Numerical Solution for Initial Value Problem

Numerical Analysis

Introduction ป�ญหาค�าเร�มต�น ค�อ ป�ญหาว�าด�วยการแก�สมการเชง

อน�พ�นธ�ท �สอดคล�องก�บเง��อนไขค�าเร�มต�นเง��อนไข วธ เชง ต�วเลขท �ไม�ได�ให�ค�าประมาณท �ต�อเน��อง แต�ให�ค�าประมาณ

ณ จ�ดท �ก)าหนดค�าประมาณอ��นๆระหว�างจ�ดท �ก)าหนดสามารถใช�การประมาณค�าในช�วง

x0 x1 x2 x3 xn

𝒅𝒚𝒅𝒙= 𝒇ሺ𝒙,𝒚ሻ, 𝒙𝟎 ≤ 𝒙≤ 𝒙𝒏, 𝒚ሺ𝒙𝟎ሻ= 𝜶

Well-posed ProblemÁºÉ° Å ªµ¤Ân¤ ´� � � � � � � 宦´ {®µnµÁ¦·É¤ o� � � � � � 𝑦′ = 𝑓ሺ𝑥,𝑦ሻ, 𝑎 ≤ 𝑥≤ 𝑏, 𝑦ሺ𝑎ሻ= 𝛼 ¤¤»·� 𝑓³ 𝑓𝑦 n°ÁºÉ° 宦´ »� � � � � � 𝑥Ä� ሾ𝑎,𝑏ሿ ´ Ê {®µnµÁ¦·É¤ o¤ Á ¥®¹ÉÁ¥ª� � � � � � � � � � � � � �𝑦ሺ𝑥ሻ 宦 � 𝑥∈ሾ𝑎,𝑏ሿ³ {®µÊÁ¦¥ªnµ {®µÂn¤ ´� � � � � � � � � (well-posed problem)

Well-posed Formula: Example¡·µ¦ µ´®µnµÁ¦·É¤ o� � � � � � � xyxy sin1 , 20 x , 00 y

Ä ÉÉ� � � xyxyxf sin1,

xyxyxf y cos, 2

� Ê� � ¼n� n°Á� ºÉ°� 宦� 2,0x ´ Ê Á� ¥�³¤Á¡¥�� Á� ¥Á�¥ªÂ³Á�È���®µ�nµÁ¦·É¤�o�Â�n¤��� � � � � � � � � � � � � � � � � � � � �

Classes of Methods¦³¤µ� � nµ� xy Ã¥ ε®� � � � x � É� »� � nµ� ÇÁ� È� ,,, 210 xxx ³Åo� � iy ÁÈ nµ¦³¤µ °� � � � � � � ixy

1. ¦³Á¥ª·¸ÊÁ¥ª� � � � � � (one step method)

¼� ¦� °� � µ¦� ¦³¤µ� � nµ� ³°¥¼nÄ� ¦¼� hyxhyy iiii ,,1 ,2,1,0i

Ã¥É� � hyx ii ,, ÁÈ¢ r nª ÉÁ ɥ¨ Á¤ºÉ°� � � � � � � � � � � � xÁ¨É¥ nµµ� � � � � ix ÁÈ� � hxi

2. ¦³Á¥ª·®µ¥Ê� � � � � (multi-step method)

ÁÈ µ¦®µ� � � 1iy Ã¥� Ä� o� ¨Á� ¥� ɦ¼o� nµÂoª¤µ� � ªnµ1 � »� ¦³Á� ¥� ª·� � ʤ¦¼� Â� � � °� ¼� ¦Á� È�

m

kkikik

m

kkiki hyxbhyay

101 ,, ,2,1,0i

Ã¥É� � maaa ,,, 10 , mbbb ,,, 10 ÁÈ nµ ª� � � � � �

Classes of Methods 宦� ¦³Á� ¥� ª·� ®µ¥� Ê� � ¹É� ¤¦¼� ¼� ¦Á� È�

m

kkikik

m

kkiki hyxbhyay

101 ,, ,2,1,0i

oµ� 0ma ®¦º° 0mb ¦³Á¥ª·¸³¤� � � � 1m � Ê� Á¡¦µ³� o°� Ä� o� o°¤¼ 1m � »� oµ� 𝒃−𝟏 = 𝟎 ³ µ¤µ¦ ®µ� � 𝒚𝒊+𝟏 � µ� � nµ� µ� � oµ� � ªµ� °� ¼� ¦Å� o� ´� � ¦³Á� ¥� ª·� � Ê� ³Á¦¥� ªnµ ¦³Á¥� �

ª·Ã¥´Âo� � � � � � (explicit) oµ� 𝒃−𝟏 ≠ 𝟎 ³¤ ªÅ¤n¦µ nµ� � � � � 𝒚𝒊+𝟏 � ¦µ� � � Ê� °� � oµ� � °� ¼� ¦ ¦³Á� ¥� ª·� � Ê� ³Á¦¥� ªnµ ¦³Á¥ª·¸� � �

Ã¥¦·¥µ¥� � (implicit)

Euler Method 宦´ {®µnµÁ¦·É¤ o� � � � � � 𝒚′ = 𝒇ሺ𝒙,𝒚ሻ, 𝒂 ≤ 𝒙≤ 𝒃, 𝒚ሺ𝒂ሻ= 𝜶 ³ ε® Ä®onµ� � � � � 𝒙 ÁnµÇ´³Å�oªnµ� � � � � 𝒙𝒊 = 𝒂+ 𝒊𝒉 𝒊 = 𝟎,𝟏,𝟐,…,𝑵

¨ nµ¦³¥³ µ¦³®ªnµ »� � � � � � � � 𝒉= 𝒃−𝒂𝑵Á¦¥ ªnµ� µ ° Ê� � � � � � �

Euler MethodÃ¥� §¬¸ ° Á¥rÁ°¦r� � � � � � � (Taylor’s Theorem)

o� µ𝑦ሺ𝑥ሻ Á� È� � Á� ¥� °� � � ®µ� nµÁ¦·É¤� o� ³ 𝑦ሺ𝑥ሻ∈𝐶2ሾ𝑎,𝑏ሿ Âoª

𝑦ሺ𝑥𝑖+1ሻ= 𝑦ሺ𝑥𝑖ሻ+ 𝑦′ሺ𝑥𝑖ሻሺ𝑥𝑖+1 − 𝑥𝑖ሻ+ 𝑦′′ሺ𝜉𝑖ሻ2! ሺ𝑥𝑖+1 − 𝑥𝑖ሻ2 Á¤ºÉ°𝑖 = 0,1,2,…,𝑛− 1 ³ µ� � 𝜉𝑖 ∈ሺ𝑥𝑖,𝑥𝑖+1ሻ Á¤ºÉ°Ä®oℎ = 𝑥𝑖+1 − 𝑥𝑖 ³ µ� � 𝑦′ = 𝑓ሺ𝑥,𝑦ሻ ³Åo� �

𝑦ሺ𝑥𝑖+1ሻ= 𝑦ሺ𝑥𝑖ሻ+ ℎ𝑓ሺ𝑥𝑖,𝑦𝑖ሻ+ ℎ22!𝑦′′ሺ𝜉𝑖ሻ

Euler Method: Formula

¼¦¦³Á¥ ª·¸� � � � Euler

𝑦0 = 𝛼 𝑦𝑖+1 = 𝑦𝑖 + ℎ𝑓ሺ𝑥𝑖,𝑦𝑖ሻ ¤ nµªµ¤ µÁ ºÉ° Á¡µ³ ·É� � � � � � � � � (Local Error) Á}� � ℎ22! 𝑦′′ሺ𝜉𝑖ሻ 宦´ µ� � � 𝜉𝑖 ∈ሺ𝑥𝑖,𝑥𝑖+1ሻ

Euler Method: Geometry Meaningµ µ¦ ÉÄ®o� � � � 𝑦𝑖 Á� È� � nµ� ¦³¤µ� � °� 𝑦ሺ𝑥𝑖ሻ Á¦µÅ� oªnµ𝑦𝑖′ ≈ 𝑦′ሺ𝑥𝑖ሻ= 𝑓ሺ𝑥𝑖,𝑦𝑖ሻ

1y

yxfy , ay

Slope ,afay

a 1x

yxfy ,

ay

1y

a 1x

2x

2y

Euler Method: ExampleÄo¦³Á¥ª·¸� � � � � � Euler Ä µ¦ ¦³¤µ Á ¥° ´®µnµÁ¦·É¤ o� � � � � � � � � � � � �

12 xyy , 20 x , 5.00 y Ã¥Äo� � 10N , 2.0h ³ ixi 2.0

µ� � 1, 2 xyyxf

Įo 5.00 y ³ 121 iiii xyhyy 9,,2,1,0 i

0i , 120001 xyhyy

8000.0105.02.05.0

1i , 121112 xyhyy

1520.112.08.02.08.0 2 2i 12

2223 xyhyy

5504.114.0152.12.0152.1 2

Euler Method: Example� Á� ¥� ÉÂ� o� ¦·� 宦 � � � ®µ� nµÁ¦·É¤� o� � Ê� È� º° xexxy 5.01 2

i x y y(x) |y-y(x)|0 0.0000000 0.5000000 0.5000000 0.00000001 0.2000000 0.8000000 0.8292986 0.02929862 0.4000000 1.1520000 1.2140877 0.06208773 0.6000000 1.5504000 1.6489406 0.09854064 0.8000000 1.9884800 2.1272295 0.13874955 1.0000000 2.4581760 2.6408591 0.18268316 1.2000000 2.9498112 3.1799415 0.23013037 1.4000000 3.4517734 3.7324000 0.28062668 1.6000000 3.9501281 4.2834838 0.33335579 1.8000000 4.4281538 4.8151763 0.3870225

10 2.0000000 4.8657845 5.3054720 0.4396875

Taylor order nµ µ¦ ¦³ µ¥° »¦¤Á¥rÁ°¦r¦° »� � � � � � � � � � � 𝒙= 𝒙𝒊 ³Åo� �

𝒚ሺ𝒙ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒚′ሺ𝒙𝒊ሻሺ𝒙− 𝒙𝒊 ሻ+ 𝒚′′ሺ𝒙𝒊ሻ𝟐! ሺ𝒙− 𝒙𝒊ሻ𝟐 + ⋯+ 𝒚ሺ𝒏ሻሺ𝒙𝒊ሻ𝒏! ሺ𝒙− 𝒙𝒊ሻ𝒏 宦´ µ� � � 𝝃𝒊 ∈ሺ𝒙𝒊,𝒙𝒊+𝟏ሻ µ� � 𝒚′ሺ𝒙𝒊ሻ= 𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯, 𝒚′′ሺ𝒙𝒊ሻ= 𝒇′൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯, ...,,𝒚ሺ𝒌ሻሺ𝒙𝒊ሻ= 𝒇ሺ𝒌−𝟏ሻ൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ ³Åo� �

𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒉𝟐𝟐 𝒇′൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ ⋯+ 𝒉𝒏𝒏!𝒇ሺ𝒏−𝟏ሻ൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒉𝒏+𝟏ሺ𝒏+ 𝟏ሻ!𝒇ሺ𝒏ሻ൫𝝃𝒊,𝒚ሺ𝝃𝒊ሻ൯

Taylor order n: Formula¦³Á¥ª·¸� � � Taylor ° ´� � � n 𝒚𝟎 = 𝜶 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝑻ሺ𝒏ሻሺ𝒙𝒊,𝒚𝒊ሻ 𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏 Á¤ºÉ°

𝑻ሺ𝒏ሻሺ𝒙𝒊,𝒚𝒊ሻ= 𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐𝒇′ሺ𝒙𝒊,𝒚𝒊ሻ+ ⋯+ 𝒉𝒏−𝟏𝒏! 𝒇ሺ𝒏−𝟏ሻሺ𝒙𝒊,𝒚𝒊ሻ nµ·¡ µÁ¡µ³ ·É º°� � � � � � � � 𝒉𝒏+𝟏

ሺ𝒏+𝟏ሻ!𝒚ሺ𝒏+𝟏ሻሺ𝝃𝒊ሻ 宦´ µ� � � 𝝃𝒊 ∈ሺ𝒙𝒊,𝒙𝒊+𝟏ሻ

Taylor order n: Example� � Äo¦³Á¥ª·¸� � � � Taylor ° ´� � � 2 ³° ´� � � 4 ´ {®µnµÁ¦·É¤ o� � � � � � � 𝒚′ = 𝒚− 𝒙𝟐 + 𝟏,𝟎≤ 𝒙≤ 𝟐,𝒚ሺ𝟎ሻ= 𝟎.𝟓 ÁºÉ° µÁ¦µÄoÁ¥rÁ°¦r° ´� � � � � � � � � 4 ¹ o° Äo° »¡´ r°� � � � � � � � � � 𝒇 ¹° ´� � � � � 3 ¹É� � 𝒇ሺ𝒙,𝒚ሻ= 𝒚−𝒙𝟐 + 𝟏 𝒇′ሺ𝒙,𝒚ሻ= 𝒅𝒅𝒙൫𝒚− 𝒙𝟐 + 𝟏൯= 𝒚′ − 𝟐𝒙= 𝒚− 𝒙𝟐 + 𝟏 − 𝟐𝒙 𝒇′′ሺ𝒙,𝒚ሻ= 𝒅𝒅𝒙൫𝒚− 𝒙𝟐 + 𝟏 − 𝟐𝒙൯= 𝒚′ − 𝟐𝒙− 𝟐 = 𝒚− 𝒙𝟐 − 𝟐𝒙− 𝟏

