1. The document discusses Fourier series and their application to solving partial differential equations like the heat equation. 2. It provides the general form of a Fourier series and conditions for a function to have a Fourier series representation. 3. As an example application, it uses Fourier series to solve the one-dimensional heat equation for a wire with given boundary conditions, expressing the temperature distribution over time and position as a Fourier series.

1. SUB-TRANSFORMS CALCULUS, FOURIER SERIES AND NUMERICAL
TECHNIQUES (21MAT31)
Under the Guidance of
PROF. SNEHA SRINIVAS
Department of Engineering Mathematics
HKBK COLLEGE OF ENGINEERING
2022-2023
HKBK COLLEGE OF ENGINEERING
22/1, Nagawara, Bengaluru – 560045.
E-mail: info@hkbk.edu.in, URL: www.hkbk.edu.in
TOPIC- TOPIC- FOURRIER SERIES &APPLICATION
SUBMITTED BY
Osman goni
DEPARTMENT OF
ELECTRONICS AND COMMUNICATION ENGINEERING
HKBK COLLEGE OF ENGINEERING

2.  FOURIER SERIES.
 APPLICATION OF FOURIER SERIES:-
 FORCED OSCILLATION.
 APPROXIMATION BY TRIGNOMETRICPOLYNOMIALS.

4.  As we know that TAYLOR SERIES representation of
functions are valid only for those functions which are
continuous and differentiable. But there are many
discontinuous periodic function which requires to express
in terms of an infinite series containing 'sine' and 'cosine'
terms.
 FOURIER SERIES, which is an infinite series representation
of such functions in terms of 'sine' and 'cosine' terms, is
useful here. Thus, FOURIER SERIES, are in certain sense,
more UNIVERSAL than TAYLOR'S SERIES as it applies to
all continuous, periodic functions and also to the functions
which are discontinuous in their values and derivatives.
FOURIER SERIES a very powerful method to solve ordinary
and partial differential equation, particularly with periodic
functions appearing as non-homogenous terms.

5. FOURRIER WAVE

6. FOURIER SERRIES can be generally written as,
Fourier series make use of the orthogonality relationships
of the sine and cosine functions.

8. CONDITIONS :-
1. f/(x) is bounded and singlevalue.
( Afunction f(x) is called single valued if each point
in the domain, it has unique value in the range.)
9. f(x) has it most, a finite no. of maxima end minima in
the interval.
3. fx} has at most, a finite no. of discontinuities in the
interval.
EXAMPLE:
𝑠𝑖𝑛−1
𝑥, we can say that the function 𝑠𝑖𝑛−1
𝑥 can’t be
expressed as Fourier series as it is not a single valued
function.
tanx, also in the interval (0,2π) cannot be
expressed as a Fourier Series because it is infinite at x=
π /2.

9. If function f/(x) is an even periodic function with the. period
2L (-L ≤9x ≤ L), then f/(x)cos(nπ x/L) is even while f/(x)sin(nπ x /L) is
odd.
Thus the Fourier series expansion of an even periodic
function f(x) with period 2L (-L ≤9x ≤ L) is given by
Even function

10. Odd function
If function f/(x) Is an even periodic function with the period
2L (-L ≤9x ≤ L), then f/(x)cos(nπx/L) is even while while
f/(x)sin(nπx /L) is odd.
Thus the Fourier series expansion of an odd periodic function
f/(x) with the period 2L (-L ≤9x ≤ L) is given by

14. APPLICATION
OF
FOURRIER SERIES

18. Heat on an insulated wire-
Let us first stay the heat equation
Suppose that we have a wire (or a thin metal
rod) of length L that is insulated except at the
end points. Let “x” denote the position along
the wire and let “t" denote time. See Figure-

19. Let u(x;t) denote the temperature at point x at time t. The
equation governing this setup is the so-called one-
dimensional heat equation
where k>0 is a constant (the thermal conductivity of the
material). That is, the change in heat at a specific point is
proportional to the second derivative of the heat along the
wire. This makes sense; if at a fixed t the graph of the heat
distribution has a maximum (the graph is concave down) then
heat flows away from the maximum. And vice versa

20. We will generally use a more convenient notation for
partial derivatives. We will write u, instead of δu/δt,
and we will write Uxx instead of δu/δ𝑥2
, With this
notation the heat equation becomes,
For the heat equation, we must also have some
boundary conditions. We assume that the ends of the
wire are either exposed and touching some body of
constant heat, or the ends are insulated. For example,
if the ends of the wire are kept at temperature 0, then
we must have the conditions.
U₁ = k.uxx
u(0; t) = 0 and u(X; t) = 0

21. Let us divide the partial differential equation
shown earlier by the positive number σ, define K/σ = α and
rename a f(x, t) as f (x, t) again. Then we have,
We begin with the homogeneous case f(x, t) = 0.
To implement the method of separation of variables we
write T(x, t)=z(t) y(x), thus expressing T(x, t) as the product
of a function of t and a function of x. Using z to denote
dz/dt and y', y" to denote dy/dx, dy/dx2, respectively, we
obtain,

22. Assuming z(t), y(x) are non-zero, we then have,
Since the left hand side is a constant with respect to x and
the right hand side is a constant with respect to t, both
sides must, in fact, be constant. It turns out that constant
should be taken to be non-positive, so we indicate it as -w²;
thus,

23. and we then have two ordinary differential equations,
We first deal with the second equation, writing it as,
The general solution of this equation takes the form,
y(x) = c coswx + d sinwx.
Since we want y(x) to be periodic with period L the
choices for w are,
z(t) = -aw² z(t), y"(x) = -w² y(x).
y" (x) + w² y(x) = 0.
and we then have two ordinary differential equations,
We first deal with the second equation, writing it as,
The general solution of this equation takes the form,
y(x) = c coswx + d sinwx.
Since we want y(x) to be periodic with period L the
choices for w are,
y" (x) + w² y(x) = 0.
w=
2πk
𝐿
, k= 0,1,2………

24. The choice k = 0 is only useful for the cosine; cos 0 = 1 Indexing the
coefficients c, d to correspond to the indicated choices of w, we have
solutions for the ‘y’ equation in the forms, Cₒ= constant.
Now, for each indicated choice ὠ = 2πk/L the z equation takes the form,
Which has the general solution,

26. It should be noted that this expression is a representation
of T (x, t) in the form of a Fourier series with coefficients
depending on the time, t:
The coefficients Ck(1), dk(t), k= 1,2,3, in the above
representation of T(x, t) remain undetermined, of course,
to precisely the extent that the constants Ck, dk remain
undetermined. In order to obtain definite values for these
coefficients it is necessary to use the initial temperature
distribution To(x). This function has a Fourier series
representation,

28. Thus we have, in fact, the heat equation,
Where, a0, ak, bk, k= 1,2,3, are Fourier coefficients of
initial temperature distribution To (x).