Numerical Analysis 10.4. Taylor's Method for Differential Equations скачать в хорошем качестве

Numerical Analysis 10.4. Taylor's Method for Differential Equations

This video is a lecture on Section 10.4: Taylor’s Method in Numerical Analysis. It explores how to generalize Euler's method to achieve a higher order of convergence for solving differential equations. Core Concept and Generalization Goal: The method aims to find an approximate solution z i+1 ​ =z i ​ +hF(t i ​ ,z i ​ ,h), where F is a function designed to provide a higher speed of convergence than the standard Euler's method [00:11]. Local Truncation Error: The accuracy of the method is determined by the local truncation error (τ i+1 ​ ), which is the difference between the exact solution's difference quotient and the function F [01:55]. Order of Convergence: If the local truncation error is of order α (i.e., proportional to h α ), then the overall approximation error will also be of order α, provided the function F is Lipschitz continuous [03:10]. Derivation using Taylor Polynomials Taylor Expansion: The method is derived by using a high-order Taylor polynomial approximation of the solution y(t) [04:36]. Computing Derivatives: To find the terms for the Taylor series, the differential equation y ′ =f(t,y) is repeatedly differentiated. The second derivative y ′′ is found by differentiating the right-hand side f(t,y) using the chain rule [05:48]. Higher-order derivatives (y ′′′ ,y (4) , etc.) become increasingly complex but can be computed if f is sufficiently differentiable [06:41]. Defining F: By factoring out the step size h from the Taylor expansion terms, a formula for the function F is established, which includes the derivatives of the composite function [09:14]. Practical Examples Second-Order Method: The instructor demonstrates a second-order method (z i+1 ​ =z i ​ +hf+ 2 h 2 ​ f (1) ) applied to a specific differential equation. Numerical results show that the error is significantly smaller than Euler's method and reduces by approximately a factor of four when the step size is halved [13:51]. Third-Order Method: A third-order method is then introduced, adding an extra term involving h 3 /6 and the second derivative of the composite function. This further reduces the error, allowing for high accuracy even with relatively large step sizes [17:04]. Conclusion and Trade-offs While Taylor’s method allows for arbitrarily high-order approximations, the instructor notes a significant drawback: the formulas for higher derivatives become very long and computationally expensive to evaluate [19:12]. Additionally, performing too many arithmetic calculations can increase the risk of rounding errors [19:55].