In this paper, we present properties and meaning of the new fundamental, called SEE change. Additionally, we use SEE change to tackle frameworks of standard differential condition

Kuffi, Emad A.

Mansour, Eman A.

Mehdi, Sadiq A.

English

Periodicals of Engineering and Natural Sciences (PEN - International University of Sarajevo)

ISSN 2303-4521 Periodicals of Engineering and Natural Sciences  Methodology/Method  Vol. 9, No. 3, July 2021, pp.277-281 © The Author 2021. This work is licensed under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/) that allows others to share and adapt the material for any purpose (even commercially), in any medium with an acknowledgement of the work's authorship and initial publication in this journal.   277  On the SEE transform and systems of ordinary differential equations  Eman A. Mansour1, Emad A. Kuffi2 Sadiq A. Mehdi3 1,3Department of Mathematics,Mustansiriyah University  2Collage of Engineering, Al-Qadisiyah University ABSTRACT   Integral transforms, have many applications in the various diciplines of natural science and engineering to solve the problems of springs, heat transfer, electronical and electrical networks, mixing problems, carbon dating problems, bending of beams, Newton’s second law of motion, signal processing, exponential growth and decay problems. Also, many phenomenon of real life (biology, neuclear physics, chemistry, and tele-communications …) can be expressed mathematically by linear or nonlinear systems and solved by using integral transform. In this paper, authers disscused the SEE (Sadiq-Emad-Eman) integral transform and systems of ordinary differential equations. SEE integral transform method is very important for the solution of the respose of a linear system governed by an ordinary differential equation in the initial conditions (data) and (or) to an external disturbance (or external input function). Also, we apply it to obtain exact solution of linear systems of ordinary differential equations. Keywords:  SEE basic change, arrangement of common differential conditions. Corresponding Author:  Eman A. Mansour Department of Mathematics, Mustansiriyah University, Baghdad, Iraq E-mail: iman.am_73@uomustansiriyah.edu.iq  1. Introduction In this writing there are various changes and widly utilized in physical science just as in designing. The fundamental change is a proficient technique to tackle the arrangement of straight standard differential conditions [2-5]. Sadiq A. Mehdi, Emad A. Kuffi and Eman A. Mansour presented another necessary change and named as SEE essential change which is characterized by, [1]:  𝑆[𝑓(𝑡)] =1𝑣𝑛∫ 𝑓(𝑡) 𝑒−𝑣𝑡𝑑𝑡∞𝑡=0 = 𝑇(𝑣)    ,    𝑣 ∈ (𝑙1, 𝑙2)     ,    𝑛 ∈ ℤ        … (1)   Furthermore, applied this fundamental change to the arrangement of arrangement of straight customary differential conditions.   