𝒇′′′ ሺ𝒙,𝒚ሻ= 𝒅𝒅𝒙൫𝒚− 𝒙𝟐 − 𝟐𝒙− 𝟏൯= 𝒚′ − 𝟐𝒙− 𝟐 = 𝒚− 𝒙𝟐 − 𝟐𝒙− 𝟏

Taylor order n: ExampleÅo� 𝑻ሺ𝟐ሻሺ𝒙𝒊,𝒚𝒊ሻ= 𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐𝒇′ሺ𝒙𝒊,𝒚𝒊ሻ = 𝒚𝒊 − 𝒙𝒊𝟐 + 𝟏 + 𝒉𝟐൫𝒚𝒊 − 𝒙𝒊𝟐 − 𝟐𝒙𝒊 + 𝟏൯= ቀ𝟏 + 𝒉𝟐ቁ൫𝒚𝒊 − 𝒙𝒊𝟐 + 𝟏൯− 𝒉𝒙𝒊 𝑻ሺ𝟐ሻሺ𝒙𝒊,𝒚𝒊ሻ= 𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐𝒇′ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐𝟔 𝒇′′ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟑𝟐𝟒𝒇′′′ ሺ𝒙𝒊,𝒚𝒊ሻ = ቀ𝟏 + 𝒉𝟐 + 𝒉𝟐𝟔 + 𝒉𝟑𝟐𝟒ቁ൫𝒚𝒊 − 𝒙𝒊𝟐൯−ቀ𝟏 + 𝒉𝟑 + 𝒉𝟐𝟏𝟐ቁ𝒉𝒙𝒊 + 𝟏 + 𝒉𝟐 − 𝒉𝟐𝟔 − 𝒉𝟑𝟐𝟒

Taylor order n: ExampleÅo ¼¦Á¥rÁ°¦r° °� � � � � � � 𝒚𝟎 = 𝟎.𝟓 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉൬𝟏 + 𝒉𝟐൰൫𝒚𝒊 − 𝒙𝒊𝟐 + 𝟏൯− 𝒉𝒙𝒊൨ ³ ¼¦Á¥rÁ°¦r° É� � � � � 𝒚𝟎 = 𝟎.𝟓 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉ቈቆ𝟏 + 𝒉𝟐+ 𝒉𝟐𝟔 + 𝒉𝟑𝟐𝟒ቇ൫𝒚𝒊 − 𝒙𝒊𝟐൯−ቆ𝟏 + 𝒉𝟑+ 𝒉𝟐𝟏𝟐ቇ𝒉𝒙𝒊 + 𝟏

+ 𝒉𝟐− 𝒉𝟐𝟔 − 𝒉𝟑𝟐𝟒 宦� 𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏

Taylor order n: ExampleÄ®o 𝒉= 𝟎.𝟐, 𝑵= 𝟏𝟎 ´ Ê� � � � 𝒙𝒊 = 𝒙𝟎 + 𝒊𝒉= 𝟎+ 𝟎.𝟐𝒊 ሺ𝒊 = 𝟏,…,𝟏𝟎ሻ µ¦³Á¥ª·Á¥rÁ°¦r° 宦´� � � � � � � � � � 𝒊 = 𝟎 Åo�

𝒚𝟏 = 𝒚𝟎 + 𝒉൬𝟏 + 𝒉𝟐൰൫𝒚𝟎− 𝒙𝟎𝟐 + 𝟏൯− 𝒉𝒙𝟎൨= 𝟎.𝟓+ 𝟎.𝟐ሾ𝟏 + 𝟎.𝟏)ሺ𝟎.𝟓− 𝟎+ 𝟏ሻ− 𝟎ሿ= 𝟎.𝟖𝟑 宦� 𝒊 = 𝟏, 𝒙𝟏 = 𝟎.𝟐

𝒚𝟐 = 𝒚𝟏 + 𝒉൬𝟏 + 𝒉𝟐൰൫𝒚𝟏 − 𝒙𝟏𝟐 + 𝟏൯− 𝒉𝒙𝟏൨= 𝟎.𝟖𝟑+ 𝟎.𝟐�𝟏 + 𝟎.𝟏)൫𝟎.𝟖𝟑− 𝟎.𝟐𝟐 + 𝟏൯− 𝟎.𝟐ሺ𝟎.𝟐ሻ൧= 𝟏.𝟐𝟏𝟓𝟖

Taylor order n: Example¦³Á¥ª·Á¥rÁ°¦r° ´� � � � � � � 4 Ä®o

宦� 𝒊 = 𝟎 𝒚𝟏 = 𝒚𝟎 + 𝒉ቈቆ𝟏 + 𝒉𝟐+ 𝒉𝟐𝟔 + 𝒉𝟑𝟐𝟒ቇ൫𝒚𝟎 − 𝒙𝟎𝟐൯−ቆ𝟏 + 𝒉𝟑+ 𝒉𝟐𝟏𝟐ቇ𝒉𝒙𝟎+ 𝟏 + 𝒉𝟐− 𝒉𝟐𝟔− 𝒉𝟑𝟐𝟒

= 𝟎.𝟓 + 𝟎.𝟐ቈቆ𝟏 + 𝟎.𝟐𝟐 +ሺ𝟎.𝟐ሻ𝟐𝟔 +ሺ𝟎.𝟐ሻ𝟑𝟐𝟒 ቇ൫𝟎.𝟓− 𝟎𝟐൯−ቆ𝟏 + 𝟎.𝟐𝟑 + 𝟎.𝟐𝟐𝟏𝟐 ቇ𝟎.𝟐ሺ𝟎ሻ+ 𝟏 + 𝟎.𝟐𝟐 −ሺ𝟎.𝟐ሻ𝟐𝟔 −ሺ𝟎.𝟐ሻ𝟑𝟐𝟒 = 𝟎.𝟖𝟐𝟗𝟑

Taylor order n: Example

宦´� 𝒊 = 𝟏, 𝒙𝟏 = 𝟎.𝟐

𝒚𝟐 = 𝒚𝟏 + 𝒉ቈቆ𝟏 + 𝒉𝟐+ 𝒉𝟐𝟔 + 𝒉𝟑𝟐𝟒ቇ൫𝒚𝟏 − 𝒙𝟏𝟐൯−ቆ𝟏 + 𝒉𝟑+ 𝒉𝟐𝟏𝟐ቇ𝒉𝒙𝟏 + 𝟏 + 𝒉𝟐− 𝒉𝟐𝟔 − 𝒉𝟑𝟐𝟒 = 𝟎.𝟖𝟐𝟗𝟑+ 𝟎.𝟐ቈቆ𝟏 + 𝟎.𝟐𝟐 +ሺ𝟎.𝟐ሻ𝟐𝟔 +ሺ𝟎.𝟐ሻ𝟑𝟐𝟒 ቇ൫𝟎.𝟖𝟐𝟗𝟑− 𝟎.𝟐𝟐൯

−ቆ𝟏 + 𝟎.𝟐𝟑 + 𝟎.𝟐𝟐𝟏𝟐 ቇ𝟎.𝟐ሺ𝟎.𝟐ሻ+ 𝟏 + 𝟎.𝟐𝟐 −ሺ𝟎.𝟐ሻ𝟐𝟔 −ሺ𝟎.𝟐ሻ𝟑𝟐𝟒 = 𝟏.𝟐𝟏𝟒𝟎𝟗𝟏𝟎𝟐

Taylor order n: Example

Exact Taylor order 2 Error

Taylor order 4 Error 𝒙𝒊 𝒚ሺ𝒙𝒊ሻ 𝒚𝒊 ȁ7𝒚ሺ𝒙𝒊ሻ− 𝒚𝒊ȁ7 𝒚𝒊 ȁ7𝒚ሺ𝒙𝒊ሻ− 𝒚𝒊ȁ7

0.00 0.5000000 0.5000000 0.0000000 0.5000000 0.0000000 0.20 0.8292986 0.8300000 0.0007014 0.8293000 0.0000014 0.40 1.2140877 1.2158000 0.0017123 1.2140910 0.0000033 0.60 1.6489406 1.6520760 0.0031354 1.6489468 0.0000062 0.80 2.1272295 2.1323327 0.0051032 2.1272396 0.0000101 1.00 2.6408591 2.6486459 0.0077868 2.6408744 0.0000153 1.20 3.1799415 3.1913480 0.0114065 3.1799640 0.0000225 1.40 3.7324000 3.7486446 0.0162446 3.7324321 0.0000321 1.60 4.2834838 4.3061464 0.0226626 4.2835285 0.0000447 1.80 4.8151763 4.8462986 0.0311223 4.8152377 0.0000614 2.00 5.3054720 5.3476843 0.0422123 5.3055554 0.0000834

Interpolate other valuesÁ¤ºÉ° o° µ¦ ¦µ nµ� � � � � � 𝒚(𝟏.𝟐𝟓) ȳ µ¤µ¦ Äoª· µ¦ ¦³¤µ nµÄ nªÁnÄo� � � � � � � � � � � � � � � ¡®»µ¤� µ¦° r� � � 2 »� � ª°¥nµÁn� � � � Ä®o𝒙𝟎 = 𝟏.𝟐³ 𝒙𝟏 = 𝟏.𝟒³ µ o°¤¼É¦³¤µÅoµ¦³Á¥ª·Á¥r� � � � � � � � � � � � �

Á°¦r° ´� � � 4 Åo� 𝒚ሺ𝟏.𝟐𝟓ሻ≈൬

𝟏.𝟐𝟓− 𝟏.𝟒𝟏.𝟐− 𝟏.𝟒൰ሺ𝟑.𝟏𝟕𝟗𝟗𝟔𝟒𝟎ሻ+൬𝟏.𝟐𝟓− 𝟏.𝟐𝟏.𝟒− 𝟏.𝟐൰ሺ𝟑.𝟕𝟑𝟐𝟒𝟑𝟐𝟏ሻ= 𝟑.𝟑𝟏𝟖𝟎𝟖𝟏

¹Énµ¦· °� � � � � � � 𝒚ሺ𝟏.𝟐𝟓ሻ= 𝟑.𝟑𝟏𝟕𝟑𝟐𝟖𝟓 ´ Ê nµ¦³¤µ ÉÅo¹¤ ªµ¤ µÁ ºÉ° Á¡¥� � � � � � � � � � � � � � � � � 0.0007525

Interpolate other values

Á¤ºÉ°Äo¡®»µ¤� � Hermite ³ o° ®µnµ¦³¤µ °� � � � � � � � 𝒚′(𝟏.𝟐)³ 𝒚′(𝟏.𝟒) ¹ÉεÅoÃ¥°µ«¥ªµ¤ ¤¡´ r°� � � � � � � � � � 𝒚′ሺ𝒙ሻ= 𝒇ሺ𝒙,𝒚ሻ ´ Ê� � � 𝒚′ሺ𝟏.𝟐ሻ= 𝒚ሺ𝟏.𝟐ሻ−ሺ𝟏.𝟐ሻ𝟐 + 𝟏 = 𝟑.𝟏𝟕𝟗𝟗𝟔𝟒𝟎−ሺ𝟏.𝟐ሻ𝟐 + 𝟏= 𝟐.𝟕𝟑𝟗𝟗𝟔𝟒𝟎 𝒚′ሺ𝟏.𝟒ሻ= 𝒚ሺ𝟏.𝟒ሻ−ሺ𝟏.𝟒ሻ𝟐 + 𝟏 = 𝟑.𝟕𝟑𝟐𝟒𝟑𝟐𝟏 −ሺ𝟏.𝟒ሻ𝟐 + 𝟏= 𝟐.𝟕𝟕𝟐𝟒𝟑𝟐𝟏

Interpolate other valuesÃ¥Äo nµ ºÁºÉ° ®µ ¤ ¦³ · ·Í° ¡®»µ¤� � � � � � � � � � � � � � Hermite ³Åoo°¤¼ µ¦µ� � � � � � �

