294-301 FRACTION IN ORDER SIMPLE DIFFERENTIAL EQUATIONS. IN CAPUTO'S SENSE 294 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us AMERICAN Journal of Public Diplomacy and International Studies Volume 01, Issue 09, 2023 ISSN (E): 2993-2157 FRACTION IN ORDER SIMPLE DIFFERENTIAL EQUATIONS. IN CAPUTO'S SENSE FRACTION IN ORDER DIFFERENTIAL EQUATIONS FOR KOSHI ISSUE Latipova Shahnoza Salim daughter Asia International University General technician Department of Sciences teacher slatipova543@gmail.com Annotation: Today's in the day differential equations and mathematics and physics equations directions wide spreading from directions one this fraction in order derivative and fraction are integral equations of order . Our life during many fields basically physics , chemistry , biology and etc in the fields processes fraction in order equations with expressing them _ in learning to us fraction in order equations help will give . In life many in processes them manage important importance occupation is enough For example , heat spread in the process something from the border the heat management Keywords: Fraction in order derivative , Caputo , Cauchy problem , Volterra integral , Mittag-Leffler function , one sexual Fraction in order simple differential equations . In Caputo's sense fraction in order differential equations for Koshi issue This In the section we mean Caputo linear fraction in order differential of Eqs sure solutions let's make In this : if when in space defined Caputo fraction in order derivative . First we when initial conditions with given Koshi the issue seeing let's go : (2.1) (2.2) Hypothesis let's do let it be In that case in space (2.1), (2.2) Cauchy issue the following to the Volterra integral equation equivalent equation will be : (2.2.3) (y)( )с aD xa + 0a > , 1[ , ]( [ ] 1),nC a b n aa g - = + 0a > ( y)( ) ( ) ( ) ( ; 1 ; ; ),с aD x y x f x a x b n n na l a l+ - = £ £ - < < Î Î  ( ) ( ) , ( ; 0,..., 1).k k ky a b b k n= Î = - ( ) [ , ] (0 1, )f x C a bg g g aÎ £ < £ 1[ , ]nC a b- 1 1 1 0 ( ) 1 ( )( ) ( ) ! ( ) ( ) ( ) ( ) x xn j j j a a b y t f ty x x a dt dt j Г x t Г x ta a l a a - - - = = - + + - -å ò ò 295 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us We have this integral equation consecutively approach method through the solution we find To this method according to as follows sequence dry we get : (2.4) (2.5) These approximations (2.4) and the following (*) from equality using , in the form of an operator writing we get can : (2.6) By the formula above the counting we can To the above similar also for the following the formula writing we get : From this , This as follows concisely to write can : (2.7) The process continue bringing the following sequence harvest we do : (2.8) m in this sequence to infinity yearning to the limit if we pass (2.3) of the integral equation to the solution we will come (2.9) Now this the solution appearance condense for citation in chapter 1 passed (2.11) 1 0 0 ( ) ( ) ! n j j j b y x x a j - = = -å 1 0 1 1 ( ) 