SEE Integral Transform of Some Elementary Functions [1],  S.N. 𝑓(𝑡) 𝑆[𝑓(𝑡)] = 𝑇(𝑣) 1.  𝐾 ≡ 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐾𝑣𝑛+1 2.  𝑡𝑚 𝑚!𝑣𝑛+𝑚+1      ,   m is a positive integer number 3.  𝑒𝑎𝑡 1𝑣𝑛(𝑣 − 𝑎)     ,    a is a constant 4.  sin(𝑎𝑡) 𝑎𝑣𝑛(𝑣2 + 𝑎2) 5.  cos(𝑎𝑡) 𝑣𝑣𝑛(𝑣2 + 𝑎2) 6.  sinh(𝑎𝑡) 𝑎𝑣𝑛(𝑣2 − 𝑎2) 7.  cosh(𝑎𝑡) 𝑣𝑣𝑛(𝑣2 − 𝑎2)  PEN Vol. 9, No. 3, July 2021, pp.277- 281 278 In this work, our motivation is to show the materialness of this intriguing new indispensable change and its productivity in tackling the straight arrangement of conventional differential conditions.  Theorem [1]: Let 𝑇(𝑣) is SEE integral transform of [𝑆[𝑓(𝑡)] = 𝑇(𝑣)].  Then:  𝑆[𝑓(𝑚)(𝑡)] =−𝑓(𝑚−1)(0)𝑣𝑛−𝑓(𝑚−2)(0)𝑣𝑛−1− ⋯ −𝑓(0)𝑣𝑛−𝑚+1+ 𝑣𝑚 𝑇(𝑣) .  2. Material and methods 2.1  Arrangement of linear ordinary differential equations SEE essential change technique is extremely viable for the arrangement of the reaction of a straight framework administered by a conventional differential condition to the underlying conditions and additionally to an outer unsettling influence (or outside input work). We look for the arrangement of framework for its state at resulting time t > 0 due to the underlying information at t equivalent to nothing and additionally to the unsettling influence applied fort > 0.   3. Results Example (I) (System of first order ordinary differential equations)  Think about the framework:   With the underlying information x_1 (0)=x_10 and x_2 (0)=x_20 where a_11 ,a_12 ,a_21 and a_22 are constants. Presenting the grids 𝑥 = (𝑥1𝑥2)  ,𝑑𝑥𝑑𝑡= (𝑑𝑥1𝑑𝑡𝑑𝑥2𝑑𝑡)  , 𝐴 = (𝑎11 𝑎12𝑎21 𝑎22)     𝑏(𝑡) = (𝑏1(𝑡)𝑏2(𝑡))   𝑎𝑛𝑑 𝑥0 = (𝑥10𝑥20) ,   We can compose the above direct framework in grid differential framework as   We take SEE change of the direct framework with the underlying conditions, we get:  {(𝑣 − 𝑎11)𝑥1̅̅ ̅ − 𝑎12𝑥2̅̅ ̅ = 𝑏1(𝑣)̅̅ ̅̅ ̅̅ ̅ +1𝑣𝑛𝑥10  ,−𝑎21𝑥1̅̅ ̅ + (𝑣 − 𝑎22)𝑥2̅̅ ̅ = 𝑏2(𝑣)̅̅ ̅̅ ̅̅ ̅ +1𝑣𝑛𝑥20  .     Where 𝑥1̅̅ ̅, 𝑥2̅̅ ̅, 𝑏1̅̅̅, 𝑏2̅̅ ̅ are SEE integral change of x_1, x_2, 𝑏1, 𝑏2 respectively.  The arrangement of these logarithmic system is:  𝑥1̅̅ ̅(𝑣) =|𝑏1̅̅ ̅(𝑣)+𝑥10𝑣𝑛−𝑎12𝑏2̅̅ ̅(𝑣)+𝑥20𝑣𝑛𝑣−𝑎22||𝑣−𝑎11 −𝑎12−𝑎21 𝑣−𝑎22|  ,   𝑥2̅̅ ̅(𝑣) =|𝑣−𝑎11 𝑏1̅̅ ̅(𝑣)+𝑥10𝑣𝑛−𝑎21 𝑏2̅̅ ̅(𝑣)+𝑥20𝑣𝑛||𝑣−𝑎11 −𝑎12−𝑎21 𝑣−𝑎22|  .   Expanding these determinates, results for 𝑥1̅̅ ̅(𝑣) and 𝑥2̅̅ ̅(𝑣) can be rearranged and the answers for x_1 (t) and x_2 (t) can be found in shut structures.   