SE3: Numercal Metods Topc 8 Ordnar Dfferental Equatons ODEs Lecture 8-36 KFUPM Term Secton 4 Read 5.-5.4 6-7- C ISE3_Topc8L
Outlne of Topc 8 Lesson : Introducton to ODEs Lesson : Talor seres metods Lesson 3: Mdpont and Heun s metod Lessons 4-5: Runge-Kutta metods Lesson 6: Solvng sstems of ODEs Lesson 7: Multple step Metods Lesson 8-9: Boundar value Problems C ISE3_Topc8L
Lecture 9 Lesson : Talor Seres Metods C ISE3_Topc8L 3
Learnng Objectves of Lesson Derve Euler formula usng te Talor seres epanson. Solve te frst order ODEs usng Euler metod. Assess te error level wen usng Euler metod. Apprecate dfferent tpes of errors n te numercal soluton of ODEs. Improve Euler metod usng ger-order Talor Seres. C ISE3_Topc8L 4
Talor Seres Metod Te problem to be solved s a frst order ODE: d d f Estmates of te soluton at dfferent base ponts: 3... are computed usng te truncated Talor seres epansons. C ISE3_Topc8L 5
C ISE3_Topc8L 6 Talor Seres Epanson Talor Seres Epanson!...!! Truncated Talor Seres Epanson n n n n k k k k d d n d d d d d d k Te n t order Talor seres metod uses te n t order Truncated Talor seres epanson.
Euler Metod Frst order Talor seres metod s known as Euler Metod. Onl te constant term and lnear term are used n te Euler metod. Te error due to te use of te truncated Talor seres s of order O. C ISE3_Topc8L 7
C ISE3_Topc8L 8 Frst Order Talor Seres Metod Frst Order Talor Seres Metod Euler Metod Euler Metod : n n n f Metod Euler f d d n Notaton O d d
Euler Metod Problem : Gven te frst order ODE : wt te ntal condton : Determne: & f for... Euler Metod : f for... C ISE3_Topc8L 9
Interpretaton of Euler Metod C ISE3_Topc8L
Interpretaton of Euler Metod Slopef f f C ISE3_Topc8L
Interpretaton of Euler Metod Slopef Slopef f f f f C ISE3_Topc8L
Eample Use Euler metod to solve te ODE: d d 4 to determne.. and.3. C ISE3_Topc8L 3
C ISE3_Topc8L 4 Eample. 4 f 9394 3... 3.9598 3 : 3.9598.. 3.98 : 3.98. 4 : Metod Euler 3 f Step f Step f Step f
Eample f 4. Summar of te result:. -4.. -3.98. -3.9595 3.3-3.9394 C ISE3_Topc8L 5
Eample f 4. Comparson wt true value: True value of. -4. -4.. -3.98-3.9799. -3.9595-3.95959 3.3-3.9394-3.9399 C ISE3_Topc8L 6
Eample f 4. A grap of te soluton of te ODE for << C ISE3_Topc8L 7
Tpes of Errors Local truncaton error: Error due to te use of truncated Talor seres to compute t n one step. Global Truncaton error: Accumulated truncaton over man steps. Round off error: Error due to fnte number of bts used n representaton of numbers. Ts error could be accumulated and magnfed n succeedng steps. C ISE3_Topc8L 8
Second Order Talor Seres Metods Gven d Second order Talor Seres metod d d d! d d d d needs f O to be derved analtcall. 3 C ISE3_Topc8L 9
Trd Order Talor Seres Metods d Gven f d Trd order Talor Seres metod d d d d! d 3! d 3 d d and d 3 d 3 3 O need to be derved analtcall. 3 4 C ISE3_Topc8L
Hg Order Talor Seres Metods d Gven f d t n order Talor Seres metod n n d d d... n d! d n! d d d 3 d 3 d O n n d... need to be derved analtcall. n d C ISE3_Topc8L
Hger Order Talor Seres Metods Hg order Talor seres metods are more accurate tan Euler metod. But te nd 3 rd and ger order dervatves need to be derved analtcall wc ma not be eas. C ISE3_Topc8L
Eample Second order Talor Seres Metod Use Second order Talor Seres metod to solve : d dt t use. Wat s : d t dt? C ISE3_Topc8L 3
C ISE3_Topc8L 4 Eample 4 4 4. solve : to metod order Talor Seres Second Use t t t dt d dt t d t dt d use t dt d
C ISE3_Topc8L 5 Eample 4. t t t t t f.976 3:.987..99 4.99...99..99 :.99 4.. : 3 Step Step Step
Eample f t t t. Summar of te results: t...99..987 3.3.976 C ISE3_Topc8L 6
Programmng Euler Metod Wrte a MATLAB program to mplement Euler metod to solve: dv v t. v dt for t.... C ISE3_Topc8L 7
Programmng Euler Metod fnlne'-*v^-t''t''v'. t v Tt; Vv; for : vv*ftv end tt; Tt; Vv; C ISE3_Topc8L 8
Programmng Euler Metod fnlne'-*v^-t''t''v'. t v Tt; Vv; for : vv*ftv end tt; Tt; Vv; Defnton of te ODE Intal condton Man loop Euler metod Storng nformaton C ISE3_Topc8L 9
Programmng Euler Metod Plot of te soluton plottv C ISE3_Topc8L 3
More n Ts Topc Lesson 3: Mdpont and Heun s metod Provde te accurac of te second order Talor seres metod wtout te need to calculate second order dervatve. Lessons 4-5: Runge-Kutta metods Provde te accurac of g order Talor seres metod wtout te need to calculate g order dervatve. C ISE3_Topc8L 3