1 ( )( ) ( ) ( ) ( ) ( ) ( ) x x m m a a y t f ty x y x dt dt Г x t Г x ta a l a a - - -= + + - -ò ò ( )m NÎ ( )a+ 1 1 ( )I ( ) ( ; ( ) 0) ( ) ( ) x a f tf x dt x a a Г x t a aa -= > Â > -ò ( ) ( )0 1( ) ( ) ( ) ( )m my x y x I y x I f xa al -= + + 1( )y x ( ) ( )1 0 0( ) ( ) ( ) ( )y x y x I y x I f xa al= + + = 1 1 1 0 0 ( ) 1 ( ) ( ) ( 1) ( ) xk k jn j j k a x ab x t f t dt Г k j Г a al a a +- - = = - = + - + +å å ò 2 ( )y x ( ) ( )2 0 1( ) ( ) ( ) ( )y x y x I y x I f xa al= + + ( ) ( )2 0 1( ) ( ) ( ) ( )y x y x I y x I f xa al= + + 11 2 2 1 2 0 0 1 ( )( ) ( ) ( ) ( 1) ( ) xk k j kn k j j k ka x ay x b x t f t dt Г k j Г k a al l a a + -- - = = = é ù- = + -ê ú+ + ë û å å åò 11 1 0 0 1 ( )( ) ( ) ( ) ( 1) ( ) xk k j kn m m k m j j k ka x ay x b x t f t dt Г k j Г k a al l a a + -- - = = = é ù- = + -ê ú+ + ë û å å åò 11 1 0 0 1 ( )( ) ( ) ( ) ( 1) ( ) xk k j kn k j j k ka x ay x b x t f t dt Г k j Г k a al l a a + -- ¥ ¥ - = = = é ù- = + -ê ú+ + ë û å å åò , 0 ( ) ( ) k k zE z Г ka b a b ¥ = = +å 296 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us Mittag-Leffler from the function we use Necessary in places this function expression if we replace it , the following in appearance to the solution have we will be : (2.10) This function (2.3) of Volterra's integral equation the solution will be and therefore , (2.1), ( 2.2) Cauchy the solution of the problem represents _ Example 2.1. Caputo's meaning to us Koshi type problem is given be : (2.11) when find the solution . Integral equation solve for don't go go away approach method we use (2.12) By the formula above the counting we can and the first approach as from we use To the above similar also for the following the formula writing we get : 1 1 , 1 , 0 ( ) ( ) ( ) ( ) ( ) ( ) xn j j j j a y x b x a E x a x t E x t f t dta a a a a al l - - + = é ù é ù= - - + - -ë û ë ûå ò ( y)( ) ( ) ( ), ( 0) ( )сD x y x f x y b ba l- = + = Î 0 1 vaa l< < Î ( y)( ) ( ) ( )сI D x I y x I f xa a a al= + ( ) ( 0) ( ) ( ) ( ) ( 0) ( ) ( ) y x y I y x I f x y x y I y x I f x a a a a l l - + = + = + + + 1( ) ( ) ( )m my x b I y x I f xa al -= + + 1( )y x 0( )y x b= 1 1 0 0 0 ( ) ( ) b ( ) ( ) ( ) 1 (x t) ( ) ( ) ( ) ( ) ( ); ( 1) x x y b I y x I f x x t bdt I f x Г b bxb I f x b I f x Г Г bxb I f x Г a a a a a a a a a a ll a l l a a a a l a -= + + = + - + = = + - = + + = = + + + + ò 2 ( )y x 2 1 1 0 2 1 1 0 0 ( ) ( ) b ( ) (t) ( ) ( ) ( 1) ( ) ( ) ( ) ( 1) ( ) x x x y b I y x I f x btx t b I f dt I f x Г Г b bb x t dt x t t dt Г Г Г a a a a a a a a a l l l a a l l a a a - - - = + + = é ù = + - + + + =ê ú+ë û = + - + - + + ò ò ò 297 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us The process continue bringing the following sequence harvest we do : m in this sequence to infinity yearning to the limit if we pass (2.11) of the integral equation to the solution we will come (2.12) Now this is the solution of (2.12). appearance condense for citation in chapter 1 passed (2.11) Mittag-Leffler from the function we use Necessary in places