Example (II) Settle the straight arrangement of differential conditions:   {𝑑𝑥𝑑𝑡= 2𝑥 − 3𝑦𝑑𝑦𝑑𝑡= 𝑦 − 2𝑥    ,   𝑡 > 0        PEN Vol. 9, No. 3, July 2021, pp.277- 281 279 With the underlying information: x(0)=8 , y(0)=3.  Take SEE change of the framework with the underlying conditions, to get:  𝑥1̅̅ ̅(𝑣) =|8𝑣𝑛33𝑣𝑛𝑣−1||𝑣−2 32 𝑣−1|=1𝑣𝑛[8𝑣−17(𝑣−4)(𝑣+1) ] .  After simple computations, we get    𝑥1̅̅ ̅ =1𝑣𝑛.3(𝑣−4)+1𝑣𝑛.5(𝑣+1)    By taking inverse SEE transform, we have  𝑥1(𝑡) = 5𝑒−𝑡 + 3𝑒4𝑡 .   Also  𝑥2̅̅ ̅(𝑣) =|𝑣−28𝑣𝑛23𝑣𝑛||𝑣−2 32 𝑣−1|=1𝑣𝑛[3𝑣−22𝑣2−3𝑣−4 ] .   After simple computations, we have:     𝑥2̅̅ ̅(𝑣) =1𝑣𝑛[−2𝑣−4] +1𝑣𝑛[5(𝑣+1)]    By taking inverse SEE integral transform, we have:  𝑥2(𝑡) = −2𝑒4𝑡 + 5𝑒−𝑡 .    Example (III) think about a direct arrangement of second request conventional differential conditions:   {𝑑2𝑥𝑑𝑡2+ 3𝑥 − 2𝑦 = 0  ,𝑑2𝑥𝑑𝑡2+𝑑2𝑦𝑑𝑡2− 3𝑥 + 5𝑦 = 0 .      … (4)     taking SEE transform of equation (4), we have  {𝑆 [𝑑2𝑥𝑑𝑡2] + 3𝑆[𝑥] − 2𝑆[𝑦] = 0                  𝑆 [𝑑2𝑥𝑑𝑡2] + 𝑆 [𝑑2𝑦𝑑𝑡2] − 3𝑆[𝑥] + 5𝑆[𝑦] = 0       … (6)    Presently, utilizing the property, SEE change of the subsidiary of the capacity in condition (6), we have   {−𝑥′(0)𝑣𝑛−𝑥(0)𝑣𝑛−1+ 𝑣2?̅? + 3?̅? − 2?̅? = 0                                       −𝑥′(0)𝑣𝑛−𝑥(0)𝑣𝑛−1+ 𝑣2?̅? −𝑦′(0)𝑣𝑛−𝑦(0)𝑣𝑛−1+ 𝑣2?̅? − 3?̅? + 5?̅? = 0        , … (7)         Using equation (5) in equation (7), we have:  {−3𝑣𝑛+ 𝑣2?̅? + 3?̅? − 2?̅? = 0−3𝑣𝑛+ 𝑣2?̅? −2𝑣𝑛+ 𝑣2?̅? − 3?̅? + 5?̅? = 0      , … (8)    Equation (8) can be written as:  {(𝑣2 + 3)?̅? − 2?̅? =3𝑣𝑛(𝑣2 − 3)?̅? + (𝑣2 + 5)?̅? =5𝑣𝑛      , … (9)    Solving equation (9) for ?̅?(𝑣) and ?̅?(𝑣), we have:  ?̅?(𝑣) =1𝑣𝑛[14𝑣2+9 +114𝑣2+1 ]    The solution is:  𝑥(𝑡) =112𝑆−1 [1𝑣𝑛.3(𝑣2+9) ] +114𝑆−1 [1𝑣𝑛(𝑣2+1)] = sin(𝑡) .     Then:   𝑥(𝑡) =112sin(3𝑡) +114sin(𝑡) .    And   PEN Vol. 9, No. 3, July 2021, pp.277- 281 280 ?̅?(𝑣) =|𝑣2+33𝑣𝑛𝑣2−35𝑣𝑛|(𝑣2+3)(𝑣2+5)+2(𝑣2−3)        So, after simple computations, the solution is  𝑦(𝑡) =114sin(𝑡) −14sin(3𝑡) .     Example (IV) think about an arrangement of direct customary differential conditions:   {𝑑𝑥𝑑𝑡+ 𝑦 = 2 cos 𝑡  𝑥 +𝑑𝑦𝑑𝑡= 0       … (10)    with 𝑥(0) = 0     𝑎𝑛𝑑   𝑦(0) = 1       … (11)   taking SEE transform of equation (10), we have  {𝑆 [𝑑𝑥𝑑𝑡] + 𝑆[𝑦] = 2𝑆[cos 𝑡]  𝑆[𝑥] + 𝑆 [𝑑𝑦𝑑𝑡] = 0       … (12)    Presently, utilizing the property, SEE necessary change of the subordinate of the capacity in condition (12), we have:  −1𝑣𝑛𝑥(0) + 𝑣?̅? + ?̅? =2𝑣𝑣𝑛(𝑣2+1)   ?̅? −1𝑣𝑛𝑦(0) + 𝑣?̅? = 0   Now, using equation (11), we have:  𝑣?̅? + ?̅? =2𝑣𝑣𝑛(𝑣2+1)?̅? + 𝑣?̅? =1𝑣𝑛 }      … (13)    Solving the equation (13) for ?̅? and ?̅?, we get:  ?̅?(𝑣) =1𝑣𝑛[1𝑣2+1 ] .    