 2 ( (i )h )( (i 2)h 2h
4 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 2 ﺷﻤﺎره 2 ﺟﻠﺪ ( )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 :[34] ( اﺳﺖ ﮐﻪ از ﻣﻌﺎدﻟﻪ زﯾﺮ ﺑﻪدﺳﺖ ﻣﯽآﯾﺪ m ) ( m ) ﯾﮏ ﻣﺎﺗﺮﯾﺲ P ﮐﻪ در آن 5 8 h P : اﯾﻦ راﺑﻄﻪ ﺑﺮرﺳﯽ ﻣﯽﺷﻮد ﺑﺮاي ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال از ﻣﺮﺗﺒﻪ ﮐﺴﺮي T I H m ( ) I h ( ), I h ( ),, I hm ( ),. : ﺑﺪﯾﻦﺻﻮرت ﺑﺴﻂ داده ﻣﯽﺷﻮد i,,, m ﺑﺮاي I hi ( ) ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ m I hi ( ) pi, j h j ( ), j : ( دارﯾﻢ 2) ﺑﺎ اﺳﺘﻔﺎده از راﺑﻄﮥ. اﺳﺖ p i, j I hi ( jh ) ﮐﻪ در آن pi, j I hi ( jh) ( ) jh ( jh t ) hi (t )dt. : را ﺑﺪﯾﻦﺻﻮرت ﺑﺎزﻧﻮﯾﺴﯽ ﮐﺮد p i, j ﻣﯽﺗﻮان hi () ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﯾﻒ j, h 2 p, j (2 3 ) j, 2 ( 3 ) h 2 2 ( j 2) ( 2) 2( j 2) 2 ( 3) 2 j ( )( 2) 3 j ( 2) 2 j j 2, : ﺑﺎﺷﺪ i m ﻓﺮد و i و اﮔﺮ 2h pi, j ( ) ( 3) 2h ( j i ) ( 2) ( j i ) 2 ( j i ) ( 2) ( j i ) 2 ( 3) j i, j i, j i, : 2 i m 2 زوج ﺑﺎﺷﺪ و i و اﮔﺮ
5 69 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي j i 2, j i, j i, j i 2, ﮐﻪ در آن ) dij 2( j i 2) 2 ( j i 2) ( 2) ( j i 2) ( 2 2( j i 2) 2 6( j i ) ( 2), و j m 2, j m, j m. ﺑﻨﺎﺑﺮاﯾﻦ ) (6 ﮐﻪ در آن, h ) 2 ( 3 h ( 2) 2 ( 3) pm, j I H m ( ) P H m ( ), i, j,,, m, P pi, j, ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال از ﻣﺮﺗﺒﻪ ﮐﺴﺮي ( m ) ( m ) ﺑﻌﺪي اﺳﺖ و ) H m ( در راﺑﻄﮥ ) (3 ﺗﻌﺮﯾﻒ ﺷﺪه اﺳﺖ. زﻣﺎﻧﯽ ﮐﻪ ﺑﺎﺷﺪ ﻣﺎﺗﺮﯾﺲ P ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ ﻣﺎﺗﺮﯾﺲ P ﺑﻨﺎﺑﺮاﯾﻦ ﻣﺎﺗﺮﯾﺲ P ﯾﮏ ﺗﻌﻤﯿﻢ ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال ﺑﺮاي ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ اﺳﺖ. ﻻزم ﺑﻪذﮐﺮ اﺳﺖ ﮐﻪ ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ P ﻣﺎﺗﺮﯾﺲ ﺗﻨﮑﯽ اﺳﺖ ﮐﻪ ﺑﺎﻋﺚ ﻣﯽﺷﻮد ﻣﺤﺎﺳﺒﺎت ﺳﺮﯾﻊ ﺗﺮ ﺷﻮد ﻫﻢﭼﻨﯿﻦ ﻣﺤﺎﺳﺒﻪ ﻣﺎﺗﺮﯾﺲ P ﯾﮏ ﺑﺎر اﻧﺠﺎم ﻣﯽﺷﻮد و ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﻣﺮﺗﺒﮥ ﮐﺴﺮي و ﻫﻢﭼﻨﯿﻦ ﻣﺮﺗﺒﻪ ﺻﺤﯿﺢ اﺳﺘﻔﺎده ﻗﺮار ﻣﯽﺷﻮد. ﮐﺎرﺑﺮد روش در اﯾﻦ ﺑﺨﺶ ﺑﻪ ﺑﯿﺎن روﺷﯽ ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻟﮥ ) ( ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ و ﻣﺎﺗﺮﯾﺲﻫﺎي ﻋﻤﻠﯿﺎﺗﯽ آن ﭘﺮداﺧﺘﻪ ﻣﯽﺷﻮد. اﺑﺘﺪا از ﺷﺮاﯾﻂ اوﻟﯿﻪ ﺑﻪﻣﻨﻈﻮر ﺗﺒﺪﯾﻞ ﻣﺴﺌﻠﻪ ﺑﺎ ﻣﻘﺪار اوﻟﯿﻪ داده ﺷﺪه ﺑﻪ ﯾﮏ ﻣﺴﺌﻠﻪ ﺑﺎ ﺷﺮاﯾﻂ اوﻟﯿﻪ ﺻﻔﺮ اﺳﺘﻔﺎده ﻣﯽﺷﻮد. ﭘﺲ از آن از ﻣﺎﺗﺮﯾﺲﻫﺎي ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال ﻣﺮﺗﺒﮥ ﮐﺴﺮي ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺮاي ﺗﺒﺪﯾﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﺑﻪ دﺳﺘﮕﺎه ﻣﻌﺎدﻻت ﻏﯿﺮﺧﻄﯽ اﺳﺘﻔﺎده ﻣﯽﺷﻮد. ﻣﻌﺎدﻟﮥ ) ( را در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ. ﺗﺎﺑﻊ f را ﻣﯽﺗﻮان ﺑﻪﺻﻮرت ) (7 ﺑﺎزﻧﻮﯾﺴﯽ ﮐﺮد : f ( ) f * ( ) u ( ), ) (7 Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 j i, h ) 2 ( 3 h ( 2) ) pi, j 2 ( 3 h ( 4) ) 2 ( 3 h dij ) 2 ( 3