this function expression if we replace it , the following in appearance to the solution have we will be : (2.13) Example 2. Caputo's meaning to us Koshi type problem is given be : (2.14) when find the solution . Equation (2.14). for those given writing we can , b=3 , =-2 Above proven (2.13) to the formula reached let's leave 1 0 2 2 2 2 22 2 0 0 ( ) (t) ( ) ( ) (x t) ( , 1) ( ) ( 1) ( ) ( ) ( ) ( 1) (2 1) ( ) ( ) ( ) ( ); ( 1) k x x k x t I f dt I f x Г b bxb B Г Г Г bx bxI I f x I f x b Г Г xI I f x I f x b I f x I f x Г k a a a a a a a a a a a a a a a a l a l l a a a a a a l ll a a ll l a - = + - + = = - - + + + é ù+ + = + + +ë û + + é ù+ + = × + +ë û + + ò å 1 0 1 1 1 0 10 ( )( ) ( ) ( 1) ( ) ( ) ( ) ; ( 1) ( ) km m k k m k k xk km m k k k xy x b I f x Г k xb x t f t dt Г k Г k a a a a l l a l l a a - = = - - = = = × + = + = × + - + å å å åò 1 0 00 ( ) ( ( ) )( ) ( ) ( ) . (2 1) ( ) ixk k k x x ty x b x t f t dt Г Г i a a al l a a a ¥ ¥ - = = - = × + - + +å åò , 0 ( ) ( ) k k zE z Г ka b a b ¥ = = +å 1 ,1 , 0 ( ) ( ) ( ) ( ) . x y x bE x x t E x t f t dta a a a a al l-é ù é ù= + - -ë û ë ûò 1 2( y)( ) 2 ( ) 10, ( 0)сD x y x y b b+ = + = Î 0 1 vaa l< < Î 1 , 2 a = ( ) 10f x = l 298 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us 3- example _ Caputo's meaning to us Koshi type problem is given be : (2.11) when find the solution . (2.12) solution in appearance will be Example 4 . Eq one sexual has been in case ( 2.13) solution as follows will be : (2.14) Example 5. this equation the solution the following in appearance will be One sexual has been case for looking after if we solution in appearance will be Example 6. This example order high has been case for seeing we go out (2.15) Koshi of the issue when the solution (2.16) with the formula is expressed . 1 ,1 , 0 1 1 1 2 2 2 1 1 1,02 2 2 1 1 1 2 2 2 1 1 3, 2 2 2 ( ) 3 ( 2 ) ( ) ( 2( ) ) ( ) 3 ( 2 ) 10 ( ) ( 2( ) ) 3 ( 2 ) 10 ( 2 ). x x y x E x x t E x t f t dt E x x t E x t dt E x x E x a a a a a a - - = - + - - - = = - + - - - = = - + - ò ò ( y)( ) ( ) ( ), ( ) ( )с aD x y x f x y a b ba l+ - = + = Î 0 1 vaa l< < Î 1 ,( ) ( ) ( ) ( ) ( ) . x a y x bE x a x t E x t f t dta a a a a al l-é ù é ù= - + - -ë û ë ûò ( y)( ) ( ) 0, ( ) ( )с aD x y x y a b ba l+ - = = Î ( ) ( )y x bE x a a a lé ù= -ë û 1 2( y)( ) ( ) ( ), ( ) ( )с aD x y x f x y a b bl+ - = + = Î 1 1 1 2 2 2 1 1 1, 2 2 2 ( ) ( ) ( ) ( ) ( ) . x a y x bE x a x t E x t f t dtl l -é ù é ù = - + - -ê ú ê ú ë û ë û ò 1 2( y)( ) ( ) 0, ( ) ( )с aD x y x y a b bl+ - = = Î 1 2 1 2 ( ) ( )y x bE x al é ù = -ê ú ë û ( y)( ) ( ) ( ), ( ) , '( )с aD x y x f x y a b y a da l+ - = = = 1 2 , ,va b da l< < Î ,2( ) ( ) ( ) ( )y x bE x a d x a E x aa a a al lé ù é ù= - + - - +ë û ë û 1 ,( ) ( ) ( ) x a x t E x t f t dta a a a l- é ù+ - -ë ûò 299 AMERICAN Journal of Public Diplomacy and International Studies www.grnjournal.us If Eq one sexual (2.17) Koshi of the issue when the solution (2.18) with the formula is expressed . 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