1.2 3.1799640 2.7399640

1.2 3.1799640 0.1118825 2.7623405 -0.3071225

1.4 3.7324321 0.0504580 2.7724321

1.4 3.7324321 𝒚ሺ𝟏.𝟐𝟓ሻ≈ 𝟑.𝟏𝟕𝟗𝟗𝟔𝟒𝟎+ሺ𝟏.𝟐𝟓− 𝟏.𝟐ሻሺ𝟐.𝟕𝟑𝟗𝟗𝟔𝟒𝟎ሻ+ሺ𝟏.𝟐𝟓− 𝟏.𝟐ሻ𝟐ሺ𝟎.𝟏𝟏𝟏𝟖𝟖𝟐𝟓ሻ+ሺ𝟏.𝟐𝟓− 𝟏.𝟐ሻ𝟐ሺ𝟏.𝟐𝟓− 𝟏.𝟒ሻሺ−𝟎.𝟑𝟎𝟕𝟏𝟐𝟐𝟓ሻ= 𝟑.𝟑𝟏𝟕𝟑𝟓𝟕𝟏

25

Runge-Kutta Method (RK)

ระเบ ยบวธ Runge-Kutta เป-นการปร�บปร�งระเบ ยบวธ เทย�เลอร� เพ��อให�พจน�ขอบเขตของค�าผดพลาดอ�นด�บส/งย�งคงร�กษาไว� ในขณะท �เราไม�จ)าเป-นต�องหาอน�พ�นธ�ย�อยอ�นด�บส/ง

26

Taylor Theory in 2 Variables

oµ� 𝒇³ »° »¡´ r¥n°¥° ´ o°¥ªnµ®¦º°Ánµ´� � � � � � � � � � � � � 𝒏 n°ÁºÉ° ÃÁ¤� � � � � � � 𝑫=ሼሺ𝒙,𝒚ሻ|𝒂 ≤ 𝒙≤ 𝒃,𝒄≤ 𝒚≤ 𝒅ሽ³ ሺ𝒙,𝒚ሻ ´� � ሺ𝒙+ 𝒉,𝒚+ 𝒌ሻ°¥¼nÄ� 𝑫 ʼnÂoª� � � 𝒇ሺ𝒙+ 𝒉,𝒚+ 𝒌ሻ= 𝒇ሺ𝒙,𝒚ሻ+ 𝒉 𝝏𝝏𝒙𝒇ሺ𝒙,𝒚ሻ+ 𝒌 𝝏𝝏𝒚𝒇ሺ𝒙,𝒚ሻ൨

+ቈ𝒉𝟐𝟐 𝝏𝟐𝝏𝒙𝟐𝒇ሺ𝒙,𝒚ሻ+ 𝒉𝒌 𝝏𝟐𝝏𝒙𝝏𝒚𝒇ሺ𝒙,𝒚ሻ+ 𝒌𝟐𝟐 𝝏𝟐𝝏𝒚𝟐𝒇ሺ𝒙,𝒚ሻ + ⋯

+ 𝟏𝒏! ቀ𝒏𝒋ቁ𝒉𝒏−𝒋𝒌𝒋𝒏

𝒋=𝟎𝝏𝒏𝝏𝒙𝒏−𝒋𝝏𝒚𝒋𝒇ሺ𝒙,𝒚ሻ

27

Runge-Kutta order 2¦³Á¥ª·Á¥rÁ°¦r° ° Åo¤µµ� � � � � � � � � � � 𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉𝒚′ሺ𝒙𝒊ሻ+ 𝒉𝟐𝟐 𝒚′′ሺ𝒙𝒊ሻ+ 𝒉𝟑𝟑! 𝒚′′′ ሺ𝝃𝒊ሻ 𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐𝟐 𝒇′ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟑𝟑! 𝒚′′′ ሺ𝝃𝒊ሻ ÁºÉ° µ� � � � 𝒇′ሺ𝒙𝒊,𝒚𝒊ሻ= 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒚′ሺ𝒙𝒊ሻ 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯³ 𝒚′ሺ𝒙𝒊ሻ=𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ Åo� 𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉൜𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯𝒉𝟐 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ൠ

+ 𝒉𝟑𝟑! 𝒚′′′ ሺ𝝃𝒊ሻ

28

Runge-Kutta order 2Á¦¥Á¥� � � �𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉൜𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯𝒉𝟐 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ൠ

+ 𝒉𝟑𝟑! 𝒚′′′ ሺ𝝃𝒊ሻ ´ µ¦ ¦³ µ¥¡ r� � � � � � � 𝒂𝒇ሺ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷ሻÃ¥Äo§¬¸ ° Á¥rÁ°¦r� � � � � � � � � � � 𝒏 = 𝟏 ³Åo� �

𝒂𝒇ሺ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷ሻ= 𝒂𝒇ሺ𝒙𝒊,𝒊 𝒚ሻ+ 𝜶 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝜷 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯൨൩ 𝒂𝒇ሺ𝒙𝒊,𝒊 𝒚ሻ+ 𝒂𝜶 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒂𝜷 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ Á¥¡ r³Åoªnµ� � � � � �

𝒂 = 𝟏, 𝜶= 𝒉𝟐³ 𝜷= 𝒉𝟐𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯

29

Runge-Kutta order 2Á¦¥Á¥� � � �𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒉൜𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒉𝟐 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯𝒉𝟐 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ൠ

+ 𝒉𝟑𝟑! 𝒚′′′ ሺ𝝃𝒊ሻ ´ µ¦ ¦³ µ¥¡ r� � � � � � � 𝒂𝒇ሺ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷ሻÃ¥Äo§¬¸ ° Á¥rÁ°¦r� � � � � � � � � � � 𝒏 = 𝟏 ³Åo� �

𝒂𝒇ሺ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷ሻ= 𝒂𝒇ሺ𝒙𝒊,𝒊 𝒚ሻ+ 𝜶 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝜷 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯൨൩ 𝒂𝒇ሺ𝒙𝒊,𝒊 𝒚ሻ+ 𝒂𝜶 𝝏𝝏𝒙𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒂𝜷 𝝏𝝏𝒚𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ Á¥¡ r³Åoªnµ� � � � � �

𝒂 = 𝟏, 𝜶= 𝒉𝟐³ 𝜷= 𝒉𝟐𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯

30

Runge-Kutta order 2

µ¦Â oª¥� � � � 𝒂𝒇ሺ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷ሻĦ³Á¥ª·Á¥rÁ°¦rŤnÄnµÁº° Á¥ª oµÄo� � � � � � � � � � � �¡ rĦ¼� � � � 𝒂𝟏𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯+ 𝒂𝟐𝒇ቀ𝒙𝒊 + 𝜶,𝒚ሺ𝒙𝒊ሻ+ 𝜷𝒇൫𝒙𝒊,𝒚ሺ𝒙𝒊ሻ൯ቁ ª�¦Á ¦·¤Ä®o ¼¦° ° εª ° ´ r¹Éº°� � � � � � � � � � � � � � � ¦¼Â ° ¼¦� � � � � � Runge-Kutta ° °� � � �(RK2) 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉 �𝒂𝟏𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒂𝟐𝒇൫𝒙𝒊 + 𝜶,𝒚𝒊 + 𝜷𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൯൧

31

RK2: Midpoint Formula

𝒂𝟏 = 𝟎, 𝒂𝟐 = 𝟏, 𝜶= 𝒉𝟐³ 𝜷= 𝒉𝟐 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝒇൭𝒙𝒊 + 𝒉𝟐,𝒚𝒊 + 𝒉𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൱

𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏

32

RK2: Modified Euler Formula

𝒂𝟏 = 𝟏𝟐, 𝒂𝟐 = 𝟏𝟐, 𝜶= 𝒉³ 𝜷= 𝒉

𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝟐 �𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒇൫𝒙𝒊+𝟏,𝒚𝒊 + 𝒉𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൯൧ 𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏

33

RK2: Heun’s Formula𝒂𝟏 = 𝟏𝟒, 𝒂𝟐 = 𝟑𝟒, 𝜶= 𝟐𝟑𝒉³ 𝜷= 𝟐𝟑𝒉

𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝟒ቈ𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝟑𝒇ቆ𝒙𝒊 + 𝟐𝟑𝒉,𝒚𝒊 + 𝟐𝟑𝒉𝒇ሺ𝒙𝒊,𝒚𝒊ሻቇ 𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏

¹É� � nµ·¡ µÁ¡µ³ ·É ° Ê µ¤ ¼¦ º°� � � � � � � � � � � � � 𝑶൫𝒉𝟑൯

34

RK2: Example

Äo¦³Á¥ª·¸� � � � � � Runge-Kutta ´ {®µnµÁ¦·É¤ o� � � � � � � 𝒚′ = 𝒚− 𝒙𝟐 + 𝟏, 𝟎≤ 𝒙≤ 𝟐, 𝒚ሺ𝟎ሻ= 𝟎.𝟓 ¡¦o°¤ oª¥� 𝑵= 𝟏𝟎,𝒉= 𝟎.𝟐, 𝒙𝒊 = 𝟎.𝟐𝒊 ³𝒚𝟎 = 𝟎.𝟓 ¼¦ µ¦ ¦³¤µ µ¦³Á¥ª·¸nµÇÅoÂn� � � � � � � � � � � � � ¼¦ »¹É µ� � � � � � � 𝒚𝒊+𝟏 = 𝟏.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟖 ¼¦� Euler ´Â¨� � � � 𝒚𝒊+𝟏 = 𝟏.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟔 ¼¦� Heun 𝒚𝒊+𝟏 = 𝟏.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟕𝟑𝟑 𝒊 = 𝟎,𝟏,𝟐,…,𝟗

35

RK2: Exampleµ� � 𝒚′ = 𝒚− 𝒙𝟐 + 𝟏, 𝟎≤ 𝒙≤ 𝟐, 𝒚ሺ𝟎ሻ= 𝟎.𝟓³ 𝑵= 𝟏𝟎,𝒉= 𝟎.𝟐, 𝒙𝒊 = 𝟎.𝟐𝒊, 𝒚𝟎 = 𝟎.𝟓 ¹ÉÅoªnµ� � � 𝒇ሺ𝒙,𝒚ሻ= 𝒚− 𝒙𝟐 + 𝟏

¼¦ »¹É µ º°� � � � � � � � 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝒇൬𝒙𝒊 + 𝒉𝟐 ,𝒚𝒊 + 𝒉𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൰ Â¥ ·Á¡µ³� � � � 𝒚𝒊 + 𝒉𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ= 𝒚𝒊 + 𝒉𝟐൫𝒚𝒊 − 𝒙𝐢𝟐 + 𝟏൯ = 𝒚𝒊 + 𝟎.𝟐𝟐 ൫𝒚𝒊 −ሺ𝟎.𝟐𝒊ሻ𝟐 + 𝟏൯= 𝒚𝒊 + 𝟎.𝟏𝒚𝒊 − 𝟎.𝟎𝟎𝟒𝒊𝟐 + 𝟎.𝟏

= 𝟏.𝟏𝒚𝒊 − 𝟎.𝟎𝟎𝟒𝒊+ 𝟎.𝟏

36

RK2: Exampleµ� � 𝒙𝒊 + 𝒉𝟐 = 𝟎.𝟐𝒊+ 𝟎.𝟏  nµ Ä� � � � � 𝒇ሺ𝒙,𝒚ሻ ³Åo� �

𝒉𝒇൭𝒙𝒊 + 𝒉𝟐,𝒚𝒊 + 𝒉𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൱= 𝟎.𝟐൛ሺ𝟏.𝟏𝒚𝒊 − 𝟎.𝟎𝟎𝟒𝒊+ 𝟎.𝟏ሻ−ሺ𝟎.𝟐𝒊+ 𝟎.𝟏ሻ𝟐 + 𝟏ൟ

= 𝟎.𝟐൛𝟏.𝟏𝒚𝒊 − 𝟎.𝟎𝟎𝟒𝒊𝟐 + 𝟎.𝟏 −൫𝟎.𝟎𝟒𝒊𝟐 + 𝟎.𝟎𝟒𝒊+ 𝟎.𝟎𝟏൯+ 𝟏ൟ = 𝟎.𝟐൛𝟏.𝟏𝒚𝒊 − 𝟎.𝟎𝟎𝟒𝒊𝟐 + 𝟎.𝟏 − 𝟎.𝟎𝟒𝒊𝟐 − 𝟎.𝟎𝟒𝒊− 𝟎.𝟎𝟏 + 𝟏ൟ = 𝟎.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟎𝟖𝒊𝟐 + 𝟎.𝟎𝟐− 𝟎.𝟎𝟎𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊− 𝟎.𝟎𝟎𝟐+ 𝟎.𝟐

= 𝟎.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟖

37

RK2: Example´ Ê ¼¦ »¹É µ 宦´ {®µÊº°� � � � � � � � � � � � � � � �

𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉𝒇൭𝒙𝒊 + 𝒉𝟐,𝒚𝒊 + 𝒉𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ൱ = 𝒚𝒊 + 𝟎.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟖

