Transforms and Partial differential equations UNIT I
I year / I sem
TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS FOURIER SERIES
9
Fourier series – series – Odd Odd and even functions – functions – Half Half range sine series – series – Half Half range cosine series – series – Complex Complex form of Fourier Series – Series – Parseval’s Parseval’s identify – Harmonic Harmonic Analysis. UNIT II
FOURIER TRANSFORM
9
Fourier integral theorem (without proof) – Sine and Cosine transforms – Properties (without Proof) – Transforms of simple functions – Convolution theorem – theorem – Parseval’s Parseval’s identity – Finite Finite Fourier transform – transform – Sine Sine and Cosine transform. UNIT III
Z -TRANSFORM AND DIFFERENCE EQUATIONS
9
Z-transform - Elementary properties (without proof) – proof) – Inverse Inverse Z – Z – transform – transform – Convolution Convolution theorem Formation of difference equations – equations – Solution Solution of difference equations using Z - transform. UNIT IV
PARTIAL DIFFERENTIAL EQUATIONS
9
Solution of First order partial differential equation reducible to standard forms – Lagrange’s linear equation – Linear partial differential equations of second order and higher order with constant coefficients. UNIT V
BOUNDARY VALUE PROBLEMS
9
Solutions of one dimensional wave equation – equation – One One dimensional heat equation – Steady Steady state solution of two-dimensional heat equation (Insulated edges excluded) – excluded) – Fourier series solutions in Cartesian coordinates. TUTORIAL :15 TOTAL: 60 TEXT BOOKS
1. Andrews, L.A., and Shivamoggi B.K., “Integral Transforms for Engineers and Applied Applied Mathematicians”, Macmillen , New York ,1988. 2. Grewal, B.S., “Higher Engineering Mathematics”, Thirty Sixth Edition, Khanna Publishers, Delhi, 2001. 3. Kandasamy, P., Thilagavathy, K., and Gunavathy, K., “Engineering Mathematics Volume III”, S. Chand & Company ltd., New Delhi, 1996. REFERENCES
1. Narayanan, S., Manicavachagom Pillay, T.K. and Ramaniah, G., “Advanced Mathematics for Engineering Students”, Volumes II and III, S. Viswanathan (Printers and Publishers) Pvt. Ltd. Chennai, 2002. 2. Churchill, R.V. and Br own, own, J.W., “Fourier Series and Boundary Value Problems”, Fourth Edition, McGraw-Hill Book Co., Singapore, 1987.
Transforms and Partial differential equations
I year / I sem
CHAPTER 1 FOURIER SERIES 1.1 PERI PERIO ODIC FUNCT UNCTIONS A f unction unction is is said to have a per per iod iod T if if f f or all r all x,
pos positive cons constant. The leas least value of of T>0 T>0 is is called the per per iod iod of of
, wher her e T is is a .
EXA XAMPLE MPLES S 1.1
We know know that 4
,6
= sin x = sin (x + 4
) =
Ther Ther ef or e the f unction unction has has per per iod iod 2
…
,
, etc. However ever , 2 is is the leas least value and ther ther ef or e is is the per per iod iod of of f( f(x x). Similar imilar ly ly cos cos x is is a per per iodic iodic f unction unction with the per per iod iod 2
and tan x has has per per iod iod
.
1.2 DIRICHLET DIRICHLET’S CONDIT NDITIONS
A f unction unction ser ies ies of of the the f or m
def def ined ined in c
x
c+2l can be expanded as as an inf inf inite inite tr tr igonometr igonometr ic ic
+
pr ovided ovided
1.
is singleingle- valued and f inite inite in (c , c+2l)
2.
is continuous continuous or or piecew piecewise continuous continuous with f inite inite number number of of f f inite inite dis discontinuities continuities in (c , c+2l).
3.
has has no or or f f inite inite number number of of maxima maxima or or minima minima in (c , c+2l).
1.3 EU EULER’S FOR FORMULAS
If a a f unction unction ser ies ies
def def ined ined in (c , c+2l) can be expanded as as the inf inf inite inite tr tr igonometr igonometr ic ic
+
[ For mulas mulas given above f or
then
and
ar ar e called Euler’s Euler’s f or mulas mulas f or F r Four our ier ier coeff coeff icients] icients]
1
