кафедра математической экономики, Бакинский государственный университет, г. Баку, Азербайджанская Республика

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1 An exsence heorem or uzzy prl derenl equon Eendyev H Rusmov L Repulc o Azerjn) Теорема о существовании нечеткого дифференциального уравнения в частных производных Эфендиева Х Д Рустамова Л А Азербайджанская Республика) Эфендиева Хeджер Джавид / Eendyev Hecer - кандидат физико-математических наук преподаватель; Рустамова Ламия Аладдин / Rusmov Lmy - кандидат физико-математических наук преподаватель кафедра математической экономики Бакинский государственный университет г Баку Азербайджанская Республика Asrc: n hs pper rs spce o uzzy numers s consruced nd sclr produc s nroduced he dervve o uzzy uncon n hs spce s dened Furher Posson s equon wh rs oundry condon or uzzy uncons s consdered I s shown h prolems d rgh hnd se nd oundry uncon) re uzzy hen soluon o hs prolem lso s uzzy uncon Аннотация: в данной статье вначале было построено пространство нечетких чисел и представлено скалярное произведение Была определена производная функции принадлежности нечеткого множества в данном пространстве Далее рассматривается уравнение Пуассона с граничными условиями первого рода для функций принадлежности нечеткого множества Показано что если данные задачи правая часть и граничная функция) содержит нечеткие числа то и решение этой задачи это функция принадлежности нечеткого множества Keywords: uzzy prl derenl equon uzzy uncon posson's equon Ключевые слова: нечеткие дифференциальные уравнения в частных производных функция принадлежности нечеткого множества уравнение Пуассона AMS Sujec Clsscon: 3A3 34K36 35R3 6E Inroducon he complexy o he world mkes he evens we ce uncern n urous orms Besdes rndomness uzzness nd s mporn uncerny whch plys n essenl role n he rel world Fuzzy se heory hs een developed very s snce ws nroduced y scens on cyernecs Zdeh [] n 965 A uzzy se chrcerzed wh s memershp uncon y Zdeh For he purpose o mesurng uzzy evens Zdeh [] presened he concep o possly mesure nd he erm o uzzy vrle n 978 o nvesge uzzy derenl equons rs one hs o nroduce he denon o he dervve o uzzy uncon hs denon mus llow one o nvesge ordnry nd prl derenl equon he concep o uzzy dervve ws rs nroduced y Chng nd Zdeh [3] nd ws ollowed up y Duos nd Prde [4] who used he exenson prncple n her pproch Oher mehods hve een dscussed y Pur nd Rlescu [5] A horough heorecl reserch o uzzy Cuchy prolems ws gven y Klev [6] Sekkl [7] Fuzzy prl derenl equons were ormuled y Buckley [8] nd Allhvrnloo [9] used numercl mehod o solve he FPDE) In he presen work nroducng he spce o uzzy numers he dervve o he uzzy uncon s deermned Usng hs pproch he mehod s proposed o nvesge uzzy prl derenl equons he spce o he prs o uzzy numers A uzzy se A s chrcerzed y generlzed chrcersc uncon A) memershp uncon dened on unverse X whch ssumes vlues n ] denoe y -cu clled [ For ny [] A x X : A x) he - cu o A Le A) s n upper semconnuous uncon nd sup p A) x X : A x) s ounded se o X A uzzy se s uzzy numer X R nd or ny [] A s convex nd he hegh o A h s sup x) xx A he hs o e equl o one hs uzzy