6 7 ﺟﻠﺪ 2 ﺷﻤﺎره 2 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 395 )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ( ﮐﻪ در آن ) f * ( ﺗﺎﺑﻌﯽ ﻣﻌﻠﻮم اﺳﺖ ﮐﻪ در اﯾﻦ ﺷﺮاﯾﻂ ﺻﺪق ﻣﯽﮐﻨﺪ : s,,,, () bs, ) (s * f و ) u ( ﺗﺎﺑﻊ ﻣﺠﻬﻮل ﺟﺪﯾﺪي اﺳﺖ. از ﻗﺮار دادن ) (7 در ) ( ﺑﺮاي ﺗﺎﺑﻊ ) u ( ﻣﺴﺌﻠﮥ ﻣﻘﺪار اوﻟﯿﻪ ﺑﺎ ﺷﺮاﯾﻂ اوﻟﯿﻪ ﺻﻔﺮ ﺑﻪﺻﻮرت ) (8 ﺑﻪدﺳﺖ ﻣﯽآﯾﺪ : r k ﺑﺎ ﺷﺮاﯾﻂ اوﻟﯿﻪ s,,,, ) (s u (), ﮐﻪ در آن r v( ) g ( ) ak ( ) D k f* ( ) D f* ( ). k ﺣﺎل ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺗﻮاﺑﻊ ) v ( و ) D u ( ﺑﺪﯾﻦﺻﻮرت ﺗﻘﺮﯾﺐ زده ﻣﯽﺷﻮد : v ( ) V T H m ( ), ) (9 D u ( ) C T H m ( ), ﮐﻪ در آن V و C ﺑﻪﺗﺮﺗﯿﺐ ﺿﺮاﯾﺐ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺗﻮاﺑﻊ ) v ( و ) D u ( ﻫﺴﺘﻨﺪ. ﺑﺎ اﺳﺘﻔﺎده از ﻣﻌﺎدﻻت ) (6 و ) (8 و ﺧﺎﺻﯿﺖ ﺣﺴﺎﺑﺎن ﮐﺴﺮي دارﯾﻢ : ) ( D k u ( ) I k D u ( ) I k C T H m () C T P k H m (), و H m () C P H m (). ) ( T T C u ( ) I D u ( ) I ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻣﻌﺎدﻻت ) ()-(9 دارﯾﻢ : r ) C T H m ( ) a ( ) W f* () C T P H m ( ) ak ( )C T P k H m ( ) V T H m ( ), (2 k ﮐﻪ در آن. ﻣﻌﺎدﻟﮥ ) (2 دﺳﺘﮕﺎه ﻣﻌﺎدﻻت ﺟﺒﺮي ﻏﯿﺮﺧﻄﯽ اﺳﺖ ﮐﻪ ﻣﯽﺗﻮان آن را ﺑﺎ دﺳﺘﻮر fsolve در ﻧﺮماﻓﺰار Matlab و ﯾﺎ از ﻃﺮﯾﻖ روشﻫﺎي ﻋﺪدي ﺗﮑﺮاري ﻣﺎﻧﻨﺪ روش ﻧﯿﻮﺗﻦ ﺣﻞ ﮐﺮد. ﭘﺲ از آن ﻣﯽﺗﻮان ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﻣﻌﺎدﻟﮥ ) ( را ﺑﺪﯾﻦﺻﻮرت ﺑﻪدﺳﺖ آورد : f ( ) f * ( ) C T P H m ( ). ﻫﻤﮕﺮاﯾﯽ و آﻧﺎﻟﯿﺰ ﺧﻄﺎ در اﯾﻦ ﺑﺨﺶ ﺧﻄﺎي روش اراﺋﻪ داده ﺷﺪه ﺑﺮرﺳﯽ ﻣﯽﺷﻮد. ﻓﺮض ﮐﻨﯿﻢ ﮐﻪ ﺳﺮي ﻗﻄﻊ ﺷﺪه ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﯾﮏ ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﺑﺮاي ﻣﻌﺎدﻟﻪ ) ( ﺑﺎﺷﺪ. ﺑﺮاي اﯾﻦ ﻣﻨﻈﻮر اﺑﺘﺪا ﺗﻌﺮﯾﻒ ﻣﯽﮐﻨﯿﻢ : f sup f ( ). ﻗﻀﯿﻪ : ﻓﺮض ﮐﻨﯿﻢ ) v m ( ﺑﺴﻂ ﺗﺎﺑﻊ ) v ( ﺑﺎ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ﺑﻪﺻﻮرت m ) vm ( ) v (ih)hi ( ﺗﻌﺮﯾﻒ ﻣﯽﺷﻮد. ﻫﻢﭼﻨﯿﻦ ﻓﺮض ﮐﻨﯿﻢ ﮐﻪ i e m v v m در اﯾﻦﺻﻮرت دارﯾﻢ : Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ) (8 [, T ], D u ( ) a ( )W( f* ( ) u ( )) ak ( ) D k u ( ) v ( ),
7 7 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي em O h3, و Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 I v I vm MT h3, 9 3 (3). اﺳﺖ v (3) M ﮐﻪ در آن ﭼﻨﺪﺟﻤﻠﻪاي دروﻧﯿﺎب درﺟﮥ دوم روي ﺑﺎزه v m ( ) ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﯾﻒ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ : اﺛﺒﺎت : ﺑﻨﺎﺑﺮاﯾﻦ. اﺳﺖ i 2, 4,, m ﺑﺮاي (i 2)h, ih i (3) v( ) vm ( ) v ( ) ( jh), 6. j i 2 : دارﯾﻢ v (3) ( ) v (3) em در i ( jh) و ﻣﺎﮐﺴﯿﻤﻢ ﻣﻘﺪار w v (3) 6 w, i i j i 2 j i 2 sup ( jh) از ﻃﺮﻓﯽ. w( ) j i 2 ( jh) ﺑﻨﺎﺑﺮاﯾﻦ ﺑﻪدﺳﺖ ﻣﯽآﯾﺪ (i i ( jh) 2 3h3 9 j i 2 از آنﺟﺎﮐﻪ 3 3 ﮐﻪ )h ﻧﻘﻄﻪ,, ﯾﺎ w 2 3h 3 9, در ﻧﺘﯿﺠﻪ em h3 v (3) 9 3 (4) Ch3. : دارﯾﻢ (2) ﻫﻢﭼﻨﯿﻦ ﺑﺎ اﺳﺘﻔﺎده از راﺑﻄﮥ I v ( ) I v m ( ) ( ) ( t ) v (t ) v m (t ) dt. : دارﯾﻢ (4) ﺑﺎ اﺳﺘﻔﺎده از راﺑﻄﮥ I v( ) I vm ( ) ( ) I v I vm 3 h v (3) ( t ) dt, 9 3 ( ) Mh3 MT h3 w ( ) ( t ) v vm h3 (3) v 9 3 w. w2 ( ) dt و w ( ) ( t ) dt ﮐﻪ در آن ( ) ( )
