Initial-Value Problems for ODEs Euler’s Method I: Introduction Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University � 2011 Brooks/Cole, Cengage Learning c
Derivation Algorithm Geometric Interpretation Example Outline Derivation of Euler’s Method 1 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 2 / 23
Derivation Algorithm Geometric Interpretation Example Outline Derivation of Euler’s Method 1 Numerical Algorithm 2 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 2 / 23
Derivation Algorithm Geometric Interpretation Example Outline Derivation of Euler’s Method 1 Numerical Algorithm 2 Geometric Interpretation 3 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 2 / 23
Derivation Algorithm Geometric Interpretation Example Outline Derivation of Euler’s Method 1 Numerical Algorithm 2 Geometric Interpretation 3 Numerical Example 4 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 2 / 23
Derivation Algorithm Geometric Interpretation Example Outline Derivation of Euler’s Method 1 Numerical Algorithm 2 Geometric Interpretation 3 Numerical Example 4 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 3 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation Obtaining Approximations Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 4 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation Obtaining Approximations The object of Euler’s method is to obtain approximations to the well-posed initial-value problem dy dt = f ( t , y ) , a ≤ t ≤ b , y ( a ) = α Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 4 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation Obtaining Approximations The object of Euler’s method is to obtain approximations to the well-posed initial-value problem dy dt = f ( t , y ) , a ≤ t ≤ b , y ( a ) = α A continuous approximation to the solution y ( t ) will not be obtained; Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 4 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation Obtaining Approximations The object of Euler’s method is to obtain approximations to the well-posed initial-value problem dy dt = f ( t , y ) , a ≤ t ≤ b , y ( a ) = α A continuous approximation to the solution y ( t ) will not be obtained; Instead, approximations to y will be generated at various values, called mesh points, in the interval [ a , b ] . Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 4 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation Obtaining Approximations The object of Euler’s method is to obtain approximations to the well-posed initial-value problem dy dt = f ( t , y ) , a ≤ t ≤ b , y ( a ) = α A continuous approximation to the solution y ( t ) will not be obtained; Instead, approximations to y will be generated at various values, called mesh points, in the interval [ a , b ] . Once the approximate solution is obtained at the points, the approximate solution at other points in the interval can be found by interpolation. Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 4 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Set up an equally-distributed mesh Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 5 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Set up an equally-distributed mesh We first make the stipulation that the mesh points are equally distributed throughout the interval [ a , b ] . Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 5 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Set up an equally-distributed mesh We first make the stipulation that the mesh points are equally distributed throughout the interval [ a , b ] . This condition is ensured by choosing a positive integer N and selecting the mesh points t i = a + ih , for each i = 0 , 1 , 2 , . . . , N . Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 5 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Set up an equally-distributed mesh We first make the stipulation that the mesh points are equally distributed throughout the interval [ a , b ] . This condition is ensured by choosing a positive integer N and selecting the mesh points t i = a + ih , for each i = 0 , 1 , 2 , . . . , N . The common distance between the points h = ( b − a ) / N = t i + 1 − t i is called the step size. Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 5 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Use Taylor’s Theorem to derive Euler’s Method Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 6 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d Use Taylor’s Theorem to derive Euler’s Method Suppose that y ( t ) , the unique solution to dy dt = f ( t , y ) , a ≤ t ≤ b , y ( a ) = α has two continuous derivatives on [ a , b ] , so that for each i = 0 , 1 , 2 , . . . , N − 1, y ( t i + 1 ) = y ( t i ) + ( t i + 1 − t i ) y ′ ( t i ) + ( t i + 1 − t i ) 2 y ′′ ( ξ i ) 2 for some number ξ i in ( t i , t i + 1 ) . Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 6 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + ( t i + 1 − t i ) y ′ ( t i ) + ( t i + 1 − t i ) 2 y ′′ ( ξ i ) 2 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 7 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + ( t i + 1 − t i ) y ′ ( t i ) + ( t i + 1 − t i ) 2 y ′′ ( ξ i ) 2 Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 7 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + ( t i + 1 − t i ) y ′ ( t i ) + ( t i + 1 − t i ) 2 y ′′ ( ξ i ) 2 Because h = t i + 1 − t i , we have y ( t i + 1 ) = y ( t i ) + hy ′ ( t i ) + h 2 2 y ′′ ( ξ i ) and, because y ( t ) satisfies the differential equation y ′ = f ( t , y ) , we write y ( t i + 1 ) = y ( t i ) + hf ( t i , y ( t i )) + h 2 2 y ′′ ( ξ i ) Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 7 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + hf ( t i , y ( t i )) + h 2 2 y ′′ ( ξ i ) Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 8 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + hf ( t i , y ( t i )) + h 2 2 y ′′ ( ξ i ) Euler’s Method Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 8 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + hf ( t i , y ( t i )) + h 2 2 y ′′ ( ξ i ) Euler’s Method Euler’s method constructs w i ≈ y ( t i ) , for each i = 1 , 2 , . . . , N , by deleting the remainder term. Thus Euler’s method is w 0 = α w i + 1 w i + hf ( t i , w i ) , for each i = 0 , 1 , . . . , N − 1 = Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 8 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Derivation (Cont’d y ( t i + 1 ) = y ( t i ) + hf ( t i , y ( t i )) + h 2 2 y ′′ ( ξ i ) Euler’s Method Euler’s method constructs w i ≈ y ( t i ) , for each i = 1 , 2 , . . . , N , by deleting the remainder term. Thus Euler’s method is w 0 = α w i + 1 w i + hf ( t i , w i ) , for each i = 0 , 1 , . . . , N − 1 = This equation is called the difference equation associated with Euler’s method. Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 8 / 23
Derivation Algorithm Geometric Interpretation Example Euler’s Method: Illustration Applying Euler’s Method Prior to introducing an algorithm for Euler’s Method, we will illustrate the steps in the technique to approximate the solution to y ′ = y − t 2 + 1 , 0 ≤ t ≤ 2 , y ( 0 ) = 0 . 5 at t = 2. using a step size of h = 0 . 5. Numerical Analysis (Chapter 5) Euler’s Method I: Introduction R L Burden & J D Faires 9 / 23
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