Taylor Series Expansion

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Example 3: Solve the differential equation y' = x/y, y(0)=1 by Euler's method to get y(1). Use the step lengths h = 0.1 and 0.2 and compare the results with the analytical solution (y 2 = 1 + x 2 ) Solution: f(x, y) = x/y, with h = 0.1 y 1 = y 0 + h f(x 0 , y 0 ) = 1.0 + 0.1*0.0/1.0 = 1.00 y 2 = y 1 + h f(x 1 , y 1 ) = 1.0 + 0.1*0.1/1.0 = 1.01 y 3 = y 2 + h f(x 2 , y 2 ) = 1.01 + 0.1*0.2/1.01 = 1.0298 y 4 = y 3 + h f(x 3 , y 3 ) = 1.0298 + 0.1*0.3/1.0298 = 1.0589 y 5 = y 4 + h f(x 4 , y 4 ) = 1.0589 + 0.1*0.4/1.0589 = 1.0967 y 6 = y 5 + h f(x 5 , y 5 ) = 1.0967 + 0.1*0.5/1.0967 = 1.1423 y 7 = y 6 + h f(x 6 , y 6 ) = 1.1423 + 0.1*0.6/1.1423 = 1.1948 y 8 = y 7 + h f(x 7 , y 7 ) = 1.1948 + 0.1*0.7/1.1948 = 1.2534 y 9 = y 8 + h f(x 8 , y 8 ) = 1.2534 + 0.1*0.8/1.2534 = 1.3172 y 10 = y 9 + h f(x 9 , y 9 ) = 1.3172 + 0.1*0.9/1.3172 = 1.3855
with h = 0.2 y 1 = y 0 + h f(x 0 , y 0 ) = 1.0 + 0.2*0.0/1.0 = 1.0 y 2 = y 1 + h f(x 1 , y 1 ) = 1.0 + 0.2*0.2/1.0 = 1.0400 y 3 = y 2 + h f(x 2 , y 2 ) = 1.0400 + 0.2*0.4/1.0400 = 1.1169 y 4 = y 3 + h f(x 3 , y 3 ) = 1.1169 + 0.2*0.6/1.1169 = 1.2243 y 5 = y 4 + h f(x 4 , y 4 ) = 1.2243 + 0.2*0.8/1.2243 = 1.3550
Page 1 and 2: The If the Taylor Series Expansion
Page 3 and 4: What If we change the interval ? If
Page 5 and 6: Finite Differences for Derivation T
Page 7 and 8: Example : Derivation of Forward Dif
Page 9 and 10: Example : Derivation of Forward Dif
Page 11 and 12: Derivatives of Bivariate & Multivar
Page 13 and 14: Second Partial Derivatives hk k y h
Page 15: Second Partial Derivatives Followin
Page 18 and 19: Taylor Series Method The problem to
Page 20 and 21: Euler Method • First order Taylor
Page 22 and 23: 22 Euler Method 1,2,... ) , ( ) ( M
Page 24 and 25: Interpretation of Euler Method y 1
Page 26 and 27: Example 1 Use Euler method to solve
Page 28 and 29: Example 1 f 2 ( x, y) 1 x , x0 1, y
Page 30 and 31: Example 1 f 2 ( x, y) 1 x , x0 1, y
Page 32 and 33: Second Order Taylor Series Methods
Page 34 and 35: 34 High Order Taylor Series Methods
Page 36 and 37: Example 2 Second order Taylor Serie
Page 38 and 39: 38 Example 2 )) 2 (1 4 1 ( 2 ) 2 (1
Page 40 and 41: Programming Euler Method Write a MA
Page 42 and 43: Programming Euler Method f=inline('
Page 44 and 45: Example 1: Find y(0.5) if y is the
Page 48: Comparison of numerical and analyti

Taylor Series Expansion

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