Numerical and Analytical Methods for Scientists and by Daniel Dubin

By Daniel Dubin

  • The digital element of the booklet is predicated at the familiar and hugely praised Mathematica software program package.
  • Each bankruptcy of the bookis a Mathematica computer with hyperlinks to web-based material.
  • The equipment are utilized to quite a number difficulties taken from physics and engineering.
  • The booklet covers straightforward and advaned numerical equipment utilized in smooth clinical computing.

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Extra resources for Numerical and Analytical Methods for Scientists and Engineers, Using Mathematica

Example text

Q ⌬ t f Ž t ny1 , ® Ž t ny1 . q O Ž ⌬ t 2 . 7 . 7. is Euler’s method. It is called a recursion relation because the value of ® at the nth step is determined by quantities from the n y 1st step, which were themselves determined in terms of variables at the n y 2nd step, and so on back to the initial condition at t s 0. Recursion relations like this are at the heart of most numerical ODE solvers. Their differences lie mainly in the degree of approximation to the integral in Eq. 6.. To see how Euler’s method works, we will write a program that can be used to solve any ODE of the form of Eq.

The resulting recursion relation is called a predictor᎐corrector method of order 2. The method is second-order accurate, because over a fixed time interval T the number of steps taken is Tr⌬ t and the total error scales as ŽTr⌬ t . ⌬ t 3 s T ⌬ t 2 . The method consists of the following two lines: an initial prediction for ® at the nth step, which we assign to a variable ®1 , and the improved correction step, given by Eq. , making use of the prediction: ®1 s ® Ž t ny1 . q ⌬ t f Ž t ny1 , ® Ž t ny1 .

S ® Ž t ny1 . q ⌬ t f Ž t ny1 , ® Ž t ny1 . q f Ž t n , ® Ž t n . q O Ž ⌬ t 3 . 11 . 2 Since the error of the method is order ⌬ t 3 in a single step, Eq. 11. is a distinct improvement over Euler’s method, Eq. 7.. However, there is a catch. Now ®Ž t n . appears on the right-hand side of the recursion relation, so we can’t use this equation as it stands to solve for ®Ž t n .. , but for general f that is nontrivial. x What we need is some way to replace ®Ž t n . , which we will then use in Eq. 11.

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