= 𝟏.𝟐𝟐𝒚𝒊 − 𝟎.𝟎𝟎𝟖𝟖𝒊𝟐 − 𝟎.𝟎𝟎𝟖𝒊+ 𝟎.𝟐𝟏𝟖

38

RK2: Example

𝒙𝒊 𝒚ሺ𝒙𝒊ሻ Midpoint Method

Error Modified Euler

Error Heun's Method

Error

0.00 0.5000000 0.5000000 0.0000000 0.5000000 0.0000000 0.5000000 0.0000000 0.20 0.8292986 0.8280000 0.0012986 0.8260000 0.0032986 0.8273333 0.0019653 0.40 1.2140877 1.2113600 0.0027277 1.2069200 0.0071677 1.2098800 0.0042077 0.60 1.6489406 1.6446592 0.0042814 1.6372424 0.0116982 1.6421869 0.0067537 0.80 2.1272295 2.1212842 0.0059453 2.1102357 0.0169938 2.1176014 0.0096281 1.00 2.6408591 2.6331668 0.0076923 2.6176876 0.0231715 2.6280070 0.0128521 1.20 3.1799415 3.1704634 0.0094781 3.1495789 0.0303627 3.1635019 0.0164396 1.40 3.7324000 3.7211654 0.0112346 3.6936862 0.0387138 3.7120057 0.0203944 1.60 4.2834838 4.2706218 0.0128620 4.2350972 0.0483866 4.2587802 0.0247035 1.80 4.8151763 4.8009586 0.0142177 4.7556185 0.0595577 4.7858452 0.0293310 2.00 5.3054720 5.2903695 0.0151025 5.2330546 0.0724173 5.2712645 0.0342074

39

Runge-Kutta order 4

𝒚𝒊+𝟏 = 𝒚𝒊 + 𝟏𝟔ሺ𝒌𝟏 + 𝟐𝒌𝟐 + 𝟐𝒌𝟑 + 𝒌𝟒ሻ 𝒌𝟏 = 𝒉𝒇ሺ𝒙𝒊,𝒚𝒊ሻ 𝒌𝟐 = 𝒉𝒇൬𝒙𝒊 + 𝒉𝟐,𝒚𝒊 + 𝒌𝟏𝟐൰

𝒌𝟑 = 𝒉𝒇൬𝒙𝒊 + 𝒉𝟐,𝒚𝒊 + 𝒌𝟐𝟐൰ 𝒌𝟒 = 𝒉𝒇ሺ𝒙𝒊+𝟏,𝒚𝒊 + 𝒌𝟑ሻ 𝒊 = 𝟎,𝟏,𝟐,…,𝑵− 𝟏 nµ·¡ µÁ¡µ³ ·É º°� � � � � � � � 𝑶൫𝒉𝟓൯

40

RK4: ExampleÄo¦³Á¥ª·¸� � � � � � Runge-Kutta ° ´� � � 4 ´ {®µnµÁ¦·É¤ o� � � � � � � 𝒚′ = 𝒚− 𝒙𝟐 + 𝟏, 𝟎≤ 𝒙≤ 𝟐, 𝒚ሺ𝟎ሻ= 𝟎.𝟓

¡¦o°¤ oª¥� 𝑵= 𝟏𝟎, 𝒉= 𝟎.𝟐, 𝒙𝒊 = 𝟎.𝟐𝒊 ³𝒚𝟎 = 𝟎.𝟓

µ� � 𝑓ሺ𝑥,𝑦ሻ= 𝑦− 𝑥2 + 1, 𝑵= 𝟏𝟎, 𝒉= 𝟎.𝟐, 𝒙𝒊 = 𝟎.𝟐𝒊 ³ 𝒚𝟎 = 𝟎.𝟓 𝑖 = 0 𝑦1 = 𝑦0 + 16ሺ𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4ሻ 𝐾1 = 𝒉𝒇ሺ𝒙𝟎,𝒚𝟎ሻ= 0.2ሺ𝑦0 − 𝑥02 + 1ሻ = 0.2ሺ0.5− 02 + 1ሻ= 0.3000000

41

RK4: Example

𝑘2 = ℎ𝑓൬𝑥0 + ℎ2,𝑦0 + 𝑘12൰= 𝟎.𝟐𝒇൬𝟎+ 𝟎.𝟐𝟐 ,𝟎.𝟓+ 𝟎.𝟑𝟐 ൰

= 𝟎.𝟐൫𝟎.𝟔𝟓− 𝟎.𝟏𝟐 + 𝟏൯= 𝟎.𝟑𝟐𝟖𝟎𝟎𝟎𝟎

𝑘3 = ℎ𝑓൬𝑥0 + ℎ2,𝑦0 + 𝑘22൰= 𝟎.𝟐𝒇൬𝟎+ 𝟎.𝟐𝟐 ,𝟎.𝟓+ 𝟎.𝟑𝟐𝟖𝟐 ൰

= 𝟎.𝟐൫𝟎.𝟔𝟔𝟒− 𝟎.𝟏𝟐 + 𝟏൯= 𝟎.𝟑𝟑𝟎𝟖𝟎𝟎𝟎 𝑘4 = ℎ𝑓ሺ𝑥1,𝑦0 + 𝑘3ሻ= 𝟎.𝟐𝒇ሺ𝟎.𝟐ሺ𝟏ሻ,𝟎.𝟓+ 𝟎.𝟑𝟑𝟎𝟖ሻ = 𝟎.𝟐൫𝟎.𝟖𝟑𝟎𝟖− 𝟎.𝟐𝟐 + 𝟏൯= 𝟎.𝟑𝟓𝟖𝟏𝟔𝟎𝟎

42

RK4: Example´ ÊÅo� � � � �

𝒚𝟏 = 𝟎.𝟓+ 𝟏𝟔ሺ𝟎.𝟑𝟎𝟎𝟎𝟎𝟎𝟎+ 𝟐ሺ𝟎.𝟑𝟐𝟖𝟎𝟎𝟎𝟎ሻ+ 𝟐ሺ𝟎.𝟑𝟐𝟖𝟎𝟎𝟎𝟎ሻ+ 𝟎.𝟑𝟓𝟖𝟏𝟔𝟎𝟎ሻ= 𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑

𝒊 = 𝟏 𝒚𝟐 = 𝒚𝟏 + 𝟏𝟔ሺ𝒌𝟏 +𝟐𝒌𝟐 +𝟐𝒌𝟑 +𝒌𝟒ሻ 𝑲𝟏 = 𝒉𝒇ሺ𝒙𝟏,𝒚𝟏ሻ= 𝟎.𝟐൫𝒚𝟏 − 𝒙𝟏𝟐 + 𝟏൯

= 𝟎.𝟐൫𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑− 𝟎.𝟐𝟐 + 𝟏൯= 𝟎.𝟑𝟓𝟕𝟖𝟓𝟖𝟕

43

RK4: Example𝑘2 = ℎ𝑓൬𝑥1 + ℎ2,𝑦1 + 𝑘12൰= 𝟎.𝟐𝒇൬𝟎.𝟐+ 𝟎.𝟐𝟐 ,𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑+ 𝟎.𝟑𝟓𝟕𝟖𝟓𝟖𝟕𝟐 ൰

= 𝟎.𝟐൫𝟏.𝟎𝟎𝟖𝟐𝟐𝟐𝟔𝟓− 𝟎.𝟑𝟐 + 𝟏൯= 𝟎.𝟑𝟖𝟑𝟔𝟒𝟒𝟓

𝑘3 = ℎ𝑓൬𝑥0 + ℎ2,𝑦0 + 𝑘22൰= 𝟎.𝟐𝒇൬𝟎.𝟐+ 𝟎.𝟐𝟐 ,𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑+ 𝟎.𝟑𝟖𝟑𝟔𝟒𝟒𝟓𝟐 ൰

= 𝟎.𝟐൫𝟏.𝟎𝟐𝟏𝟏𝟏𝟓𝟓𝟓− 𝟎.𝟑𝟐 + 𝟏൯= 𝟎.𝟑𝟖𝟔𝟐𝟐𝟑𝟏 𝑘4 = ℎ𝑓ሺ𝑥1,𝑦0 + 𝑘3ሻ= 𝟎.𝟐𝒇ሺ𝟎.𝟐ሺ𝟐ሻ,𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑+ 𝟎.𝟑𝟖𝟔𝟐𝟐𝟑𝟏ሻ = 𝟎.𝟐൫𝟏.𝟐𝟏𝟓𝟓𝟔𝟒− 𝟎.𝟒𝟐 + 𝟏൯= 𝟎.𝟒𝟏𝟏𝟏𝟎𝟑𝟑

Åo� 𝒚𝟏 = 𝟎.𝟖𝟐𝟗𝟐𝟗𝟑𝟑+ 𝟏𝟔ሺ𝟎.𝟑𝟓𝟕𝟖𝟓𝟖𝟕+ 𝟐ሺ𝟎.𝟑𝟖𝟑𝟔𝟒𝟒𝟓ሻ+ 𝟐ሺ𝟎.𝟑𝟖𝟔𝟐𝟐𝟑𝟏ሻ+𝟎.𝟒𝟏𝟏𝟏𝟎𝟑𝟑ሻ= 𝟏.𝟐𝟏𝟒𝟎𝟕𝟔𝟐

44

RK4: Example

𝒙𝒊 RK4 𝒚ሺ𝒙𝒊ሻ Error 𝒌𝟏 𝒌𝟐 𝒌𝟑 𝒌𝟒 0.00 0.5000000 0.5000000 0.0000000 0.3000000 0.3280000 0.3308000 0.3581600 0.20 0.8292933 0.8292986 0.0000053 0.3578587 0.3836445 0.3862231 0.4111033 0.40 1.2140762 1.2140877 0.0000114 0.4108152 0.4338968 0.4362049 0.4580562 0.60 1.6489220 1.6489406 0.0000186 0.4577844 0.4775628 0.4795407 0.4976925 0.80 2.1272027 2.1272295 0.0000269 0.4974405 0.5131846 0.5147590 0.5283923 1.00 2.6408227 2.6408591 0.0000364 0.5281645 0.5389810 0.5400626 0.5481771 1.20 3.1798942 3.1799415 0.0000474 0.5479788 0.5527767 0.5532565 0.5546301 1.40 3.7323401 3.7324000 0.0000599 0.5544680 0.5519148 0.5516595 0.5447999 1.60 4.2834095 4.2834838 0.0000743 0.5446819 0.5331501 0.5319969 0.5150813 1.80 4.8150857 4.8151763 0.0000906 0.5150171 0.4925189 0.4902690 0.4610709 2.00 5.3053630 5.3054720 0.0001089

45

RK4 vs Other Methods¦³Á¥ª·¸� � � Runge-Kutta ° ´� � � 4 Äoµ¦®µnµ¢{r� � � � � � � 4 ¦Ên° Ê °� � � � � � � °µÁ¥Åo� � � � � �¦³Á¥ª·¸� � � Euler ɤ µ ³Â¦� � � � � � � ( Ê °� � � � )Á}� � 𝟏𝟒 ¦³Á¥ª·¸� � � Runge-Kutta ° ´� � � 4 ªnµ¦³Á¥ª·¸� � � � � Runge-Kutta ° ´� � � 2 ( ɤ µ °� � � � � �Ê °� � � � 𝒉𝟐 ) Á¡¦µ³¦³Á¥ª·¸� � � Runge-Kutta ° ´� � � 4Äoεª µ¦®µnµ¤µ ªnµÁ}� � � � � � � � � � 2 Ánµ�° Ân³ Ê °� � � � � � �

46

RK4 vs Other Methods¡·µ¦ µ{®µ� � � � 𝒚′ = 𝒚− 𝒙𝟐 + 𝟏, 𝟎≤ 𝒙≤ 𝟐, 𝒚ሺ𝟎ሻ= 𝟎.𝟓 Ã¥� Äo¦³Á¥ª·¸� � � � Euler ¡¦o°¤ oª¥� 𝒉= 𝟎.𝟎𝟐𝟓¦³Á¥ª·¸� � � Euler ´Â¨� � � � (RK2)¡¦o°¤oª¥� 𝒉= 𝟎.𝟎𝟓 ³¦³Á¥ª·¸� � � Runge-Kutta 4 (RK4) ´� � 𝒉= 𝟎.𝟏 Á¦¥Á¥»�� � � � � � � 𝒙= 𝟎.𝟏,𝟎.𝟐,𝟎.𝟑,𝟎.𝟒,𝟎.𝟓

