ﻣﻌﺮﻓﯽ ﻣﺸﺘﻖ ﮐﻮاﻧﺘﻮﻣﯽ ﮐﺴﺮي ﻓﺎزي و ﺧﻮاص آن

Introduction of fuzzy q-fractional derivative and its properties

ﻣﻄﺎﻟﻌﻪي ﺣﺴﺎب ﮐﻮاﻧﺘﻮﻣﯽ ﯾﺎ ﮐﯿﻮ ﺣﺴﺎب از اواﯾﻞ ﻗﺮن ﺑﯿﺴﺘﻢ ﺗﻮﺳﻂ ﺟﮑﺴﻮن آﻏﺎز ﺷﺪ؛ اﻣﺎاﻣﺮوزه ﻣﻄﺎﻟﻌﺎﺗﯽ زﯾﺎدي در ﻣﻮرد ﻣﺤﺎﺳﺒﺎت ﮐﻮاﻧﺘﻮﻣﯽ ﺻﻮرت ﮔﺮﻓﺘﻪ؛ ﮐﻪ ﺑﺎﻋﺚ ﺗﻮﺟﻪ ﺑﯿﺸﺘﺮ ﻋﻼﻗﻤﻨﺪان در اﯾﻦ زﻣﯿﻨﻪ ﺷﺪه اﺳﺖ. ﺣﺴﺎب ﮐﻮاﻧﺘﻮﻣﯽ ﯾﮑﯽ از ﻋﻠﻮمﻫﺎي ﮐﺎرﺑﺮدي و ﺑﯿﻦ رﺷﺘﻪ اي اﺳﺖ ﮐﻪ ﺑﻪ دﻟﯿﻞ داﺷﺘﻦ وﯾﮋﮔﯽﻫﺎي ﺧﺎص از ﺟﻤﻠﻪ، ﺗﻌﺮﯾﻒ ﻣﺸﺘﻖ ﺑﺪون وﺟﻮد ﺣﺪ، ﺑﺎﻋﺚ ﻣﺰﯾﺘﺶ ﻧﺴﺒﺖ ﺑﻪ ﺣﺴﺎب ﻣﻌﻤﻮﻟﯽ ﺷﺪه اﺳﺖ و ﮐﺎر ﺑﺎ ﺣﺴﺎب ﮐﻮاﻧﺘﻮﻣﯽ از ﻧﻈﺮ ﻋﺪدي ﺳﺮﯾﻌﺘﺮ و راﺣﺖﺗﺮ از ﺣﺴﺎب اﺳﺘﺎﻧﺪارد اﺳﺖ . از آﻧﺠﺎ ﮐﻪ اﮐﺜﺮ ﻣﺴﺎﺋﻞ ﻣﻮﺟﻮد در ﻃﺒﯿﻌﺖ ﻣﻨﺠﺮ ﺑﻪ ﻣﻮاﺟﻬﻪ ﺑﺎ ﻣﻌﺎدﻻت ﻓﺎزي ﺷﺎﻣﻞ ﻣﺸﺘﻘﺎﺗﯽ از ﻣﺮﺗﺒﻪ ﮐﺴﺮي ﻣﯽﺷﻮﻧﺪ؛ در اﯾﻦ ﭘﮋوﻫﺶ ﺑﻌﺪ از ﻣﻌﺮﻓﯽ ﻣﺸﺘﻖ ﮐﻮاﻧﺘﻮﻣﯽ )ﺑﻪ اﺧﺘﺼﺎرq -ﻣﺸﺘﻖ( ﻓﺎزي ﺑﺮ - q ﻣﺒﻨﺎي ﺗﻔﺎﺿﻞ ﻫﺎﮐﻮﻫﺎراي ﺗﻌﻤﯿﻢﯾﺎﻓﺘﻪ، ﻣﺸﺘﻖ ﮐﻮاﻧﺘﻮﻣﯽ ﮐﺎﭘﻮﺗﻮي ﮐﺴﺮي ﻓﺎزي و اﻧﺘﮕﺮال ﮐﻮاﻧﺘﻮﻣﯽ )ﺑﻪ اﺧﺘﺼﺎر اﻧﺘﮕﺮال( رﯾﻤﻦ-ﻟﯿﻮول ﮐﺴﺮي ﻓﺎزي ﻣﻌﺮﻓﯽ ﻣﯽ ﮐﻨﯿﻢ. ﺳﭙﺲ ﺑﻪ ﺑﺮرﺳﯽ ﻗﻀﺎﯾﺎي اﺳﺎﺳﯽ و ﺑﯿﺎن ﺗﻌﺎرﯾﻒ ﻣﻬﻢ در راﺑﻄﻪ ﺑﺎ q -ﻣﺸﺘﻖ ﮐﺎﭘﻮﺗﻮي ﮐﺴﺮي ﻓﺎزي و q -اﻧﺘﮕﺮال رﯾﻤﻦ-ﻟﯿﻮول ﮐﺴﺮي ﻓﺎزي ﻣﯽﭘﺮدازﯾﻢ. ﮐﻪ اﯾﻦ ﻧﺘﺎﯾﺞ در ﺑﺴﯿﺎري از برنامه هاي كاربردي مانند فيزيك، نظريه كوانتومي، نظريه اعداد، مكانيك آماري و غيره رخ مي دهد

The quantum calculus or q-calculus begins with F. H. Jackson in the early twentieth century, but only recently it has aroused interest, due to high demand of mathematics that is modeling quantum computing and it has been an important subject for applied sciences . The quantum calculus is one of the applied and inter disciplinary sciences, which is more important than the classical calculus because in the standard calculus the definition of the derivative depends on the existence of limit but the quantum derivative in quantum calculus works without the definition of limit and for this reason the work with a quantum calculus is numerically faster and easier than the standard calculus. In this paper, fuzzy quantum derivative, fuzzy quantum fractional derivative in Caputo sense by using generalized Hukuhara difference and fuzzy quantum fractional integral of the Riemann-Liouville type are introduced, then the related theorems and properties are provided in details.These results occur in many applications as physics, quantum theory, number theory, statistical mechanics, etc.