1 LECTURE NOTES OF APPLIED MATHEMATICS–III (Sub Code: BTC 305) COURSE CONTENT 1) Numerical Analysis 2) Fourier Series 3) Fourier Transforms & Z-transforms 4) Partial Differential Equations 5) Linear Algebra 6) Calculus of Variations Text Book: Higher Engineering Mathematics by Dr. B.S.Grewal (36th Edition – 2002) Khanna Publishers,New Delhi Reference Book: Advanced Engineering Mathematics by E. Kreyszig (8th Edition – 2001) John Wiley & Sons, INC. New York FOURIER SERIES DEFINITIONS : A function y = f(x) is said to be even, if f(-x) = f(x). The graph of the even function is always symmetrical about the y-axis. A function y=f(x) is said to be odd, if f(-x) = - f(x). The graph of the odd function is always symmetrical about the origin. For example, the function f(x) = x in [-1,1] is even as f(-x) = x x = f(x) and the function f(x) = x in [-1,1] is odd as f(-x) = -x = -f(x). The graphs of these functions are shown below :
2 Graph of f(x) = x Graph of f(x) = x Note that the graph of f(x) = x is symmetrical about the y-axis and the graph of f(x) = x is symmetrical about the origin. 1. If f(x) is even and g(x) is odd, then h(x) = f(x) x g(x) is odd h(x) = f(x) x f(x) is even h(x) = g(x) x g(x) is even For example, 1. h(x) = x2 cosx is even, since both x2 and cosx are even functions 2. h(x) = xsinx is even, since x and sinx are odd functions 3. h(x) = x2 sinx is odd, since x2 is even and sinx is odd. a 2. If f(x) is even, then a a f ( x)dx 2 f ( x)dx 0 a f ( x)dx 0 3. If f(x) is odd, then a For example, a a a 0 cos xdx 2 cos xdx, a and as cosx is even sin xdx 0, as sinx is odd a
3 PERIODIC FUNCTIONS :A periodic function has a basic shape which is repeated over and over again. The fundamental range is the time (or sometimes distance) over which the basic shape is defined. The length of the fundamental range is called the period. A general periodic function f(x) of period T satisfies the condition f(x+T) = f(x) Here f(x) is a real-valued function and T is a positive real number. As a consequence, it follows that f(x) = f(x+T) = f(x+2T) = f(x+3T) = ….. = f(x+nT) Thus, f(x) = f(x+nT), n=1,2,3,….. The function f(x) = sinx is periodic of period 2 since Sin(x+2n) = sinx, n=1,2,3,…….. The graph of the function is shown below : Note that the graph of the function between 0 and 2 is the same as that between 2 and 4 and so on. It may be verified that a linear combination of periodic functions is also periodic. FOURIER SERIES A Fourier series of a periodic function consists of a sum of sine and cosine terms. Sines and cosines are the most fundamental periodic functions. The Fourier series is named after the French Mathematician and Physicist Jacques Fourier (1768 – 1830). Fourier series has its application in problems pertaining to Heat conduction, acoustics, etc. The subject matter may be divided into the following sub topics.
4 FOURIER SERIES Series with arbitrary period Half-range series Complex series Harmonic Analysis FORMULA FOR FOURIER SERIES Consider a real-valued function f(x) which obeys the following conditions called Dirichlet’s conditions : 1. f(x) is defined in an interval (a,a+2l), and f(x+2l) = f(x) so that f(x) is a periodic function of period 2l. 2. f(x) is continuous or has only a finite number of discontinuities in the interval (a,a+2l). 3. f(x) has no or only a finite number of maxima or minima in the interval (a,a+2l). Also, let a0 an bn 1 l 1 l 1 l a 2l f ( x)dx (1) a a 2l a a 2l a n f ( x) cos l xdx, n 1,2,3,..... (2) n f ( x) sin xdx, l n 1,2,3,...... (3) Then, the infinite series a0 n n (4) an cos x bn sin x 2 n 1 l l is called the Fourier series of f(x) in the interval (a,a+2l). Also, the real numbers a0, a1, a2, ….an, and b1, b2 , ….bn are called the Fourier coefficients of f(x). The formulae (1), (2) and (3) are called Euler’s formulae. It can be proved that the sum of the series (4) is f(x) if f(x) is continuous at x. Thus we have a n n f(x) = 0 an cos (5) x bn sin x ……. 2 n 1 l l Suppose f(x) is discontinuous at x, then the sum of the series (4) would be 1 f ( x ) f ( x ) 2 where f(x+) and f(x-) are the values of f(x) immediately to the right and to the left of f(x) respectively.