47

RK4 vs Other Methods

𝒙𝒊 Exact Euler 𝒉= 𝟎.𝟎𝟐𝟓 Modified Euler 𝒉= 𝟎.𝟎𝟓 Runge-Kutta order 4 𝒉= 𝟎.𝟏

0.0 0.5000000 0.5000000 0.5000000 0.5000000 0.1 0.6574145 0.6554982 0.6573085 0.6574144 0.2 0.8292986 0.8253385 0.8290778 0.8292983 0.3 1.0150706 1.0089334 1.0147254 1.0150701 0.4 1.2140877 1.2056345 1.2136079 1.2140869 0.5 1.4256394 1.4147264 1.4250141 1.4256384

48

Multi-Step Methodoµ°· ·Á¦ ¤ µ¦Á·° »¡´ r� � � � � � � � � � � 𝒅𝒚𝒅𝒙 = 𝒇ሺ𝒙,𝒚ሻÁ®º° nª� � � ሾ𝒙𝒊,𝒙𝒊+𝟏ሿ ³Åo� �

𝒚ሺ𝒙𝒊+𝟏ሻ− 𝒚ሺ𝒙𝒊ሻ= 𝒚′ሺ𝒙ሻ𝒅𝒙𝒙𝒊+𝟏𝒙𝒊 = 𝒇ሺ𝒙,𝒚ሻ𝒅𝒙𝒙𝒊+𝟏𝒙𝒊

´ Ê� � � � 𝒚ሺ𝒙𝒊+𝟏ሻ= 𝒚ሺ𝒙𝒊ሻ+ 𝒇ሺ𝒙,𝒚ሻ𝒅𝒙𝒙𝒊+𝟏𝒙𝒊

Â� � 𝒚ሺ𝒙ሻ oª ¡®»µ¤ ª ¦³¤µ nµÄ nª� � � � � � � � � � 𝑷ሺ𝒙ሻÁ¦µ³ µ¤µ¦ °· ·Á¦ Åo¹ÉÁ}¡®»� � � � � � � � � � �µ¤ ¦¸� � � 𝒎®µÅoµ µ » o°¤¼� � � � � � � � ሺ𝒙𝒊−𝒎,𝒚𝒊−𝒎ሻ, ሺ𝒙𝒊−𝒎+𝟏,𝒚𝒊−𝒎+𝟏ሻ, ... , ሺ𝒙𝒊,𝒚𝒊ሻ³𝒚ሺ𝒙𝒊ሻ≈ 𝒚𝒊 ³Åo� �

𝒚ሺ𝒙𝒊+𝟏ሻ≈ 𝒚𝒊 + 𝒇൫𝒙,𝑷ሺ𝒙ሻ൯𝒅𝒙𝒙𝒊+𝟏𝒙𝒊

49

Multi-Step Method

¦¼Â ° ¡®»µ¤ ª ¦³¤µÄÇ µ¤µ¦ ÄoÂ� � � � � � � � � � � � � � 𝑷ሺ𝒙ሻÅoÂn� � É ³ ª É » º° µ¦Äo� � � � � � � �¨ nµ ºÁºÉ° ¥o° ® ° ·ª ´� � � � � � � � � � � � �

¦³Á¥ª·®µ¥Ê µ¤µ¦ ÂnÅoÁ}� � � � � � � � � � � 2 »n¤ º°� �

1. ¦³Á¥ª·Ã¥´Âo� � � � � � � � Á}¦³Á¥ª·¸É� � � � � � 𝒚𝒊+𝟏 Ťn¹Ê´ µ¦®µnµ� � � � � � 𝒇ሺ𝒙𝒊+𝟏,𝒚𝒊+𝟏ሻ 2. ¦³Á¥ª·Ã¥¦·¥µ¥� � � � � ¹É� � ¤ µ nª ¹Ê´� � � � � � � 𝒇ሺ𝒙𝒊+𝟏,𝒚𝒊+𝟏ሻ

50

Multi-Step Method: Explicit case¦³Á¥ª·®µ¥ÊÃ¥´Âo� � � � � � � � � � ¦ ¸É� � � 𝒑= 𝟎 ¼¦ °� � � Adams-Bashforth ° ´ nµÇĦ¼Â� � � � � � � � �

𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉σ 𝒃𝒎𝒌𝒇ሺ𝒙𝒊−𝒌,𝒚𝒊−𝒌ሻ𝒎𝒌=𝟎

° ´� � � 𝒎 𝒃𝒎𝟎 𝒃𝒎𝟏 𝒃𝒎𝟐 𝒃𝒎𝟑 𝑬𝒎

1 0 1 𝒉𝟐𝟐 𝒚′′ሺ𝜼𝟎ሻ 2 1 3/2 -1/2 𝟓𝒉𝟑𝟏𝟐 𝒚′′′ ሺ𝜼𝟏ሻ 3 2 23/12 -16/12 5/12 𝟑𝒉𝟒𝟖 𝒚ሺ𝟒ሻሺ𝜼𝟐ሻ 4 3 55/24 -59/24 37/24 -9/24 𝟐𝟓𝟏𝒉𝟓𝟕𝟐𝟎 𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ

51

Adams-Bashforth FormulaAdams-Bashforth ° ´� � � 3 (AB3) 𝒚𝟎 = 𝜶𝟎,𝒚𝟏 = 𝜶𝟏,𝒚𝟐 = 𝜶𝟐 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉ቂ𝟐𝟑𝟏𝟐𝒇ሺ𝒙𝒊,𝒚𝒊ሻ− 𝟏𝟔𝟏𝟐𝒇ሺ𝒙𝒊−𝟏,𝒚𝒊−𝟏ሻ+ 𝟓𝟏𝟐𝒇ሺ𝒙𝒊−𝟐,𝒚𝒊−𝟐ሻቃ

ሺ𝒊 = 𝟏,𝟐,…,𝑵− 𝟏ሻ nµ·¡ µÁ¡µ³ ·Éº°� � � � � � � � 𝟑𝒉𝟒𝟖 𝒚ሺ𝟒ሻሺ𝜼𝒊ሻ 宦´ µ� � � 𝜼𝒊 ∈ሺ𝒙𝒊−𝟐,𝒙𝒊+𝟏ሻ

Adams-Bashforth ° ´� � � 4 (AB4) 𝒚𝟎 = 𝜶𝟎,𝒚𝟏 = 𝜶𝟏,𝒚𝟐 = 𝜶𝟐,𝒚𝟑 = 𝜶𝟑 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉ቂ𝟓𝟓𝟐𝟒𝒇ሺ𝒙𝒊,𝒚𝒊ሻ− 𝟓𝟗𝟐𝟒𝒇ሺ𝒙𝒊−𝟏,𝒚𝒊−𝟏ሻ+ 𝟑𝟕𝟐𝟒𝒇ሺ𝒙𝒊−𝟐,𝒚𝒊−𝟐ሻ−𝟗𝟐𝟒𝒇ሺ𝒙𝒊−𝟑,𝒚𝒊−𝟑ሻቃ ሺ𝒊 = 𝟏,𝟐,…,𝑵− 𝟏ሻ

nµ·¡ µÁ¡µ³ ·Éº°� � � � � � � � 𝟐𝟓𝟏𝒉𝟓𝟕𝟐𝟎 𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ 宦´ µ� � � 𝜼𝒊 ∈ሺ𝒙𝒊−𝟑,𝒙𝒊+𝟏ሻ

52

Multi-Step Method: Explicit case

¦³Á¥ª·®µ¥ÊÃ¥´Âo� � � � � � � � � � ¦ ¸É� � � 𝒑≠ 𝟎 (𝟎≤ 𝒑≤ 𝒎) µ µ¦ ¦³ µ¥¨ nµ ºÁºÉ° ³Å�o¦¼�Â���°� ¼�¦Á�}�� � � � � � � � � � � � � � � � � � � � �

𝒚𝒊+𝟏 = 𝒚𝒊−𝒑+ 𝒉σ 𝒃𝒎𝒌𝒇ሺ𝒙𝒊−𝒌,𝒚𝒊−𝒌ሻ𝒎𝒌=𝟎

𝒎 𝒑 𝒃𝒎𝟎 𝒃𝒎𝟏 𝒃𝒎𝟐 𝒃𝒎𝟑 𝑬𝒎

1 1 2 𝒉𝟑𝟔 𝒚′′′ ሺ𝜼𝟏ሻ 3 3 8/3 -4/3 8/3 𝟏𝟒𝒉𝟓𝟒𝟓 𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ

53

Multi-Step Method: Explicit case ¼¦ °� � � Milne 𝒚𝒊+𝟏 = 𝒚𝒊−𝟑 + 𝒉ቂ𝟖𝟑𝒇ሺ𝒙𝒊,𝒚𝒊ሻ− 𝟒𝟑𝒇ሺ𝒙𝒊−𝟏,𝒚𝒊−𝟏ሻ+ 𝟖𝟑𝒇ሺ𝒙𝒊−𝟐,𝒚𝒊−𝟐ሻቃ ሺ𝒊 = 𝟏,𝟐,…,𝑵− 𝟏ሻ nµ·¡ µÁ¡µ³ ·É º°� � � � � � � � 𝟏𝟒𝒉𝟓𝟒𝟓 𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ 宦´ µ� � � 𝜼𝟑 ∈ሺ𝒙𝒊−𝟐,𝒙𝒊+𝟏ሻ

Á¤ºÉ°𝒎Á}εª É³Áº°� � � � � � � 𝒑= 𝒎 ¼¦ 宦´ ¦ ¸� � � � 𝒑≠ 𝟎 ³Äoεª °� � � � � � � 𝒇 o°¥�ªnµ ¼¦ 宦´ ¦ ¸� � � � � 𝒑= 𝟎 ɤ° ´Á¥ª ´� � � � � � �

54

Multi-Step Method: Implicit case¦³Á¥ª·Ã¥¦·¥µ¥� � � � �

¦ ¸É� � � 𝒑= 𝟎 : 𝒚𝒊+𝟏 = 𝒚𝒊 + 𝒉σ 𝒃𝒎𝒌𝒇ሺ𝒙𝒊−𝒌,𝒚𝒊−𝒌ሻ𝒎−𝟏𝒌=−𝟏

¼¦ ÊÁ¦¥ªnµ� � � ¼¦ °� � � Adams-Moulton ° ´ nµÇ� � � � � ° ´� � � 𝒎 𝒃𝒎,−𝟏 𝒃𝒎𝟎 𝒃𝒎𝟏 𝒃𝒎𝟐 𝑬𝒎

1 0 1 −𝒉𝟐𝟐 𝒚′′ሺ𝜼𝟎ሻ 2 1 1/2 1/2 − 𝒉𝟑𝟏𝟐𝒚′′′ ሺ𝜼𝟏ሻ 3 2 5/12 8/12 -1/12 − 𝒉𝟒𝟐𝟒𝒚ሺ𝟒ሻሺ𝜼𝟐ሻ 4 3 9/24 19/24 -5/24 1/24 −𝟏𝟗𝒉𝟓𝟕𝟐𝟎 𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ

Á¤ºÉ°𝒙𝒊−𝒎+𝟏 < 𝜼𝟑 < 𝒙𝒊+𝟏

55

Multi-Step Method: Implicit case¦³Á¥ª·Ã¥¦·¥µ¥� � � � �

¦ ¸É� � � 𝒑≠ 𝟎 ¼¦ Énµ Ä º° ¼¦� � � � � � � Simpson (𝒎= 𝟑, 𝒑= 𝟏) 𝒚𝒊+𝟏 = 𝒚𝒊−𝟏 + 𝒉𝟑ሾ𝒇ሺ𝒙𝒊+𝟏,𝒚𝒊+𝟏ሻ− 𝟒𝒇ሺ𝒙𝒊,𝒚𝒊ሻ+ 𝒇ሺ𝒙𝒊−𝟏,𝒚𝒊−𝟏ሻሿ ሺ𝒊 = 𝟏,𝟐,…,𝑵− 𝟏ሻ nµ·¡ µÁ¡µ³ ·É º°� � � � � � � � − 𝒉𝟓𝟗𝟎𝒚ሺ𝟓ሻሺ𝜼𝟑ሻ 宦´ µ� � � 𝜼𝟑 ∈ሺ𝒙𝒊−𝟏,𝒙𝒊+𝟏ሻ

Á¡ºÉ° ɳÄo ¼¦Ã¥´Âo³å¦·¥µ¥� � � � � � � � � � � εÁ}o° ®µnµ� � � � � � 𝒚−𝒎,𝒚−𝒎+𝟏,…,𝒚𝟎 Á ¥n° oª¥¦³Á¥ª·¸ÊÁ¥ª� � � � � � � � � (single step method) Á ¥n° ÁnÄo¦³Á¥ª· °� � � � � � � � � �Runge-Kutta order 4

