Taylor Series Expansion

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Example 1: Find y(0.5) if y is the solution of IVP y' = -2x-y, y(0) = -1 using Euler's method with step length 0.1. Also find the error in the approximation. Solution: f(x, y) = -2x - y, y 1 = y 0 + h f(x 0 , y 0 ) = -1 + 0.1* (-2*0 - (-1)) = -0.8999 y 2 = y 1 + h f(x 1 , y 1 ) = -0.8999 + 0.1* (-2*0.1 - (-0.8999)) = -0.8299 y 3 = y 2 + h f(x 2 , y 2 ) = -0.8299 + 0.1* (-2*0.2 - (-0.8299)) = -0.7869 y 4 = y 3 + h f(x 3 , y 3 ) = -0.7869 + 0.1* (-2*0.3 - (-0.7869)) = -0.7683 y 5 = y 4 + h f(x 4 , y 4 ) = -0.7683 + 0.1* (-2*0.4 - (-0.7683)) = -0.7715
Example 2: Use Eulers method to solve for y[0.1] from y' = x + y + xy, y(0) = 1 with h = 0.01 also estimate how small h would need to obtain four-decimal accuracy. Solution : f(x, y) = x + y + xy, y 1 = y 0 + h f(x 0 , y 0 ) = 1.0 + .01*(0 + 1 + 0*1) = 1.01 y 2 = y 1 + h f(x 1 , y 1 ) = 1.01 + .01*(0.01 + 1.01 + 0.01*1.01) =1.02 y 3 = y 2 + h f(x 2 , y 2 ) = 1.02 + .01*(0.02 + 1.02 + 0.02*1.02) =1.031 y 4 = y 3 + h f(x 3 , y 3 ) = 1.031 + .01*(0.03 + 1.031 + 0.03*1.031) =1.042 y 5 = y 4 + h f(x 4 , y 4 ) = 1.042 + .01*(0.04 + 1.042 + 0.04*1.042) = 1.053 y 6 = y 5 + h f(x 5 , y 5 ) = 1.053 + .01*(0.05 + 1.053 + 0.05*1.053) = 1.065 y 7 = y 6 + h f(x 6 , y 6 ) = 1.065 + .01*(0.06 + 1.065 + 0.06*1.065) = 1.076 y 8 = y 8 + h f(x 7 , y 7 ) = 1.076 + .01*(0.07 + 1.076 + 0.07*1.076) = 1.089 y 9 = y 9 + h f(x 8 , y 8 ) = 1.089 + .01*(0.08 + 1.089 + 0.08*1.089) = 1.101 y 10 = y 10 + h f(x 9 , y 9 ) = 1.101 + .01*(0.09 + 1.101 + 0.09*1.101) = 1.114
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 46 and 47: Example 3: Solve the differential e
Page 48: Comparison of numerical and analyti

Taylor Series Expansion

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