2 numer usully s clled convex norml uzzy numer F Le's dene y F he clss o convex norml uzzy numers hen or ny he se o [ ) s dened [7]) Le -cu o uzzy numer he nervl L R )] [] F F nd [ L ) R )] [ L ) R )] uzzy numer nd k hen -cu o k dene s [ L ) L ) R ) R )] nd k [ kl ) kr )] respecvely Noe h F s no lner spce he operon o surcon s no dened n F ) We consder he se o prs F F mulplcon nd equvlency s ) ) ) k k k k ) ) ) nd dene he operon o ddon ) As zero elemen o hs spce s ken he pr ) e he se o elemens F ) For ny ) From ls relon ) we ge ) x x ) I s cler h x x) ) ) he se o ll prs F F orms srucure o lner spce Le x ) F F y ) F F hen [ L ) R )] L ) R For ny x y F F dene he sclr produc s [ )] [] x y R [ L ) L ) R )) L )) R ) L ) R )) ))] d I my e shown h hs denon sses ll requremens o he sclr produc We denoe hs spce y LF Norm n hs spce s dened s x [ L ) L )) R ) R )) ] d 3) F nd F s We dene dsnce eween wo uzzy numers x y 4) where x ) y ) 3 Dervve o he uzzy uncon Now le s consder uzzy uncon he uncon ) For ny [] [ L ) R )] [] F or ech ) ) nd dene dervve o 5)

3 s clled -cu o he uncon ) F F h Δ ) ) lm ) ) Δ Δ ) ) F F s clled dervve o he uncon ) Denon Le here exss such hen he pr pon hs denon my e wren n he ollowng orm Δ ) ) lm ) Δ Δ re -cu or he uncons ) ) 6) 7) he where )) I s shown h L ) R ) s connuous derenle relvely hen ) s derenle Ech uncon ) my e consdered s n elemen ) F F hen ) )) ) ) Now le ) e pr o uzzy uncons e ) ) ) rom 8) From relon )) )) )) we see h he dervve o he uncon ) For ny ) u )) lso s pr rom F F F F whch F F dened y he ormul ) I cn e shown h consder he sclr produc 9) One my show h hs dervve sses he "necessry nurl" condons Exmple Le ) e uzzy uncon whose -cu s dened s ollows: ) ) d ) ) ) ) d ) [ ) ) ] ) where [ ) )] Exmple Le [ In hs cse ] [ ] y y y hen s no dcul o show h Anlogclly we cn dene prl derenl on n Exmple 3 Le F F e uzzy numer nd y y ) y y y R e uzzy uncon I s no dcul o show h or

4 y u y In overse y ) u y ) y Also s cler h u ) y 4 Fuzzy ellpc equon n Le D R e gven ounded domn wh smooh oundry S nd uzzy uncon y F F y y y y ) n e u u ) depends on he prmeer D u u y) y D We ll wre u CD) he uncon uy) connues on y n D Anlogclly we cn dene U C D) Consder he oundry prolem u y) y D ) u ) g ) S ) Le g ) g ) g )) F F D y) y) y)) F F y D In he derence o rdonl prolems here soluon o he prolem 6) 7) s uzzy uncon u u y) F or pr o he uzzy uncon u y) u y) u y)) F F For he o smplcy hs ype uncons we ll cll uzzy uncon Equon ) nd oundry condon ) we undersnd s equly pr o he domns heorem Le C D) C D) nd C S) hen here exss unequl soluon u y) u y) u y)) F F y D I s neresng o nvesge prolem ) ) when y) g ) y) F g ) F y D S o he prolem ) ) g re uzzy uncons e here s h n hs cse soluon o he prolem ) ) lso s uzzy uncon rom F heorem Le or ny y D nd S e uzzy uncon nd C D) C D) g C S) hen here exss unequl uzzy uncon u u y) F o he prolem ) ) Reerences soluon Zdeh L A Fuzzy ses Inormon nd conrol 965 v 8 p Zdeh L A Fuzzy ses s ss or heory o possly Fuzzy se nd sysems 978 v p Chng S I Zdeh L A On uzzy mppng nd conrol IEEE rns Sysems Mn Cyerne 97; : Duos D Prde H owrds uzzy derenl clculus III Derenon Fuzzy Ses nd Sysems 98; 8 3): Pur M L Rlescu D A Derenls or uzzy uncon J Mh Anl Appl 978; 64: Klev O Fuzzy derenl equons uzzy Ses nd Sysems 4 987) 3-37

5 7 Sekkl S On he uzzy nl vlue prolem Fuzzy Ses nd Sysems 987; 4 3): Buckley J J Feurng Fuzzy derenl equons Fuzzy Ses nd Sysems vol pp Allhvrnloo Derence mehods or uzzy prl derenl equons Compuonl mehods n ppled mhemcs ) No 3 PP ) 6-33