56

Predictor-Corrector Method

ระเบ ยบวธ น 1 เป-นการใช�ส/ตร 2 ส/ตรร�วมก�น โดยส/ตรหน3�งจะเป-นส/ตรแบบช�ดแจ�ง เร ยกว�า ส/ตรต�วท)านาย (Predictor Formula) และอ กส/ตรหน3�งเป-นส/ตรโดยปรยาย ท �ม ความแม�นย)าส/งกว�า เร ยกว�า ส/ตรต�วแก� (Corrector Formula)

การใช�ส/ตรต�วแก� เร ยกว�า การท)าซำ)1าภายใน (inner iteration)

ต�องใช�ค/�ต�วท)านายและต�วแก� ท �ม ค�าคลาดเคล��อนอย/�ในอ�นด�บใกล�เค ยงก�น มฉะน�1นแล�วการท)าซำ)1าภายในอาจจะไม�ล/�เข�า

57

Predictor-Corrector Method ¼¦ °� � � Adams-Bashforth-Moulton (ABM4)

Predictor 𝒚𝒊+𝟏ሺ𝟎ሻ = 𝒚𝒊 + 𝒉𝟐𝟒ሾ𝟓𝟓𝒇𝒊 − 𝟓𝟗𝒇𝒊−𝟏 + 𝟑𝟕𝒇𝒊−𝟐 − 𝟗𝒇𝒊−𝟑ሿ Corrector 𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ= 𝒚𝒊 + 𝒉𝟐𝟒ቂ𝟗𝒇𝒊+𝟏ሺ𝒌ሻ + 𝟏𝟗𝒇𝒊 − 𝟓𝒇𝒊−𝟏 + 𝒇𝒊−𝟐ቃ Äo¦³Á¥ª·Â ÊÁ¥ªÁn� � � � � � � � � � � RK Á¡ºÉ°®µnµ� 𝒚𝟏,𝒚𝟐,𝒚𝟑³ 𝒇𝟎,𝒇𝟏,𝒇𝟐,𝒇𝟑 1. εª� � � 𝒚𝒊+𝟏 oª¥ ¼¦ ª� � � εµ¥� � Ä®onµÎµª Á}� � � � � � 𝒚𝒊+𝟏ሺ𝟎ሻ 2. Äonµ� � 𝒚𝒊+𝟏ሺ𝟎ሻ Á¡ºÉ° εª� � � 𝒇𝒊+𝟏ሺ𝟎ሻ 3. εª� � � 𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ oª¥ ¼¦ ª� � � Âo� 4. εª nµ� � � � 𝒇𝒊+𝟏ሺ𝒌ሻ³ εÊ뵀 o°� � � � 3 ¦³ É� � � � � ቚ𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ− 𝒚𝒊+𝟏ሺ𝒌ሻቚ< 𝜀

58

Predictor-Corrector Method¼n ¼¦ ª εµ¥� � � � � - ªÂo°ºÉÇ� � �

¼¦ ª εµ¥Äo ¼¦ °� � � � � � � � Milne

𝒚𝒊+𝟏ሺ𝟎ሻ = 𝒚𝒊−𝟑 + 𝟒𝒉𝟑 ሾ𝟐𝒇𝒊 − 𝒇𝒊−𝟏 + 𝟐𝒇𝒊−𝟐ሿ ¼¦ ªÂoÄo ¼¦� � � � � Simpson

𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ= 𝒚𝒊−𝟏 + 𝒉𝟑ቂ𝒇𝒊+𝟏ሺ𝒌ሻ + 𝟒𝒇𝒊 + 𝒇𝒊−𝟏ቃ ®¦º° ¼¦ ªÂoÄo ¼¦� � � � � Hamming

𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ= 𝟏𝟖ቀ𝟗𝒚𝒊 − 𝒚𝒊−𝟐 + 𝟑𝒉ቂ𝒇𝒊+𝟏ሺ𝒌ሻ + 𝟐𝒇𝒊 − 𝒇𝒊−𝟏ቃቁ

59

Predictor-Corrector Method: Example

Äo¦³Á¥ª·¸ª εµ¥� � � � � � � � � - ªÂo®µnµ°� � � � � 𝒚 É»� � � 𝒙= 𝟏.𝟐 ° {®µnµÁ¦·É¤ o� � � � � � �𝒚′ = 𝟏 + 𝒚𝒙 ɤ� 𝒚= 𝟐Á¤ºÉ° 𝒙= 𝟏Ã¥Äo� � 𝒉= 𝟎.𝟏

ε® Ä®o� � � ¼n ¼¦ ÉÄoº°� � � � �

¼¦ ª εµ¥� � � � 𝒚𝒊+𝟏ሺ𝟎ሻ = 𝒚𝒊 + 𝒉𝒇ሺ𝒙𝒊,𝒚𝒊ሻ ¼¦ ªÂo� � � 𝒚𝒊+𝟏ሺ𝒌+𝟏ሻ= 𝒚𝒊 + 𝒉𝟐ቂ𝒇ቀ𝒙𝒊+𝟏,𝒚𝒊+𝟏ሺ𝒌ሻቁ+ 𝒇ሺ𝒙𝒊,𝒚𝒊ሻቃ

60

Predictor-Corrector Method: Example

𝒙𝟎 = 𝟏,𝒚𝟎 = 𝟐,𝒇ሺ𝒙𝟎,𝒚𝟎ሻ= 𝟑

𝒚𝟏ሺ𝟎ሻ= 𝟐+ሺ𝟎.𝟏ሻቀ𝟏 + 𝟐𝟏ቁ= 𝟐.𝟑

𝒇ቀ𝒙𝟏,𝒚𝟏ሺ𝟎ሻቁ= 𝟏 + 𝟐.𝟑𝟏.𝟏 = 𝟑.𝟎𝟗𝟎𝟗

𝒌= 𝟏 𝒚𝟏ሺ𝟏ሻ= 𝟐+ 𝟎.𝟏𝟐 ሾ𝟑.𝟎𝟗𝟎𝟗+ 𝟑ሿ= 𝟐.𝟑𝟎𝟒𝟓

61

Predictor-Corrector Method: Example

𝒇ቀ𝒙𝟏,𝒚𝟏ሺ𝟏ሻቁ= 𝟏 + 𝟐.𝟑𝟎𝟒𝟓𝟏.𝟏 = 𝟑.𝟎𝟗𝟓

𝒌= 𝟐 𝒚𝟏ሺ𝟐ሻ= 𝟐+ 𝟎.𝟏𝟐 ሾ𝟑.𝟎𝟗𝟗𝟓+ 𝟑ሿ= 𝟐.𝟑𝟎𝟒𝟖 𝒌= 𝟑 𝒚𝟏ሺ𝟑ሻ= 𝟐.𝟑𝟎𝟒𝟖 𝒙𝟏 = 𝟏.𝟏, 𝒚𝟏 = 𝟐.𝟑𝟎𝟒𝟖≈ 𝒚ሺ𝟏.𝟏ሻ

62

Predictor-Corrector Method: Example

³Åo� 𝒚𝟐ሺ𝟎ሻ= 𝟐.𝟔𝟏𝟒𝟑

𝒚𝟐ሺ𝟏ሻ= 𝟐.𝟔𝟏𝟖𝟓

𝒚𝟐ሺ𝟐ሻ= 𝟐.𝟔𝟏𝟖𝟕

𝒚𝟐ሺ𝟑ሻ= 𝟐.𝟔𝟏𝟖𝟕 ≈ 𝒚ሺ𝟏.𝟐ሻ ªµ¤ µÁ ºÉ° ° ¼¦ ªÂoº°� � � � � � � � � � � − 𝒉𝟑𝟏𝟐𝒚′′′ ሺ𝜼𝟏ሻ Á¤ºÉ°𝒙𝒊 < 𝜼𝟏 < 𝒙𝒊+𝟏

63

Predictor-Corrector Method: Example

¦³¤µ� � 𝒚′′′ ሺ𝜼𝟏ሻ oª¥� 𝒚′′′ ሺ𝟏ሻ= −𝟏³ 𝒉= 𝟎.𝟏Åoªµ¤ µÁ ºÉ°� � � � � � ŤnÁ·� � ቚ−𝟎.𝟏𝟑𝟏𝟐 ሺ𝟏ሻቚ≈ 𝟎.𝟖𝟑× 𝟏𝟎−𝟒  ªnµ Á ¥¤ ªµ¤Â¤n� � � � � � 3 D.P.

𝒚′ = 𝒇ሺ𝒙,𝒚ሻ= 𝟏 + 𝒚𝒙

𝒚′′ = 𝒅𝒚′𝒅𝒙 = 𝟎+ 𝒙𝒅𝒚𝒅𝒙+ 𝒚𝒙𝟐 = 𝒙ቀ𝟏 + 𝒚𝒙ቁ− 𝒚𝒙𝟐 = 𝟏𝒙 𝒚′′′ = 𝒅𝒚′′𝒅𝒙 = − 𝟏𝒙𝟐

High ODE and System of ODE

{®µnµÁ¦·É¤ o ɤ ¤ µ¦Á·° »¡´ r° ´� � � � � � � � � � � � � � � 𝒏åɪŤ¦¼Â Á}� � � � � � � � 𝒚ሺ𝒏ሻ= 𝒇൫𝒙,𝒚,𝒚′,…,𝒚ሺ𝒏−𝟏ሻ൯ Ã¥¤ÁºÉ° Å nµÁ¦·É¤ o� � � � � � � 𝒚ሺ𝒙𝟎ሻ= 𝜶𝟏, 𝒚′ሺ𝒙𝟎ሻ= 𝜶𝟐, ..., 𝒚ሺ𝒏−𝟏ሻሺ𝒙𝟎ሻ= 𝜶𝒏

High ODE and System of ODE

Á¤ºÉ°Ä®o 𝒚= 𝒖𝟏, 𝒚′ = 𝒖𝟐, ..., 𝒚ሺ 𝒏−𝟏ሻ= 𝒖𝒏 ¨ ÉÅoº°¦³ ¤ µ¦� � � � � � �

𝒖𝟏′ = 𝒖𝟐 𝒖𝟐′ = 𝒖𝟑

𝒖𝟑′ = 𝒖𝟒

⋮ 𝒖𝒏−𝟏′ = 𝒖𝒏

𝒖𝒏′ = 𝒇ሺ𝒙;𝒖𝟏,𝒖𝟐,…,𝒖𝒏ሻ ³ÁºÉ° Å nµÁ¦·É¤ o� � � � � � 𝒖𝟏ሺ𝒙𝟎ሻ= 𝜶𝟏, 𝒖𝟐ሺ𝒙𝟎ሻ= 𝜶𝟐, ..., 𝒖𝒏ሺ𝒙𝟎ሻ= 𝜶𝒏Á¤ºÉ° 𝒙= 𝒙𝟎

High ODE and System of ODE

Á¥ÅoĦ¼Â ¤ µ¦Áª Á°¦rÁ·° »¡´ r� � � � � � � � � � � � � � � ÅoÁ}� � � 𝒀′ሺ𝒙ሻ= 𝑭൫𝒙,𝒀൯ Ã¥¤�

𝒀= 𝒀𝟎 Á¤ºÉ° 𝒙= 𝒙𝟎

å� �

𝒀ሺ𝒙ሻ= ൦

𝒖𝟏ሺ𝒙ሻ𝒖𝟐ሺ𝒙ሻ⋮𝒖𝒏ሺ𝒙ሻ൪ ; 𝑭൫𝒙,𝒀൯=ۏێێێ𝒇𝟏൫𝒙;𝒀൯𝒇𝟐൫𝒙;𝒀൯⋮𝒇𝒏൫𝒙;𝒀൯ۍ ے

ۑۑۑ=ې ൦

𝒖𝟏′ ሺ𝒙ሻ𝒖𝟐′ ሺ𝒙ሻ⋮𝒇ሺ𝒙;𝒖𝟏,𝒖𝟐,…,𝒖𝒏ሻ൪

𝒀ሺ𝒙𝟎ሻ= 𝜶𝟏𝜶𝟐⋮𝜶𝒏

High ODE and System of ODE

¦³Á¥ª·¸ÉÄo®µ Á ¥{®µnµÁ¦·É¤ o� � � � � � � � � � � � 𝒚′ = 𝒇ሺ𝒙,𝒚ሻ ɤ� 𝒚= 𝒚𝟎Á¤ºÉ° 𝒙= 𝒙𝟎 o°� �Á É¥ ¦·¤µ Áµ¦r� � � � � 𝒚,𝒚′,𝒇ሺ𝒙,𝒚ሻ,𝒚𝟎 ³𝒌𝟏,𝒌𝟐,𝒌𝟑,𝒌𝟒Ä ¼¦Â� � � � Runge-Kutta

