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Analysis of Taylor's Method vs. Euler's Method for Solving Initial Value Problems

This report investigates the application of Taylor's theorem in solving initial value problems by computing Taylor polynomials of degree 4 for given differential equations. The findings compare the approximations of solutions at x = 1 using Taylor polynomials with those obtained from Euler's method. Three specific problems are analyzed, including their corresponding solutions and numerical results. The research demonstrates that Taylor's method yields closer approximations than Euler's method, highlighting the effectiveness of Taylor series in numerical analysis. Other methods like the Improved Euler and Runge-Kutta methods are also discussed.

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Analysis of Taylor's Method vs. Euler's Method for Solving Initial Value Problems

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  1. Taylor‘s Method 수학과 2004023371 이창혁 수학과 2008019455 김종현 수학과 2008057903 조예원

  2. Main Theorem Taylor’s Theorem

  3. Subject for Inquiry • Problem 1 • Compute the Taylor polynomials of degree 4 for the solutions to the given initial value problems. Use these Taylor polynomials to approximate the solution at x = 1. • (i) dy/dx = x – 2y; y(0) = 1. (ii) dy/dx = y(2 – y); y(0) = 4. • Problem 2 Compare the use of Euler’s method with that of Taylor series to approximate the solution Ф(x) to the initial value problem dy/dx + y = cos x – sin x; y(0) = 0.

  4. Result of Research (1/12) • (i) dy/dx = x – 2y; y(0) = 1

  5. Result of Research (2/12) • Graphs

  6. Result of Research (3/12) • Codes

  7. Result of Research (4/12) • (ii) dy/dx = y(2 – y); y(0) = 4.

  8. Result of Research (5/12) • Graphs

  9. Result of Research (6/12) • Codes

  10. Result of Research (7/12) • dy/dx + y = cos x – sin x; y(0) = 0.

  11. Result of Research (8/12) • X = 1 • X = 3

  12. Result of Research (9/12) • Codes

  13. Result of Research (10/12) • Results

  14. Result of Research (11/12) • [ ] : error

  15. Result of Research (12/12) • [ ] : error

  16. Conclusion & Discussion • Taylor Method is approaching the exact value • closer than Euler Method. • Other approaching method… •  Improved Euler Method •  RungeKutta Method

  17. Thank You!

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