8 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 2 ﺷﻤﺎره 2 ﺟﻠﺪ ( )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ W( f ) ﻫﻢﭼﻨﯿﻦ ﻓﺮض ﮐﻨﯿﻢ ﺗﺎﺑﻊ. ( ﺑﺎﺷﻨﺪ ) ﺟﻮاب دﻗﯿﻖ و ﺗﻘﺮﯾﺒﯽ ﻣﻌﺎدﻟﻪ ﺑﻪﺗﺮﺗﯿﺐ f m و f ﻓﺮض ﮐﻨﯿﻢ :2 ﻗﻀﯿﻪ L ﯾﻌﻨﯽ ﺑﺮاي ﯾﮏ در ﺷﺮط ﻟﯿﭗ ﺷﯿﺘﺲ ﺻﺪق ﮐﻨﺪ W( f ( )) W( f 2 ( )) L f ( ) f 2 ( ),. Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 : ﻫﻢﭼﻨﯿﻦ ﻓﺮض ﮐﻨﯿﻢ ak k,,, r, Nk, و N k T k LN T. k k r (5) در اﯾﻦ ﺻﻮرت Em f f m MT h 3, (6) N kt k LN T 9 3 k k (3). اﺳﺖ v ﮐﺮان ﺑﺎﻻي M ﮐﻪ در آن r :: دارﯾﻢ (7) ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ راﺑﻄﮥ : اﺛﺒﺎت Em f f m f * u f * um u um. : دارﯾﻢ (8) از راﺑﻄﻪ D u ( ) D um ( ) a ( ) W( f* ( ) u ( )) W( f* ( ) um ( )) r ak ( ) D k u ( ) D k um ( ) v( ) vm ( ). (7) k : دارﯾﻢ (7) ﺣﺎل از ﺳﺎده ﺳﺎزي ﻣﻌﺎدﻟﻪ u ( ) um ( ) I a ( ) W( f* ( ) u ( )) W( f* ( ) um ( )) r I ak ( ) D k u ( ) D k um ( ) I v( ) I vm ( ). k ﺑﻨﺎﺑﺮاﯾﻦ Em I a W( f* u ) W( f* um ) r I ak D k u D k u m k I v I vm, : ﯾﺎ ﺑﻪﻋﺒﺎرﺗﯽ Em a I W( f* u ) I W( f* um ) r ak k I k u I k um MT h ( ) (8) : دارﯾﻢ (2) ﺣﺎل ﺑﺎ اﺳﺘﻔﺎده از I k u ( ) I k um ( ) ( t ) k (u (t ) u m (t )) dt. k
9 73 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي در ﻧﺘﯿﺠﻪ ( t ) k u (t ) um (t ) dt k I k u ( ) I k u m ( ) Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 u um k k ( t ) k dt u um k T k u um k, ﺑﻨﺎﺑﺮاﯾﻦ I k u I k um k T u um. k (9) ﻫﻢﭼﻨﯿﻦ I W( f ( )) I W( f m ( )) ( t ) W( f (t)) W( f m (t)) dt, ( ) در ﻧﺘﯿﺠﻪ I W( f ( )) I W( f m ( )) ( t ) W( f (t)) W( f m (t)) dt ( ) ( t ) L f (t) f m (t) dt ( ) L f fm ( ) L ( ) ( t ) dt f fm LT f fm ( ), : از ﻃﺮﻓﯽ ﻣﯽداﻧﯿﻢ I W(f ( )) I W( f m ( )) I W(f * ( ) u ( )) I W(f * ( ) u m ( )), و f f m u u m, ﺑﻨﺎﺑﺮاﯾﻦ I W( f ) I W( f m ) LT u um ( ). (2) : ﻧﺘﯿﺠﻪ ﻣﯽﮔﯿﺮﯾﻢ (2) ( و 9) (3) از رواﺑﻂ LN T N kt MT h 3 Em Em Em. 9 3 k k r k
10 74 ﺟﻠﺪ 2 ﺷﻤﺎره 2 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 395 )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ( در ﻧﺘﯿﺠﻪ Em O h3 اﺳﺖ و اﺛﺒﺎت ﻗﻀﯿﻪ ﺗﮑﻤﯿﻞ ﺷﺪ. ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ اﯾﻦﮐﻪ ﻣﺮﺗﺒﻪ ﻫﻢﮔﺮاﯾﯽ ﺗﻮاﺑﻊ ﮐﻼﻫﯽ O h 2 و ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ O h3 اﺳﺖ اﺳﺘﻔﺎده از ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ در ﺣﻞ ﻣﺴﺎﺋﻞ ﺟﻮاب را ﺑﺎ دﻗﺖ ﺑﻬﺘﺮي ﺑﻪدﺳﺖ ﻣﯽآورد. در ﺑﺨﺶ ﺑﻌﺪ ﻣﻘﺎﯾﺴﮥ ﻧﺘﺎﯾﺞ ﻋﺪدي ﻣﺜﺎلﻫﺎي داده ﺷﺪه ﮔﻮاﻫﯽ ﺑﺮ اﯾﻦ ﻧﮑﺘﻪ ﻫﺴﺘﻨﺪ. در اﯾﻦ ﺑﺨﺶ ﺑﺎ ﭼﻨﺪ ﻣﺜﺎل ﮐﺎراﯾﯽ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﻧﺸﺎن داده ﻣﯽﺷﻮد. ﻫﻢﭼﻨﯿﻦ ﺑﻪﻣﻘﺎﯾﺴﮥ روشﻫﺎي ﺗﻮاﺑﻊ ﮐﻼﻫﯽ و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﻣﯽﭘﺮدازﯾﻢ. ﻣﺜﺎل : ﻣﻌﺎدﻟﻪ ﻧﻮﺳﺎن ﮐﺴﺮي ) (2 را ﺑﺮرﺳﯽ ﻣﯽﮐﻨﯿﻢ ] :[2 D f ( ) D f ( ) f ( ) 8, [,], 2, ) (2 f (), f (). 2 ﺟﻮاب دﻗﯿﻖ ﺑﻪﺻﻮرت ﯾﮏ ﺳﺮي در ] [35 داده ﺷﺪه اﺳﺖ. در ﺟﺪول ﻣﻘﺎﯾﺴﮥ ﺟﻮاب دﻗﯿﻖ ﺑﺎ ﺟﻮاب ﺣﺎﺻﻞ از روش اراﺋﻪ ﺷﺪه و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ] [2 ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. در ﺟﺪول 2 ﻣﻘﺎﯾﺴﮥ ﺑﯿﻦ ﺧﻄﺎﻫﺎي ﻣﻄﻠﻖ ﺑﻪدﺳﺖ آﻣﺪه ﺑﺎ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ روش ﺗﻮاﺑﻊ ﺑﻼك ﭘﺎﻟﺲ ] [27 و روش ﺗﻔﺎﺿﻼت ﮐﺴﺮي ] [35 ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. ﺑﺮاي ﺟﺰﺋﯿﺎت ﺑﯿﺶﺗﺮ ﺧﻄﺎﻫﺎي ﻣﻄﻠﻖ اﯾﻦ ﻣﺜﺎل ﺑﺎ اﺳﺘﻔﺎده از روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ در ﺷﮑﻞ رﺳﻢ ﺷﺪه اﺳﺖ. ﺟﺪول. ﻧﺘﺎﯾﺞ ﻋﺪدي ﺑﺮاي ﻣﻌﺎدﻟﮥ ) (2 ﺑﺎ m=.5.75 روش ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺟﻮاب دﻗﯿﻖ روش ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺟﻮاب دﻗﯿﻖ / / / / / / / / / / / / / /39757 / / / / / / /2 / / / / / / / / / / / / /3 /4 / / / / / / / / / / / / / / / / / / /5 /6 /7 2/ / / / / / / / 675 2/ / / / 676 /8 /9 2/ / / /9572 3/9529 3/9574 / ﺟﺪول.2 ﻣﻘﺎﯾﺴﻪ ﻣﺎﮐﺰﯾﻤﻢ ﺧﻄﺎي ﻣﻄﻠﻖ ﺑﺮاي ﻣﺜﺎل ) ( ﺑﺎ m= و =/5 /75 روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ روش ﺗﻮاﺑﻊ ﺑﻼك ﭘﺎﻟﺲ روش ﺗﻔﺎﺿﻼت ﮐﺴﺮي /7e-9 8/9e-9 4/e-7 3/2e-7 2/5e-3 2/3e-3 4/5e-3 3/e-3 /5 /75 Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﻣﺜﺎلﻫﺎي ﻋﺪدي
11 75 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻣﺜﺎل :2 ﻣﻌﺎدﻟﮥ دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻏﯿﺮﺧﻄﯽ ) (22 را ﺑﺮرﺳﯽ ﻣﯽﮐﻨﯿﻢ ] :[2 ad f ( ) bd f ( ) cd f ( ) ef 3 ( ) g ( ), ) (22 f () 2, f (), f (), [,], 2 2, 2 2/ a 2b 2c g ( ) و ﺟﻮاب دﻗﯿﻖ ﺑﺮاﺑﺮ ﮐﻪ در آن e ) ( / 8 ) (4 2 ) ( f ( ) اﺳﺖ ﺑﺎ 3 ﻓﺮض ﮐﻨﯿﺪ.75 a b c e و در ﺟﺪول 3 ﻣﻘﺎﯾﺴﮥ ﺟﻮاب دﻗﯿﻖ ﺑﺎ ﺟﻮاب ﺣﺎﺻﻞ از روش اراﺋﻪ ﺷﺪه و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. در ﺟﺪول 4 ﻣﻘﺎﯾﺴﮥ ﺑﯿﻦ ﺧﻄﺎﻫﺎي ﻣﻄﻠﻖ ﺑﻪدﺳﺖ آﻣﺪه ﺑﺎ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ روش ﺗﻮاﺑﻊ ﺑﻼك ﭘﺎﻟﺲ و روش ﻣﻮﺟﮏ ﻫﺎر ] [26 ﺑﺮاي m 6,32,64 ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. ﺑﺮاي ﺟﺰﺋﯿﺎت ﺑﯿﺶﺗﺮ ﺧﻄﺎﻫﺎي ﻣﻄﻠﻖ اﯾﻦ ﻣﺜﺎل ﺑﺎ اﺳﺘﻔﺎده از روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ در ﺷﮑﻞ 2 رﺳﻢ ﺷﺪه اﺳﺖ. ﺟﺪول.3 ﻧﺘﺎﯾﺞ ﻋﺪدي ﺑﺮاي ﻣﻌﺎدﻟﮥ ) (22 m 64 روش ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ m 32 روش ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ m 6 ﺟﻮاب دﻗﯿﻖ ﯾﺎﻓﺘﻪ / / / / / / / / / / /64559 /59537 /75576 /64442 /5949 / /634 /5674 /7599 /62674 /55977 /75768 /649 /59775 /73888 /654 /52833 /75783 /25 /25 /375 / / /46395 /83469 /46445 /83594 /45773 /82692 /45778 / /446 /8449 / /8382 /5 /625 /46325 / / / / / /46572 / / /44956 / / /45959 / / / / / /4625 / / /75 /875 / ﺟﺪول.4 ﻣﻘﺎﯾﺴﮥ ﻣﺎﮐﺴﯿﻤﻢ ﺧﻄﺎي ﻣﻄﻠﻖ ﺑﺮاي ﻣﺜﺎل 2 ﺑﺎ m= روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ روش ﻣﻮﺟﮏ ﻫﺎر روش ﺗﻮاﺑﻊ ﺑﻼك ﭘﺎﻟﺲ m 2/3e-4 3/5e-5 5/e-3 2/3e-4 /9e-3 5/4e-3 2/7e-2 /2e /3e-5 8/4e-5 /3e-4 2/8e-5 64 Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﺷﮑﻞ. ﺧﻄﺎي ﻣﻄﻠﻖ )در ﻣﻘﯿﺎس ﻟﮕﺎرﯾﺘﻤﯽ( ﺑﺮاي ﻣﺜﺎل ﺑﺎ m= و =/5 /5
12 76 ﺟﻠﺪ 2 ﺷﻤﺎره 2 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 395 )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ( ﻣﺜﺎل :3 ﻣﻌﺎدﻟﮥ رﯾﮑﺎﺗﯽ ﮐﺴﺮي ﻏﯿﺮﺧﻄﯽ زﯾﺮ را ﺑﺮرﺳﯽ ﻣﯽ ﮐﻨﯿﻢ ] :[36 D f ( ) b ( )f ( ) c ( )f ( ) e ( ), [,], 2 2, ) (23 f () k. 2 ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ رﯾﮑﺎﺗﯽ ﮐﺎرﺑﺮدﻫﺎي زﯾﺎدي در ﻣﻬﻨﺪﺳﯽ و ﻋﻠﻮم ﮐﺎرﺑﺮدي از ﺟﻤﻠﻪ ﻗﻮاﻧﯿﻦ ﻧﻮﺳﺎﻧﺎت رﺋﻮﻟﻮژي ﻓﺮاﯾﻨﺪﻫﺎي اﻧﺘﺸﺎر ﭘﺪﯾﺪه ﻫﺎي ﺧﻂ اﻧﺘﻘﺎل ﻣﺴﺎﺋﻞ ﻧﻈﺮﯾﻪ ﮐﻨﺘﺮل ﺑﻬﯿﻨﻪ و ﻏﯿﺮه ] [42]-[37 دارد. از اﻟﮕﻮرﯾﺘﻢ ارﺋﻪ داده ﺷﺪه در ﺑﺨﺶ 6 ﺑﺮاي ﺣﻞ اﯾﻦ ﻣﻌﺎدﻟﻪ ﺑﺎ e( ) c ( ) 2 b ( ) و k اﺳﺘﻔﺎده ﻣﯽﮐﻨﯿﻢ. ﺟﻮاب دﻗﯿﻖ ﺑﺮاي از اﯾﻦ راﺑﻄﻪ ﺑﻪدﺳﺖ ﻣﯽآﯾﺪ : 2 f ( ) 2 tanh 2 log, 2 2 و ﻣﯽﺗﻮان ﻣﺸﺎﻫﺪه ﮐﺮد ﮐﻪ ﻫﺮﮔﺎه آنﮔﺎه. f ( ) 2 = در ﺟﺪول 5 ﻣﻘﺎﯾﺴﮥ ﺟﻮاب دﻗﯿﻖ ﺑﺎ ﺟﻮاب ﺣﺎﺻﻞ از روش اراﺋﻪ ﺷﺪه و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ] [2 ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. ﺑﺮاي ﺟﺰﺋﯿﺎت ﺑﯿﺶﺗﺮ ﺧﻄﺎﻫﺎي ﻣﻄﻠﻖ اﯾﻦ ﻣﺜﺎل ﺑﺎ اﺳﺘﻔﺎده از روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ و روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ در ﺷﮑﻞ 3 رﺳﻢ ﺷﺪه اﺳﺖ. در ﺷﮑﻞ 4 رﻓﺘﺎر ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﺑﺮاي ﻣﻘﺎدﯾﺮ ﻣﺨﺘﻠﻒ (.5,.75, ) و m 64 ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. ﺟﺪول.5 ﻧﺘﺎﯾﺞ ﻋﺪدي ﺑﺮاي ﻣﻌﺎدﻟﮥ ) (23 m 64 ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ m 32 ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ m 6 ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺟﻮاب دﻗﯿﻖ / /29464 / /36967 / /29379 / /5953 / /29698 / /48698 / /2952 / / / / / / / / / / / / / / / / /2 /3 /56785 / / / / / / / / / / / / / / / / / / / / /4 /5 /6 / / / / / / / / / / /53727 / / / /7 /8 / / / / / / / / / / / / /52693 / /9 / Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﺷﮑﻞ.2 ﺧﻄﺎي ﻣﻄﻠﻖ )در ﻣﻘﯿﺎس ﻟﮕﺎرﯾﺘﻤﯽ( ﺑﺮاي ﻣﺜﺎل 2 ﺑﺎ m=