®¦º° ¼¦°ºÉÇ� � Á} ¦·¤µÁª Á°¦r� � � � � � 𝒀,𝒀′,𝑭൫𝒙;𝒀൯,𝒀𝟎³ 𝑲𝟏,𝑲𝟐,𝑲𝟑,𝑲𝟒

RK4 in vector form ¼¦Â� � � RK4 º°�

𝒀𝒊+𝟏 = 𝒀𝒊 + 𝟏𝟔൫𝑲𝟏 + 𝟐𝑲𝟐 + 𝟐𝑲𝟑 + 𝑲𝟒൯ 𝑲𝟏 = 𝒉𝑭൫𝒙𝒊,𝒀𝒊൯ 𝑲𝟐 = 𝒉𝑭൬𝒙𝒊 + 𝒉𝟐,𝒀𝒊 + 𝑲𝟏𝟐൰

𝑲𝟑 = 𝒉𝑭൬𝒙𝒊 + 𝒉𝟐,𝒀𝒊 + 𝑲𝟐𝟐൰ 𝑲𝟒 = 𝒉𝑭൫𝒙𝒊+𝟏,𝒀𝒊 + 𝑲𝟑൯ ሺ𝒊 = 𝟎,𝟏,…,𝑵− 𝟏ሻ

ABM4 in vector form

¼¦ ª εµ¥� � � � 𝒀𝒊+𝟏ሺ𝒑ሻ = 𝒀𝒊 + 𝒉𝟐𝟒 �𝟓𝟓𝑭𝒊 − 𝟓𝟗𝑭𝒊−𝟏 + 𝟑𝟕𝑭𝒊−𝟐 − 𝟗𝑭𝒊−𝟑൧ ¼¦ ªÂo� � � 𝒀𝒊+𝟏ሺ𝒄ሻ = 𝒀𝒊 + 𝒉𝟐𝟒 �𝟗𝑭𝒊+𝟏 + 𝟏𝟗𝑭𝒊 − 𝟓𝑭𝒊−𝟏 + 𝑭𝒊−𝟐൧ Ã¥É� � 𝑭𝒊 = 𝑭ቀ𝒙𝒊;𝒀ሺ𝒙𝒊ሻቁ

High ODE and System of ODE: Example

µ {®µnµÁ¦·É¤ o� � � � � � � 𝒚′′ = 𝒙𝒚𝟐,𝒚ሺ𝟎ሻ= 𝟏,𝒚′ሺ𝟎ሻ= −𝟏 ®µnµ¦³¤µ °� � � � � � �𝒚ሺ𝟎.𝟏ሻ oª¥¦³Á¥ª·¸� � � � RK4 ε® Ä®o� � � 𝒚′ = 𝒖

´ Ê� � � � 𝒚′′ = 𝒖′ = 𝒙𝒚𝟐 𝒀ሺ𝒙ሻ= 𝒚ሺ𝒙ሻ𝒖ሺ𝒙ሻ൨, 𝑭൫𝒙,𝒀൯= 𝒖ሺ𝒙ሻ𝒙𝒚𝟐൨, 𝒀𝟎ሺ𝒙ሻ= 𝒚ሺ𝟎ሻ𝒖ሺ𝟎ሻ൨= ቂ

𝟏−𝟏ቃ

High ODE and System of ODE: Example

Åo¦³ ¤ µ¦� � � � 𝒀′ = 𝑭൫𝒙,𝒀൯ ¡¦o°¤ÁºÉ° Å� � � 𝒀𝟎ሺ𝒙ሻ= ቂ

𝟏−𝟏ቃ Ã¥¦³Á¥ª·¸� � � � Runge-Kutta ° ´� � � 4

𝒀𝒊+𝟏 = 𝒀𝒊 + 𝟏𝟔൫𝑲𝟏 + 𝟐𝑲𝟐 + 𝟐𝑲𝟑 + 𝑲𝟒൯ 𝑲𝟏 = 𝒉𝑭൫𝒙𝟎,𝒀𝟎൯= ሺ𝟎.𝟏ሻ −𝟏

ሺ𝟎ሻሺ𝟏ሻ𝟐൨= ቂ−𝟎.𝟏𝟎 ቃ

𝒀𝟎 + 𝑲𝟏𝟐 = ቂ𝟏−𝟏ቃ+ 𝟏𝟐ቂ−𝟎.𝟏𝟎 ቃ= ቂ

𝟎.𝟗𝟓−𝟏 ቃ 𝒚ሺ𝒙ሻ𝒖ሺ𝒙ሻ൨

High ODE and System of ODE: Example

𝑲𝟐 = 𝒉𝑭൬𝒙𝟎+ 𝒉𝟐,𝒀𝟎 + 𝑲𝟏𝟐൰= ሺ𝟎.𝟏ሻ −𝟏ሺ𝟎.𝟎𝟓ሻሺ𝟎.𝟗𝟓ሻ𝟐൨= ቂ

−𝟎.𝟏𝟎.𝟎𝟎𝟒𝟓ቃ 𝒀𝟎 + 𝑲𝟏𝟐 = ቂ

𝟎.𝟗𝟓−𝟏 ቃ 𝒚ሺ𝒙ሻ𝒖ሺ𝒙ሻ൨

𝒀𝟎 + 𝑲𝟐𝟐 = ቂ𝟏−𝟏ቃ+ 𝟏𝟐ቂ −𝟎.𝟏𝟎.𝟎𝟎𝟒𝟓ቃ= ቂ

𝟎.𝟗𝟓−𝟎.𝟗𝟗𝟕𝟖ቃ 𝒚ሺ𝒙ሻ𝒖ሺ𝒙ሻ൨

𝑲𝟑 = 𝒉𝑭൬𝒙𝟎 + 𝒉𝟐,𝒀𝟎 + 𝑲𝟐𝟐൰= ሺ𝟎.𝟏ሻ −𝟎.𝟗𝟗𝟕𝟖ሺ𝟎.𝟎𝟓ሻሺ𝟎.𝟗𝟓ሻ𝟐൨= ቂ

−𝟎.𝟎𝟗𝟗𝟖𝟎.𝟎𝟎𝟒𝟓 ቃ

High ODE and System of ODE: Example

𝑲𝟒 = 𝒉𝑭൫𝒙𝟏,𝒀𝟎 + 𝑲𝟑൯= ሺ𝟎.𝟏ሻ −𝟎.𝟗𝟗𝟓𝟓ሺ𝟎.𝟏ሻሺ𝟎.𝟗𝟎𝟎𝟐ሻ𝟐൨= ቂ

−𝟎.𝟎𝟗𝟗𝟔𝟎.𝟎𝟎𝟖𝟏 ቃ

𝒀𝟎 + 𝑲𝟑 = ቂ𝟏−𝟏ቃ+ቂ

−𝟎.𝟎𝟗𝟗𝟖𝟎.𝟎𝟎𝟒𝟓 ቃ= ቂ𝟎.𝟗𝟎𝟎𝟐−𝟎.𝟗𝟗𝟓𝟓ቃ 𝒚ሺ𝒙ሻ𝒖ሺ𝒙ሻ൨

µ� � 𝒀𝟏 = 𝒀𝟎 + 𝟏𝟔൫𝑲𝟏 + 𝟐𝑲𝟐 + 𝟐𝑲𝟑 + 𝑲𝟒൯ = ቂ

𝟏−𝟏ቃ+ 𝟏𝟔ቄቂ−𝟎.𝟏𝟎 ቃ+ 𝟐ቂ −𝟎.𝟏𝟎.𝟎𝟎𝟒𝟓ቃ+ 𝟐ቂ−𝟎.𝟎𝟗𝟗𝟖𝟎.𝟎𝟎𝟒𝟓 ቃ+ቂ−𝟎.𝟎𝟗𝟗𝟔𝟎.𝟎𝟎𝟖𝟏 ቃቅ= ቂ

𝟎.𝟗𝟎𝟎𝟏𝟑−𝟎.𝟗𝟗𝟓𝟔𝟓ቃ ´ Ê� � � � 𝒚ሺ𝟎.𝟏ሻ= 𝟎.𝟗𝟎𝟎𝟏𝟑³ 𝒚′ሺ𝟎.𝟏ሻ= −𝟎.𝟗𝟗𝟓𝟔𝟓

High ODE and System of ODE: Example

Äo¦³Á¥ª·¸ª εµ¥� � � � � � � � � - ªÂoÁ¡ºÉ°®µnµ� � � 𝒚Á¤ºÉ° 𝒙= 𝟎.𝟒 ε® Ä®o� � � 𝒚′′ + 𝒚= 𝟎 ³ 𝒙 𝒚 𝒚′ = 𝒖 0.0 1 -1

0.1 0.895171 -1.094838

0.2 0.781397 -1.1787736

0.3 0.659816 -1.250857

High ODE and System of ODE: Example

ε® Ä®o� � � 𝒙𝟎 = 𝟎,𝒙𝒊 = 𝒊𝒉,𝒉= 𝟎.𝟏

𝒚′ = 𝒖 𝒚′′ = 𝒖′ = −𝒚

µ� � 𝒀= ቂ𝒚𝒖ቃ 𝒀𝟎 = ቂ

𝟏−𝟏ቃ, 𝒀𝟏 = ቂ𝟎.𝟖𝟗𝟓𝟏𝟕𝟏−𝟏.𝟎𝟗𝟒𝟖𝟑𝟖ቃ, 𝒀𝟐 = ቂ𝟎.𝟕𝟖𝟏𝟑𝟗𝟕−𝟏.𝟏𝟕𝟖𝟕𝟑𝟔ቃ, 𝒀𝟑 = ቂ𝟎.𝟔𝟓𝟗𝟖𝟏𝟔−𝟏.𝟐𝟓𝟎𝟖𝟓𝟕ቃ

µ� � 𝑭= ቂ𝒖−𝒚ቃ 𝑭𝟎 = ቂ

−𝟏−𝟏ቃ, 𝑭𝟏 = ቂ−𝟏.𝟎𝟗𝟒𝟖𝟑𝟖−𝟎.𝟖𝟗𝟓𝟏𝟕𝟏ቃ, 𝑭𝟐 = ቂ−𝟏.𝟏𝟕𝟖𝟕𝟑𝟔−𝟎.𝟕𝟖𝟏𝟑𝟗𝟕ቃ, 𝑭𝟑 = ቂ−𝟏.𝟐𝟓𝟎𝟖𝟓𝟕−𝟎.𝟔𝟓𝟗𝟖𝟏𝟔ቃ

High ODE and System of ODE: Example

Äo ¼¦ ª εµ¥� � � � � - ªÂo� � ABM4

¼¦ ª εµ¥� � � � 𝒀𝒊+𝟏ሺ𝒑ሻ = 𝒀𝒊 + 𝒉𝟐𝟒 �𝟓𝟓𝑭𝒊 − 𝟓𝟗𝑭𝒊−𝟏 + 𝟑𝟕𝑭𝒊−𝟐 − 𝟗𝑭𝒊−𝟑൧

𝒀𝟒ሺ𝟎ሻ= 𝒀𝟑 + 𝒉𝟐𝟒 �𝟓𝟓𝑭𝟑 − 𝟓𝟗𝑭𝟐 + 𝟑𝟕𝑭𝟏 − 𝟗𝑭𝟎൧= ቂ𝟎.𝟔𝟓𝟗𝟖𝟏𝟔−𝟏.𝟐𝟓𝟎𝟖𝟓𝟕ቃ+ 𝟎.𝟏𝟐𝟒ቄ𝟓𝟓ቂ−𝟏.𝟐𝟓𝟎𝟖𝟓𝟕−𝟎.𝟔𝟓𝟗𝟖𝟏𝟔ቃ− 𝟓𝟗ቂ−𝟏.𝟏𝟕𝟖𝟕𝟑𝟔−𝟎.𝟕𝟖𝟏𝟑𝟗𝟕ቃ+ 𝟑𝟕ቂ−𝟏.𝟎𝟗𝟒𝟖𝟑𝟖−𝟎.𝟖𝟗𝟓𝟏𝟕𝟏ቃ− 𝟗ቂ−𝟏−𝟏ቃቅ= ቂ

𝟎.𝟒𝟒𝟎𝟒𝟏𝟎−𝟏.𝟑𝟖𝟓𝟎𝟕𝟒ቃ

High ODE and System of ODE: Example

𝑭𝟒ሺ𝟎ሻ= ቂ−𝟏.𝟑𝟖𝟓𝟎𝟕𝟒−𝟎.𝟒𝟒𝟎𝟒𝟏𝟎ቃ

µ� � 𝒀𝟒ሺ𝟎ሻ= ቂ𝟎.𝟒𝟒𝟎𝟒𝟏𝟎−𝟏.𝟑𝟖𝟓𝟎𝟕𝟒ቃ ³ 𝑭= ቂ

𝒖−𝒚ቃ

¼¦ ªÂo� � � 𝒀𝒊+𝟏ሺ𝒄ሻ = 𝒀𝒊 + 𝒉𝟐𝟒 �𝟗𝑭𝒊+𝟏 + 𝟏𝟗𝑭𝒊 − 𝟓𝑭𝒊−𝟏 + 𝑭𝒊−𝟐൧ 𝒀𝟒ሺ𝟏ሻ= 𝒀𝟑 + 𝟎.𝟏𝟐𝟒ቂ𝟗𝑭𝟒ሺ𝟎ሻ+ 𝟏𝟗𝑭𝟑 − 𝟓𝑭𝟐 + 𝑭𝟏ቃ= ቂ

