Mathematical Methods for Engineers and Scientists 3: Fourier by Kwong-Tin Tang

By Kwong-Tin Tang

Pedagogical insights won via 30 years of educating utilized arithmetic led the writer to write down this set of pupil orientated books. subject matters akin to complicated research, matrix idea, vector and tensor research, Fourier research, critical transforms, traditional and partial differential equations are provided in a discursive sort that's readable and straightforward to keep on with. a number of sincerely said, thoroughly labored out examples including rigorously chosen challenge units with solutions are used to augment scholars' realizing and manipulative ability. The target is to make scholars cozy and assured in utilizing complicated mathematical instruments in junior, senior, and starting graduate courses.

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Extra resources for Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods

Example text

Sin 5 (2n − 1) L n=1 This series is rapidly convergent due to the fifth power of n in the denominator. 2 Periodically Driven Oscillator Consider a damped spring–mass system driven by an external periodic forcing function. The differential equation describing this motion is m d2 x dx +c + kx = F (t). 35) We recall that if the external forcing function F (t) is a sine or cosine function, then the steady state solution of the system is an oscillatory motion with the same frequency of the input function.

37) This square-wave repeats itself in the time interval of 2L. The number of times that it repeats itself in 1 s is called frequency ν. Clearly ν = 1/(2L). Recall that the angular frequency ω is defined as 2πν. Therefore ω = 2π π 1 = . 2L L Often ω is just referred to as frequency. 50 1 Fourier Series Now as we have shown, the Fourier series expansion of F (t) is given by ∞ F (t) = bn sin nπ t, L n=1 ⎧ ⎨ 4 n = odd, bn = nπ ⎩ 0 n = even. It is seen that the first term is a pure sine wave with the same frequency as the input square-wave.

7. Extension of a function. (a) The function is defined only between 0 and L. (b) A symmetrical extension yields an even function with a periodicity of 2L. 1. The function f (t) is defined only over the range 0 < t < 1 to be f (t) = t − t2 . Find the half-range cosine and sine Fourier expansions of f (t). 1. (a) Let the interval (0,1) be half period of the symmetrically extended function, so that 2L = 2 or L = 1. 4 Fourier Series of Nonperiodic Functions in Limited Range f (t) = 25 1 a0 + an cos nπt 2 n=1 with 1 (t − t2 )dt = a0 = 2 0 1 , 3 1 (t − t2 ) cos nπt dt, an = 2 n = 0.

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