Then, the solution is: 𝑥(𝑡) = sin(𝑡) .     And  ?̅?(𝑣)  =1𝑣𝑛[𝑣𝑣2+1]    Then, the solution is: 𝑦(𝑡) = cos(𝑡) .     4. Discussion  SEE integral transform method is very efficient to solve a linear system governed by an ordinary differential equation to the initial condition (data) and|or to an external disturbance (or external input function). Also, we seek the solution of a linear differential system for its state at sub sequence time 𝑡 > 0 due to the initial point at 𝑡 = 0 and|or to the disturbance applied for 𝑡 > 0.  5. Conclusions In the present paper, the SEE (Sadiq- Emad- Eman) integral transform and linear systems of ordinary differential equations are established successfully. A new integral transform “SEE integral transform” is a convenant tool for solving linear system of ordinary differential equations in the time domain. In future, using this integral transform, we can easly solved many advanced systems (linear or nonlinear) of modern era such as system of drug distribution in the body, arms, race systems or (models), health problems such as detection of diabetes). Also, in this paper we introduce the solution of general linear systems of first order ordinary differential equations, as well as, solved examples of linear systems of first and second order ordinary differential equations.  References  [1] S. Aggarwal, N. Sharma, R. Chaudhary, and A. R. Gupta, “A comparative study of Mohand and Kamal transforms,” Glob. J. Eng. Sci. Res., vol. 6, no. 2, pp. 113–123, 2019. [2] S. Aggarwal and R. Chauhan, “A comparative study of Mohand and Aboodh transforms,” Int. J. Res.  PEN Vol. 9, No. 3, July 2021, pp.277- 281 281 Advent Technol., vol. 7, no. 1, pp. 520–529, 2019. [3] T. M. Elzaki, “On the ELzaki Transform and Systems of Ordinary Differential Equations,” Glob. J. Pure Appl. Math., vol. 7, no. 1, pp. 113–119, 2011. [4] E. A. Mansour, E. A. Kuffi, and S. A. Mehdi, “The new integral transform ‘SEE transform’ and its applications,” Period. Eng. Nat. Sci., vol. 9, no. 2, pp. 1016–1029, 2021. [5] J. Zhang, “A Sumudu based algorithm for solving differential equations,” Comput. Sci. J. Mold., vol. 45, no. 3, pp. 303–313, 2007. [6]       Sudhanshu Aggarwal, Nidhi Sharma, Raman Chauhan, Duality relations of Kamal Transform with Laplace, Laplace-Carson, Aboodh, Sumudu, Elzaki, Mohanad and Sawi transforms. SN Applied Science, (2020).      

On the SEE transform and systems of ordinary differential equations

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On the SEE transform and systems of ordinary differential equations

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