Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai.
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
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).
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.
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 2 2 2 2 1 0 () n n n a a a a a a y F
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.
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, )
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
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
Iniial value ODE f (, y); y( ) y 0 0 Boundary Value ODE
Acual classificaion of ODE problems: Propagaion problems Equilibrium problems Eigen problems
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 4 4 0 0 This is an iniial value ODE for which iniial condiion has been menioned.
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.
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
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.
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
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
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( )
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.