By A C King; J Billingham; S R Otto

The authors concentrate on developing suggestions analytically, and studying their that means; MATLAB is used widely to demonstrate the cloth. the various labored examples, in keeping with fascinating genuine international difficulties, the massive choice of workouts, together with a number of lengthier initiatives, the huge assurance, and transparent and concise presentation will attract undergraduates.

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**Extra info for Differential equations : linear, nonlinear, ordinary, partial**

**Example text**

We note that (x − 1) (x − 1)(1 + x) (x − 1)(1 + x) 1+x Q = = = . P (x2 − x) x(x − 1) x At this point we need to recall that we want information near x = 1, so we rewrite x as 1 − (1 − x), and hence (x − 1) 2 − (1 − x) Q = . P {1 − (1 − x)} Expanding in powers of (1 − x) using the binomial expansion gives (x − 1) Q 1−x =2 1− P 2 {1 + (1 − x) + (1 − x)2 + · · · }, 25 26 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS which is a power series in (x − 1) that is convergent provided |x − 1| < 1.

2 The general solution is this particular integral plus the complementary function (yp (x) + yh (x)). 2 The Generating Function for Pn (x) In order to make a more systematic study of the Legendre polynomials, it is helpful to introduce a generating function, G(x, t). This function is deﬁned in such a way that the coeﬃcients of the Taylor series of G(x, t) around t = 0 are Pn (x). We start with the assertion that G(x, t) = (1 − 2xt + t2 )−1/2 = ∞ Pn (x)tn . 3) 35 36 LEGENDRE FUNCTIONS Just to motivate this formula, let’s consider the ﬁrst terms in the Taylor series expansion about t = 0.

The total momentum in the direction of i is the blooddensity times the integral ∇φ · idV = I= V (sφ)|r=R dS, S where S is the surface of V . Hence show that ∞ I = 2πR2 1 cn n=0 −1 sPn (s)ds = 4 2 3 πR c1 = V 3 2 1 sf (s) ds. −1 EXERCISES (e) Use the answer to part (c) to show that 4I = (1−a)(a2 +4+3a)V U . (f) Explain how this model could be made more realistic in terms of the ﬂuid mechanics and the physiology. You may like to refer to Pedley (1980) for some ideas on this. 57 CHAPTER THREE Bessel Functions In this chapter, we will discuss a class of functions known as Bessel functions.