Ordinary Differential Equations

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1 Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai.

2 In naure, mos of he phenomena ha can be mahemaically described are in he form of differenial equaions. For example, in hydrology, you may have sudied he classic conservaion equaion ds I( ) Q( ) where S is he waer in he reservoir or waer resource sysem I () is he inflow o he sysem Q( ) is he ouflow from he sysem

3 Similarly, here are many phenomena ha can be represened in differenial equaion form. Definiion of differenial equaion: I is an equaion consising of derivaives of a dependen variable wih respec o independen variable(s).

4 If a dependen variable is varying wih respec o more han one independen variable, hen he governing equaions formed are usually parial differenial equaions. If in cerain siuaions, we can approximae he dependen variable o be varying wih respec o only one independen variable, hen, he equaion is an ordinary differenial equaion.

5 f (, y) Wha is he order of he above ODE? Order of an ODE is he highes order derivaive presen in ha ODE. A general nh order ODE can be represened as n n 1 n 2 2 d y d y d y d y n n 1 1 n () n n n a a a a a a y F

6 Wha is a linear ODE? An ODE in which all he derivaives appear in linear form and he coefficiens do no depend on he dependen variable is known as a linear ODE. y f ( y) where is eiher a consan coefficien or a funcion of. However, if (, y), hen he above equaion is a non linear ODE.

7 Homogeneous ODE y 0 Non-Homogeneous ODE y f () Sysem of ODE dx f ( x, y, z, ) g( x, y, z, ) dz h( x, y, z, )

8 Discussion in his lecure shall be peraining o General non-linear firs order ODE General non-linear second order ODE 2 f (, y) d y P( y, ) Q( y, ) y f ( ) 2

9 Classificaion of ODEs: Based on he condiions given o he applicaion of an ODE, hey can be classified as Iniial value ODE Boundary value ODE

10 Iniial value ODE f (, y); y( ) y 0 0 Boundary Value ODE

11 Acual classificaion of ODE problems: Propagaion problems Equilibrium problems Eigen problems

12 One Dimensional Iniial value ODE: As has been described earlier, mos of he governing equaions are differenial equaions (ODE and PDE). Some of he simplified cases can be described using ODEs. For example, he hea ransfer o he surrounding is described using he firs order ODE: dt T T, T (0) T This is an iniial value ODE for which iniial condiion has been menioned.

13 Finie difference mehod: Objecive of he finie difference mehod (FDM) is o conver he ODE ino algebraic form. The following seps are followed in FDM: Discreize he coninuous domain (spaial or emporal) o discree finie-difference grid. Approximae he derivaives in ODE by finie difference approximaions. Subsiue hese approximaions in ODEs a any insan or locaion. Obain algebraic equaions. Solve he resuling algebraic equaions.

14 Discreizaion of emporal domain: s f ( y, ) s s (s 1) s O( ) (Forward difference) s y y s ( s 1) O( ) (Backward difference) s y y

15 y( s 1) y( s 1) 2 (Cenral difference) Subsiuing he value of he derivaive according o he forward difference scheme in he differenial equaion, we have y or s ( s 1) y s 2 f ( y, ) y( s 1) ys f ( ys, s) s s O( ) This is a finie difference algebraic equaion.

16 Care should be aken ha he funcions involved in FDM soluions are coninuous and smooh. Else, i can give error or flucuaion. While using FDM, following errors can creep: Error in iniial daa Algebraic errors Truncaion errors Round off errors Inheried errors

17 Firs order approximaions for For f (, y); y( ) y 0 0 f (, y) n n n f (, y ) If we use forward difference formula, we will ( n 1) n have y y n

18 So, from he equaion, n n f f (, y ) n f n y ( n 1) y n where So, y y f ; O( ) ( n 1) n 2 n This is known as firs order explici Euler mehod. On repeiive applicaion of Euler explici mehod, he order of approximaion reduces o O( )

19 Implici Euler Mehod: ( n 1) n y y n 1 Since n 1 f n 1 n 1 (, y ) ( n 1) n n 1 n 1 2 y y f (, y ) ; O( ) This is he firs order implici Euler mehod.

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