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Notes-PDE pt2 - Second Order Linear Partial Differential...

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Second Order Linear Partial Differential Equations Part II Fourier series; Euler-Fourier formulas; Fourier Convergence Theorem; Even and odd functions; Cosine and Sine Series Extensions; Particular solution of the heat conduction equation Fourier Series Suppose f is a periodic function with a period T = 2 L . Then the Fourier series representation of f is a trigonometric series (that is, it is an infinite series consists of sine and cosine terms) of the form = + + = 1 0 sin cos 2 ) ( n n n L x n b L x n a a x f π π Where the coefficients are given by the Euler-Fourier formulas : - = L L m dx L x m x f L a π cos ) ( 1 , m = 0, 1, 2, 3, … - = L L n dx L x n x f L b π sin ) ( 1 , n = 1, 2, 3, … The coefficients a ’s are called the Fourier cosine coefficients (including a 0 , the constant term, which is in reality the 0-th cosine term), and b ’s are called the Fourier sine coefficients .
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Note 1: If f is piecewise continuous, then the definite integrals in the Euler- Fourier formulas always exist (i.e. even in the cases where they are improper integrals, the integrals will converge). On the other hand, f needs not to be piecewise continuous to have a Fourier series. It just needs to be periodic. However, if f is not piecewise continuous, then there is no guarantee that we could find its Fourier coefficients, because some of the integrals used to compute them could be improper integrals which are divergent. Note 2: Even though that the “=” sign is usually used to equate a periodic function and its Fourier series, we need to be a little careful. The function f and its Fourier series “representation” are only equal to each other if, and whenever, f is continuous. Hence, if f is continuous for - < x < ∞, then f is exactly equal to its Fourier series; but if f is piecewise continuous, then it disagrees with its Fourier series at every discontinuity. (See the Fourier Convergence Theorem below for what happens to the Fourier series at a discontinuity of f .) Note 3: Recall that a function f is said to be periodic if there exists a positive number T , such that f ( x + T ) = f ( x ), for all x in its domain. In such a case the number T is called a period of f . A period is not unique, since if f ( x + T ) = f ( x ), then f ( x + 2 T ) = f ( x ) and f ( x + 3 T ) = f ( x ) and so on. That is, every integer-multiple of a period is again another period. The smallest such T is called the fundamental period of the given function f . A special case is the constant functions. Every constant function is clearly a periodic function, with an arbitrary period. It, however, has no fundamental period, because its period can be an arbitrarily small real number. The Fourier series representation defined above is unique for each function with a fixed period T = 2 L . However, since a periodic function has infinitely many (non- fundamental) periods, it can have many different Fourier series by using different values of L in the definition above. The difference, however, is really in a technical sense. After simplification they would look the same.
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