13 77 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﺷﮑﻞ.4 رﻓﺘﺎر ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﺑﺎ ﻣﻘﺎدﯾﺮ ﻣﺨﺘﻠﻒ و m=64 ﺑﺮاي ﻣﺜﺎل 3 ﻧﺘﯿﺠﻪ در اﯾﻦ ﻣﻘﺎﻟﻪ روﺷﯽ ﻣﺤﺎﺳﺒﺎﺗﯽ ﺑﺮ اﺳﺎس ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي ﻏﯿﺮﺧﻄﯽ اراﺋﻪ ﺷﺪه اﺳﺖ. ﻣﺎﺗﺮﯾﺲﻫﺎي ﻋﻤﻠﯿﺎﺗﯽ اﻧﺘﮕﺮال ﮐﺴﺮي ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ را ﺑﻪدﺳﺖ آورده و ﺑﺮاي ﺗﺒﺪﯾﻞ ﭼﻨﯿﻦ ﻣﺴﺎﺋﻠﯽ دردﺳﺘﮕﺎه ﻣﻌﺎدﻻت ﺟﺒﺮي ﻏﯿﺮﺧﻄﯽ اﺳﺘﻔﺎده ﮐﺮدﯾﻢ. ﻋﻼوه ﺑﺮ اﯾﻦ ﺗﺠﺰﯾﻪ و ﺗﺤﻠﯿﻞ ﺧﻄﺎ ﺑﺮاي روش ﮐﻼﻫﯽ ﺑﻬﺒﻮد ﯾﺎﻓﺘﻪ اراﺋﻪ ﺷﺪه اﺳﺖ. ﻣﻬﻢﺗﺮﯾﻦ ﻣﺰﯾﺖ اﯾﻦ روش ﮐﻢﺑﻮدن ﻫﺰﯾﻨﻪ راهاﻧﺪازي دﺳﺘﮕﺎه ﺑﺪون اﺳﺘﻔﺎده از روشﻫﺎي ﻃﺮحرﯾﺰي ﻣﺎﻧﻨﺪ روش ﮔﺎﻟﺮﮐﯿﻦ و ﯾﺎ اﻧﺘﮕﺮالﮔﯿﺮي اﺳﺖ. ﻫﻢﭼﻨﯿﻦ ﻣﺎﺗﺮﯾﺲ P ﯾﮏ ﻣﺎﺗﺮﯾﺲ ﺑﺎﻻ ﻫﺴﻨﺒﺮﮔﯽ اﺳﺖ ﮐﻪ ﻣﯽﺗﻮان آن را ﯾﮏ ﺑﺎر ﺑﺮاي ﻣﻘﺎدﯾﺮ ﻣﺨﺘﻠﻒ و m ﺑﻪدﺳﺖ آورد و ﺑﺮاي ﻣﺴﺎﺋﻞ ﻣﺨﺘﻠﻒ آن را اﺳﺘﻔﺎده ﮐﺮد. ﺑﻨﺎﺑﺮاﯾﻦ ﻫﺰﯾﻨﮥ ﻣﺤﺎﺳﺒﺎﺗﯽ روش ﮐﻢ اﺳﺖ و اﯾﻦ ﻣﺰاﯾﺎ اﯾﻦ روش ﻣﺤﺎﺳﺒﺎﺗﯽ را ﺑﺴﯿﺎر ﺟﺬاب ﺳﺎده و ﻣﻘﺮون ﺑﻪ ﺻﺮﻓﻪ ﻣﯽﮐﻨﺪ. دﻗﺖ و ﮐﺎراﯾﯽ روش ﺑﺎ ﭼﻨﺪ ﻣﺜﺎل ﻧﺸﺎن داده ﺷﺪه اﺳﺖ. ﻫﻢﭼﻨﯿﻦ ﻧﺘﺎﯾﺞ ﺑﻪدﺳﺖ آﻣﺪه ﺑﺎ ﺟﻮاب دﻗﯿﻖ و ﺟﻮابﻫﺎي ﻋﺪدي ﺑﻪدﺳﺖ آﻣﺪه ﺑﺎ ﺑﺮﺧﯽ Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﺷﮑﻞ.3 ﺧﻄﺎي ﻣﻄﻠﻖ )در ﻣﻘﯿﺎس ﻟﮕﺎرﯾﺘﻤﯽ( ﺑﺮاي ﻣﺜﺎل 3 ﺑﺎ m=
14 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 2 ﺷﻤﺎره 2 ﺟﻠﺪ ( )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ ﺗﻮﺟﻪ داﺷﺘﻪ ﺑﺎﺷﯿﺪ ﮐﻪ اﯾﻦ روش را در ﭘﺎﯾﺎن. از روشﻫﺎي ﻋﺪدي دﯾﮕﺮ از ﺟﻤﻠﻪ روش ﺗﻮاﺑﻊ ﮐﻼﻫﯽ ﻣﻘﺎﯾﺴﻪ ﺷﺪه اﺳﺖ. ﻣﯽﺗﻮان ﺑﻪراﺣﺘﯽ ﺑﺮاي ﺣﻞ ﻣﻌﺎدﻻت اﻧﺘﮕﺮال ﭼﻨﺪ ﺑﻌﺪي ﺗﻌﻤﯿﻢ داد و ﺑﻪﮐﺎر ﺑﺮد ﺗﻘﺪﯾﺮ و ﺗﺸﮑﺮ Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 ﮐﻤﺎل ﺗﺸﮑﺮ و از داوران ﻣﺤﺘﺮم ﺑﻪ ﺧﺎﻃﺮ ﭘﯿﺸﻨﻬﺎدﻫﺎي ﺳﺎزﻧﺪهﺷﺎن ﮐﻪ ﻣﻮﺟﺐ ﺑﻬﺒﻮد ﮐﯿﻔﯿﺖ و ﭘﺮﺑﺎرﺗﺮ ﺷﺪن ﻣﻘﺎﻟﻪ ﺷﺪ ﻣﻨﺎﺑﻊ. ﻗﺪرداﻧﯽ را دارﯾﻢ. Diethelm K., Ford N.J., "Analysis of fractional differential equations", J. Math. Anal. Appl., 265 (22) Diethelm K., Ford N.J., "Multi-order fractional differential equations and their numerical solution", Appl. Math. Comput., 54 (24) Kiryakova V., "Generalized fractional calculus and applications", in: Pitman Res. Notes Math. Ser., 3, Longman-Wiley, New York (994). 4. Podlubny I., "Fractional differential equations", Academic Press, San Diego (999). 5. Samko S.G., Kilbas A.A., Marichev O.I., "Fractional integrals and derivatives", Theory and Applications, Gordon and Breach, Yverdon (993). 6. Podlubny I., "Fractional differential equations: an introduction to fractional derivatives", fractional differential equations, to methods of their solution and some of their applications. New York: Academic Press (999). 7. Gaul L., Klein P., Kemple S., "Damping description involving fractional operators", Mech. Syst. Signal. Pr., 5 (99) Suarez L., Shokooh A., "An eigenvector epansion method for the solution of motion containing fractional derivatives", J. Appl. Mech., 64 (997) Momani S., "An algorithm for solving the fractional convection-diffusion equation with nonlinear source term", Commun. Nonlinear