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6. Elementary Numerical Analysis by Atkinson-Han, Wiley Student Edition

ii) Advanced Engineering Mathematics by Michael Greenberg -Pearson publishers,

iii) Introductory Methods of Numerical Analysis by S.S. Sastry, PHI

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126. partial differential equations and all have plenty of real life applications

For example

® Fluid mechanies is used 10 understand how the ciculaory system works, how to
rockets and plans o fly and even to some extent how the weather be
Heat and mass transfer is used 10 understand how drug delivery devices work, ho
Kidney dialysis works, and how to control heat for temperatte-senshive things, I
probably also explains why thermoses work!
+ Electromagnetism is used for all electricity out there, and everything that inv

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178. ¡Solution First, we assume that the

We separate x function of only Lon one side and a funcion of onl

[CASEN

In his ease we know th solution tothe differential equation is

179. MATHEMATICS «I PARTIAL DIFFERENTIAL EQUATIONS

applying the second boundary condition, and using the above rest. gi

ni tis case

Applying he boundary conditions give

0=¢0)=a 0=9(L)

in this case the only solution isthe trivial solution and so À = 0 is not ane

sinh (LA)

ans Sin(L-4)0. We therefore we m
an only get the trivia inthis ease.

or, there will be no negative eigen values for this boundary value problem. The

complete ist of eigenvalues and eigen functions for this problem are then
(Ey we

the im.

DEPARTMENT OF HUMANITIES &

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180. MATHEMATICS «I PARTIAL DIFFERENTIAL EQUATIONS

dí

we can finally write down a solution, Note however that we have in fat found infinitely
‘many solutions sine there are infinitely many solutions (ie, eigen functions) 10 the spatial
problem

4x)
un (2,1) = e

The product solution ua 10 acknowledge that each value ofn will ici a different solution

Also note that we've changed the ce in the solution tothe time problem to Ba to denote the
fact hat it will probably be different foreach value

DEPARTMENT OF HUMANITIES &

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181. DEPARTMENT OF HUMANITIES & SCIENCE

182. MATHEMATICS «I DOUBLE AND TRIPLE INTEGRALS

Introduction
The multiple integral i «definite integral ofa funcion of more than one rel variab
instance, Ar, y) or, y, 2. Integrals of a function of two variables over a n
called Double integrals, and integrals of a function of three variables over
called Triple ing
Just asthe definite integral of a positive function of one variable rp
region between the graph of the function and the saxis, the double integral of a postive
function of two variables represents the volume of the region between the surface defined by

the function (onthe Ihree-dimensional Cartesian plane where :=/Ux, and the plane which

contains its domain, If there are more variables, a multiple integral wil yield hyper olumes of

multidimensional functions. Double integrals are used to ealulate the arca of a region, th
volume under a surface, and the average value of a function of two variables o
angular region.

Definition of double integral : suppose we have a region inthe plane R and a function

(99). then double integral ff, f(t,y)dA is defined as follows

Divide the region R into small pieces, numbered from 1 to n.Let AA; be the area ofthe 1%

piece and also pik a point (x,y. in that piece

nthe sum ES. (7) À)

DEPARTMENT OF HUMANITIES & SCIENCES [JBMROEE|EAMGERCODE MRD)

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184. DEPARTMENT OF HUMANITIES & SCIENCES.