𝟎.𝟓𝟐𝟖𝟖𝟒𝟓−𝟏.𝟑𝟎𝟕𝟎𝟓𝟖ቃ 𝑭𝟒ሺ𝟏ሻ= ቂ−𝟏.𝟑𝟎𝟕𝟎𝟓𝟖−𝟎.𝟓𝟐𝟖𝟖𝟒𝟓ቃ

𝒀𝟒ሺ𝟐ሻ= 𝒀𝟑 + 𝟎.𝟏𝟐𝟒ቂ𝟗𝑭𝟒ሺ𝟏ሻ+ 𝟏𝟗𝑭𝟑 − 𝟓𝑭𝟐 + 𝑭𝟏ቃ= ቂ𝟎.𝟓𝟑𝟏𝟕𝟕𝟎−𝟏.𝟑𝟏𝟎𝟑𝟕𝟓ቃ

High ODE and System of ODE: Example

𝒀𝟒ሺ𝟑ሻ= ቂ𝟎.𝟓𝟑𝟏𝟔𝟒𝟔−𝟏.𝟑𝟏𝟎𝟒𝟖𝟒ቃ 𝒀𝟒ሺ𝟒ሻ= ቂ𝟎.𝟓𝟑𝟏𝟔𝟒𝟐−𝟏.𝟑𝟏𝟎𝟒𝟖𝟎ቃ

ቛ𝒀𝟒ሺ𝟒ሻ− 𝒀𝟒ሺ𝟑ሻቛ∞ = 𝟎.𝟒× 𝟏𝟎−𝟓

 ªnµ Á ¥� � � � ¤ ªµ¤Â¤n¥Îµ¹ « ·¥¤ ε®nÉ� � � � � � � � � � 5 (5 D.P.) 𝒚ሺ𝟎.𝟒ሻ= 𝟎.𝟓𝟑𝟏𝟔𝟒𝟐 ³ 𝒚′ሺ𝟎.𝟒ሻ= −𝟏.𝟑𝟏𝟎𝟒𝟖

Divided Different Method for BVP

{®µnµ°� � � � � (BVP) ° ¤ µ¦Á·° »¡´ rÁ·Á o� � � � � � � � � � � 𝒚′′ሺ𝒙ሻ+ 𝒑ሺ𝒙ሻ𝒚′ሺ𝒙ሻ+ 𝒒ሺ𝒙ሻ𝒚ሺ𝒙ሻ= 𝒈ሺ𝒙ሻ 𝒂 < 𝑥< 𝑏 𝒚ሺ𝒂ሻ= 𝜶, 𝒚ሺ𝒃ሻ= 𝜷

µ¦®µ Á ¥Á· ªÁ ³ o° Á É¥ {®µÄ¦³ n°ÁºÉ° Ä®oÁ} {®µÄ¦³ Ťn� � � � � � � � � � � � � � � � � � � � � � � � � � �n°ÁºÉ° Ã¥� � � �

1. Á É¥ÃÁ¤ n°ÁºÉ° °� � � � � � � � � 𝒙Ä nª� � � ሾ𝒂,𝒃ሿÁ}Á ° »Ânnªº°� � � � � � � � � � � � � 𝒙𝟎,𝒙𝟏,…,𝒙𝒏 2. Á É¥ ¤ µ¦Á·° »¡´ rÁ} ¤ µ¦ ¨ nµ ºÁºÉ°� � � � � � � � � � � � � � � � � 3. Á É¥ÁºÉ° Å nµ° Éε® Ä®oÁ}ÁºÉ° ÅÁÉ¥ª ´ nµ°� � � � � � � � � � � � � � � � � � � � � � � 𝒚 É»Ânnª nµÇ� � � � � � � � �

Differential Equation to Divided Different Form

𝒚′ሺ𝒙𝒊ሻ≅ 𝒚𝒊′ = 𝟏𝟐𝒉ሺ𝒚𝒊+𝟏 − 𝒚𝒊−𝟏ሻ 𝒚′′ሺ𝒙𝒊ሻ≅ 𝒚𝒊′′ = 𝟏𝒉𝟐ሺ𝒚𝒊+𝟏 − 𝟐𝒚𝒊 + 𝒚𝒊−𝟏ሻ 𝒚ሺ𝟒ሻሺ𝒙𝒊ሻ≅ 𝒚𝒊ሺ𝟒ሻ= 𝟏𝒉𝟒ሺ𝒚𝒊+𝟐 − 𝟒𝒚𝒊+𝟏 + 𝟔𝒚𝒊 − 𝟒𝒚𝒊−𝟏 + 𝒚𝒊−𝟐ሻ

µ� � 𝒚′′ሺ𝒙ሻ+ 𝒑ሺ𝒙ሻ𝒚′ሺ𝒙ሻ+ 𝒒ሺ𝒙ሻ𝒚ሺ𝒙ሻ= 𝒈ሺ𝒙ሻ Á¤ºÉ°Â¨ Á}¨ nµ ºÁºÉ° ³Åo� � � � � � � � � � � � 𝟏𝒉𝟐ሺ𝒚𝒊+𝟏 − 𝟐𝒚𝒊 + 𝒚𝒊−𝟏ሻ+ 𝒑ሺ𝒙𝒊ሻ 𝟏𝟐𝒉ሺ𝒚𝒊+𝟏 − 𝒚𝒊−𝟏ሻ+ 𝒒ሺ𝒙𝒊ሻ𝒚𝒊 = 𝒈ሺ𝒙𝒊ሻ

ሺ𝒊 = 𝟎,𝟏,…,𝒏− 𝟏ሻ

Divided Different Formula for Linear BVP

Ä®o 𝒑𝒊 = 𝒑ሺ𝒙𝒊ሻ, 𝒒𝒊 = 𝒒ሺ𝒙𝒊ሻ ³ 𝒈𝒊 = 𝒈ሺ𝒙𝒊ሻ ¦¼Ä®¤nÅo� � � � ൬𝟏 − 𝒉𝟐𝒑𝒊൰𝒚𝒊−𝟏 +൫−𝟐+ 𝒉𝟐𝒒𝒊൯𝒚𝒊 +൬𝟏 + 𝒉𝟐𝒑𝒊൰𝒚𝒊+𝟏 = 𝒉𝟐𝒈𝒊 𝒊 = 𝟎,𝟏,…,𝒏− 𝟏

Á¤ºÉ°Â nµ´ ¸³Åo¦³ ¤ µ¦Á·Á o¤Á¤ ¦· r° ¤ ¦³ · ·ÍÁ}Á¤ ¦· r µ¤Âª� � � � � � � � � � � � � � � � � � � � � � � � � � � �Á¥ ɦµ nµ» ª¦³ ¤ µ¦ ʤ� � � � � � � � � � � � � 𝒏− 𝟏 ¤ µ¦Â³¤ ªÅ¤n¦µ nµÎµª� � � � � � � � 𝒏− 𝟏 ª�º°� 𝒚𝟏,𝒚𝟐,…,𝒚𝒏−𝟏Ä ³ É� � � � 𝒚𝟎 = 𝜶³ 𝒚𝒏 = 𝜷

Existing of Solution

oµÁº° µ ° nª� � � � � � � � � 𝒉 ÉεĮo� � 𝒉< 𝟐𝑴Ã¥É� � 𝑴= 𝐦𝐚𝐱𝒂≤𝒙≤𝒃ȁ7𝒑ሺ𝒙ሻȁ7Á¤ ¦· r°� � � � � ¤ ¦³ · ·ÍĦ³ ¤ µ¦� � � � � � � ቀ𝟏 − 𝒉𝟐𝒑𝒊ቁ𝒚𝒊−𝟏 +൫−𝟐+ 𝒉𝟐𝒒𝒊൯𝒚𝒊 +ቀ𝟏 +𝒉𝟐𝒑𝒊ቁ𝒚𝒊+𝟏 = 𝒉𝟐𝒈𝒊, ሺ𝒊 = 𝟎,𝟏,…,𝒏− 𝟏ሻ ³¤Âª Â¥¤»¤ n¤Âo¹¤ Á ¥� � � � � � � � � �Ân° ³¤Á¡¥ Á ¥Á¥ª� � � � � � �

Divided Different for BVP: Example

®µ� � 𝒚 É� 𝒙= 𝟎.𝟓 µ {®µnµ°� � � � � � � 𝒚′′ + 𝒙𝒚′ + 𝒚= 𝒙, 𝒚ሺ𝟎ሻ= 𝟎, 𝒚ሺ𝟏ሻ= 𝟎Á¤ºÉ°Äo� 𝒉= 𝟏𝟒

𝒙𝟎 = 𝟎 𝒙𝟏 = 𝟏𝟒 𝒙𝟐 = 𝟏𝟐 𝒙𝟒 = 𝟏 𝒙𝟑 = 𝟑𝟒

Â� � 𝒑𝒊,𝒒𝒊,𝒈𝒊Ä� ቀ𝟏 − 𝒉𝟐𝒑𝒊ቁ𝒚𝒊−𝟏 +൫−𝟐+ 𝒉𝟐𝒒𝒊൯𝒚𝒊 +ቀ𝟏 + 𝒉𝟐𝒑𝒊ቁ𝒚𝒊+𝟏 = 𝒉𝟐𝒈𝒊 ³Åo� �

ቀ𝟏 − 𝒙𝒊𝒉𝟐 ቁ𝒚𝒊−𝟏 +൫−𝟐+ 𝒉𝟐൯𝒚𝒊 +ቀ𝟏 + 𝒙𝒊𝒉𝟐 ቁ𝒚𝒊+𝟏 = 𝒉𝟐𝒙𝒊

Divided Different for BVP: Example

宦´Ân³ nµ� � � 𝒊 ³Åo� � 𝒊 = 𝟏 ቀ𝟏 − 𝒙𝟏𝒉𝟐 ቁ𝒚𝟎 +൫𝒉𝟐 − 𝟐൯𝒚𝟏 +ቀ𝟏 + 𝒙𝟏𝒉𝟐 ቁ𝒚𝟐 = 𝒉𝟐𝒙𝟏 𝒊 = 𝟐 ቀ𝟏 − 𝒙𝟐𝒉𝟐 ቁ𝒚𝟏 +൫𝒉𝟐 − 𝟐൯𝒚𝟐 +ቀ𝟏 + 𝒙𝟐𝒉𝟐 ቁ𝒚𝟑 = 𝒉𝟐𝒙𝟐 𝒊 = 𝟑 ቀ𝟏 − 𝒙𝟑𝒉𝟐 ቁ𝒚𝟐 +൫𝒉𝟐 − 𝟐൯𝒚𝟑 +ቀ𝟏 + 𝒙𝟑𝒉𝟐 ቁ𝒚𝟒 = 𝒉𝟐𝒙𝟑

 nµ� � � 𝒉= 𝟏𝟒, 𝒙𝟏 = 𝟏𝟒, 𝒙𝟐 = 𝟏𝟐, 𝒙𝟑 = 𝟑𝟒, 𝒚ሺ𝟎ሻ= 𝟎, 𝒚ሺ𝟏ሻ= 𝟎

³Åo¦³ ¤ µ¦� � � � �

−𝟏.𝟗𝟑𝟕𝟓 𝟏.𝟎𝟑𝟏𝟐𝟓 𝟎𝟎.𝟗𝟑𝟕𝟓 −𝟏.𝟗𝟑𝟕𝟓 𝟏.𝟎𝟔𝟐𝟓𝟎 𝟎.𝟗𝟎𝟔𝟐𝟓 −𝟏.𝟗𝟑𝟕𝟓൩𝒚𝟏𝒚𝟐𝒚𝟑൩= 𝟎.𝟎𝟏𝟓𝟔𝟐𝟓𝟎.𝟎𝟑𝟏𝟐𝟓𝟎𝟎.𝟎𝟒𝟔𝟖𝟕𝟓൩

Divided Different for BVP: Example

®µ Á ¥° ¦³ ¤ µ¦Á·Á o� � � � � � � � � � ÅoÁ ¥� � � 𝒚𝟑 = −𝟎.𝟎𝟓𝟔𝟐𝟒𝟒𝟏, 𝒚𝟐 = −𝟎.𝟎𝟔𝟖𝟓𝟐𝟏𝟖, 𝒚𝟏 = 𝟎.𝟎𝟒𝟒𝟓𝟑𝟓