Mathematical Tools for Physics
10—Partial Differential Equations 313 10.30 Sum the series Eq. ( 24 ) to get a closed-form analytic expression for the temperatu ...
10—Partial Differential Equations 314 10.37( From the preceding problem you can have a potential, a solution of Laplace’s equati ...
Numerical Analysis You could say that some of the equations that you encounter in describing physical systems can’t be solved in ...
11—Numerical Analysis 316 where the last term gives an estimate of the error:+h^2 f′′(0)/ 8. As an example, interpolate the func ...
11—Numerical Analysis 317 − 3 k −k 0 k 3 k f(k) =f(0) +kf′(0) + 1 2 k^2 f′′(0) + 1 6 k^3 f′′′(0) +···. (2) I want to isolatef(0) ...
11—Numerical Analysis 318 11.2 Solving equations Example:sinx−x/2 = 0 From the first graph, the equation clearly has three real ...
11—Numerical Analysis 319 Such iterative procedures are ideal for use on a computer, but use them with caution, as a simple exam ...
11—Numerical Analysis 320 W is a factor that can be chosen greater than one to increase the correction or less than one to decre ...
11—Numerical Analysis 321 As usual, the geometric approach doesn’t indicate the size of the error, so it’s back to Taylor’s seri ...
11—Numerical Analysis 322 To eliminate the largest source of error, theh^3 term, multiply the first equation by 8 and subtract t ...
11—Numerical Analysis 323 The previous example of the derivative ofsinxatx= 0. 2 withh= 0. 1 gives, using this formula: 1 2. 4 [ ...
11—Numerical Analysis 324 11.4 Integration The basic definition of an integral is a limit of the sum, ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 ∑ f(ξi ...
11—Numerical Analysis 325 The error for expression (b) requires another expansion, error (b) =hf(h)− ∫h 0 dxf(x) =h [ f(0) +hf′( ...
11—Numerical Analysis 326 This is known as Simpson’s rule. Simpson’s Rule Before applying this last result, I’ll go back and der ...
11—Numerical Analysis 327 The error term (the “truncation error”) is h 3 [ f(−h) + 4f(0) +f(−h) ] − ∫h −h dxf(x)≈ 1 12 .^1 3 h^5 ...
11—Numerical Analysis 328 Gaussian Integration If the integrand is known at all points of the interval and not just at discrete ...
11—Numerical Analysis 329 the Legendre polynomial of second order. This approach to integration, known as Gaussian integration, ...
11—Numerical Analysis 330 You can now iterate this procedure using this newly found value ofyas a new starting condition to go f ...
11—Numerical Analysis 331 wherejandkare the order ofh. Note that because this term appears in an expression multiplied byh^2 , i ...
11—Numerical Analysis 332 You can look up a fancier version of this called the Runge-Kutta-Fehlberg method. It’s one of the bett ...
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