Sci. Numer. Simul., 2 (7) (27) Jafari H., Seifi S., "Solving a system of nonlinear fractional partial differential equations using homotopy analysis method", Commun. Nonlinear Sci. Numer. Simul., 4 (5) (29) Sweilam N.H., Khader M.M., Al-Bar R.F., "Numerical studies for a multi-order fractional differential equation", Phys. Lett. A, 37(2) (27) Das S., "Analytical solution of a fractional diffusion equation by variational iteration method, Comput", Math. Appl., 57 (3) (29) Arikoglu A., Ozkol I., "Solution of fractional integro-differential equations by using fractional differential transform method", Chaos Solitons Fract., 4 (2) (29)
15 79 اﺳﺘﻔﺎده از ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺮاي ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﮐﺴﺮي 4. Erturk V.S., Momani S., Odibat Z., "Application of generalized differential transform method to multi-order fractional differential equations", Commun. Nonlinear Sci. Numer. Simul., 3 (8) (28) Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st Meerschaert M., Tadjeran C., "Finite difference approimations for two-sided spacefractional partial differential equations", Appl. Numer. Math., 56 () (26) Pandey R.K., Singh O.P., Baranwal V.K., "An analytic algorithm for the space-time fractional advection-dispersion equation", Comput. Phys. Commun., 82 (2) Saadatmandi A., Dehghan M., "A new operational matri for solving fractional order differential equations", Comput. Math. Appl., 59 (2) Odibat Z., Momani S., Erturk V.S., "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 97 (28) Baranwal V.K., Pandey R.K., Tripathi M.P., Singh O.P., "Analytic solution of fractionalorder heat and wave-like equations using generalized n-dimensional differential transform method", Z. Naturforsch, 66a (2) Cuesta E., Lubich Ch., Palencia C., "Convolution quadrature time discretization of fractional diffusion-wave equation", Math. Comput, 75 (26) Tripathi M.P., Baranwa V.K., Pandey R.K., Singh O.P., "A new numerical algorithm to solve fractional differential equations based on operational matri of generalized hat functions, Commun", Nonlinear Sci. Numer. Simulat, 8 (23) Odibat Z., Shawagfeh N., "Generalized Taylors formula", Appl. Math. Comput., 86 () (27) Li Y., "Solving a nonlinear fractional differential equation using Chebyshev wavelets", Commun. Nonlinear Sci. Numer, Simulat, 5 (2) Arikoglu A., Ozkol I., "Solution of fractional differential equations by using differential transform method", Chao. Solitons Fract., 34 (27) Rehman M.u., Ali Khan R., "The Legendre wavelet method for solving fractional differential equations", Commun, Nonlinear Sci. Numer, Simulat, 6 (2) Li Y., Zhao W., "Haar wavelet operational matri of fractional order integration and its applications in solving the fractional order differential equations", Appl. Math. Comput., 26 (2) Li Y., Sun N., "Numerical solution of fractional differential equations using the generalized block pulse operational matri", Comput. Math. Appl., 62 (2)
16 395 ﭘﺎﯾﯿﺰ و زﻣﺴﺘﺎن 2 ﺷﻤﺎره 2 ﺟﻠﺪ ( )ﻧﺸﺮﯾﻪ ﻋﻠﻮم داﻧﺸﮕﺎه ﺧﻮارزﻣﯽ Wu J.L., "A wavelet operational method for solving fractional partial differential equations numerically", Appl. Math. Comput., 24 () (9) Lepik Ü., "Solving fractional integral equations by the Haar wavelet method", Appl. Math. Downloaded from mmr.khu.ac.ir at 5:28 IRDT on Tuesday May st 28 Comput., 24 (2) (29) Delbosco D., Rodino L., "Eistence and Uniqueness for a Nonlinear Fractional Differential Equation", J. Math. Anal. Appl., 24 (996) Tenreiro Machado J.A., "Fractional derivatives: probability interpretation and frequency response of rational approimations", Commun. Nonlinear Sci. Numer. Simul., 4 (9-) (29) Atkinson K.E., "The numerical solution of integral equations of the second kind", Cambridge University Press, Cambridge (997). 33. Mirzaee F., Hadadiyan E., "Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions", Appl. Math. Comput., 25 (25) Mirzaee F., Hadadiyan E., "Approimation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions", J. Comput. Appl. Math., 32 (26) Momani S., Odibat Z., "Numerical comparison of methods for solving linear differential equations of fractional order", Chao. Solitons Fract., 3 (27) Odibat Z., Momani Sh., "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order", Chaos Solitons Fractals, 36 (28) Khan N.A., Jamil M., Ara A., Das S., "Eplicit solution of time-fractional batch reactor system", Int. J. Chem. React. Eng., 9 (2) Article ID A F-Batlle V., Perez R., Rodriguez L., "Fractional robust control of main irrigation canals with variable dynamic parameters", Control Eng. Pract., 5 (27) Podlubny I., "Fractional-Order Systems and Controllers", IEEE Trans. Auto. Control, 44 (999) Garrappa R., "On some eplicit Adams multistep methods for fractional differential equations", J. Comput. Appl. Math., 229 (29) Jamil M., Khan N.A., "Slip effects on fractional viscoelastic fluids", Int. J. Differ. Equ., 2 (2) Article ID Mohammadi F., Hosseini M.M., "A comparative study of numerical methods for solving quadratic Riccati differential equation", J. Frank. Inst., 348 (2) (2)
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