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in the last lecture we were looking at how
to analyse convergence of non-linear procedures
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for solving nonlinear algebraic equations
iterative procedures and we said that in general
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we could write any iterative method for solving
nonlinear algebraic equations as 1 equation
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i want to solve for f of x=0 x belongs to
rn and f is a nx1 vector this is nx1 function
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vector
any iterative method to solve this problem
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numerically can be written as xk+1=g of xk
so the old guess generates a new guess and
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this process is continued till differences
between 2 successive solutions become negligible
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or norm of f of x goes close to 0 if you look
carefully this is a nonlinear difference equation
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the index here is iteration index k so the
guess is generated from the old guess g is
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the transformation i showed you that all the
methods that we are looking at iterative methods
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can be expressed in this form now just like
we had conditions for analysing linear difference
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equations
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earlier we had looked at equations of this
type=b xk and for this particular case we
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had derived necessary and sufficient condition
for norm xk to go to 0 as k goes to infinity
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in this case we had a very very powerful result
that is spectral radius of b is strictly less
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than 1 this was the situation for the linear
difference equation we had got this kind of
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a generic form by analysing iterative methods
for solving linear algebraic equations
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we could derive a very very powerful result
here based on the eigen value of matrix b
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we wanted all eigen values of matrix b to
be inside the unit circle now coming to nonlinear
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equations it is not possible to prove so strong
result we can only give sufficient conditions
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it is not possible to come up with necessary
and sufficient conditions for a general nonlinear
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difference equation of that form we have to
come up with some kind of local condition
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these local conditions i described through
contraction mapping theorem or contracting
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mapping principle which forms the foundation
of analyzing iterative schemes and 1 special
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that we saw was the operator g
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g is something g maps a ball around x not
of radius r to where r was a special radius
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it should be >= a certain number that we had
defined yesterday so if is a mapping which
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maps a ball of radius r*itself and if g is
a contraction map 1 simple way of finding
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out whether g is contraction map over u was
to see whether dou g/dou x was strictly <1
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or <= theta which is <1 for all x
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if the partial derivative of g with respect
to x has any induced norm strictly <1 everywhere
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then we know that map g is the contraction
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if map g is contraction then in neighborhood
of x not of radius r we were assured of existence
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of a solution we are assured that any sequence
starting from any point in this region would
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converse to the solution so solution of this
problem is x*=g of x* x* is the solution and
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if this condition is met everywhere in this
ball then it is sufficient condition to say
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that any sequence generated by this difference
equation will converse to this solution
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solution is x*=g(x*) just to draw the parallel
i am writing this just to draw parallel we
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had a sufficient condition here that if norm
of b is strictly <1 then also this condition
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holds that xk goes to 0 as k goes to infinity
so we said this is the weaker condition than
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this necessary and sufficient condition but
this condition helped us to analyse to come
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up with diagonal dominance and all kinds of
other theorems which were used to analyse
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iterative schemes
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likewise analogous to this when i come here
this contraction mapping principle tells us
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very very important things 1 is that if g
is a contraction map if its local derivative
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has if you g (x) to be dx then local derivative
of g with respect to x will be matrix b and
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any induced norm of matrix b being strictly
less than 1 is the condition that we are looking
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for there so they coincide this particular
equation only difference there was the solution
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the point where we wanted to reach was 000
origin
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in this case we want to reach a solution x*=g(x*)
it is possible to make everything in terms
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of 000 if you redefine or shift the origin
to x* then you can make the 2 problems almost
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equivalent but that is not important it is
just matter of shifting the origin what is
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important is that there is an analogous sufficient
condition here for nonlinear difference equations
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it does not help us here to look at the spectral
radius of this matrix
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it does not help here the reasons which are
difficult to explain as a part of this course
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but we have to use only norm and any induced
norm if any induced norm is strictly < 1 in
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some region then you are guaranteed that there
exist a solution to this difference equation
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in that region the solution is unique and
the third point which was very very important
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start from any initial guess you will converse
to that solution
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start from any initial guess in that region
you will converse to the solution x*=g(x*)
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so these are very very important findings
of this particular theorem in general it is
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more difficult to apply this theorem for a
complex real problem nevertheless it gives
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us some insights for example you can try and
make the sufficient conditions meet by ensuring
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that dou g/dou x has induced norm <1 you can
try to do this
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if there is some problem in solving some nonlinear
equations we can these are sufficient conditions
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remember that if this conditions are violated
even then the conversions can occur these
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are not necessary conditions but this happens
convergence will occur just like in this case
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when we were talking about linear algebraic
equations if norm of these <1 spectral radius
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is <1 it is a sufficient condition
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but if norm of b is >1 even then convergence
can occur because convergence depends upon
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the spectral radius spectral radius can be
<1 similarly contraction mapping principle
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gives us a sufficient condition for convergence
it is not a necessary condition if you meet
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the sufficient condition you are guaranteed
to converge so this gives at least some handle
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to understand how the convergence occurs from
that view point this is important
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those of you who are solving large algebraic
equations as a part of your research m tech
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or phd and hit into problems you should look
at the norm of the jacobian i mean at least
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that much you should remember look at the
norm of the jacobian i try to see whether
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you can make the norm of the jacobian <1 you
have good chances of convergence just to illustrate
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this idea of contraction map i just give you
1 example here
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i want to solve simultaneously these are 2
nonlinear algebraic equations which i want
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to solve simultaneously if i write this – this=0
and this-this=0 then this is f(x)=0 there
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are 2 functions f1zy and f2zy i want to find
out a solution for this particular problem
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i am formulating an iteration scheme here
zk+1=1/16-1/4 yk square and
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i have just formed 1 iteration scheme this
is not the only way to form iteration scheme
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i am showing you 1 possible way of forming
the iteration scheme this is a jacoby type
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iteration scheme what would be the gauss-seidel
kind of iteration scheme if i were to use
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zk+1 here it will become gauss-seidel type
iteration scheme this is the jacoby type iteration
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scheme now what i am going to do here is
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i have this scheme which is yz=g(yz) where
g i this right hand side function
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i am considering this unit ball let us say
x not my initial guess is 0 1 no no my initial
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guess
is x0=0 0 and i am considering this unit ball
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of radius 1 in the neighborhood of 0 0 so
i am looking at
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now what is
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this infinite norm i am taking some point
xi and some point xj x here is x consist of
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y and z x is the vector consisting of 2 elements
y and z now i am looking at this
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what is the infinite norm infinite norm is
maximum of the absolute value of the elements
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what i am doing is i am taking xi-gxz it has
2 elements i am just taking the maximum of
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these 2 absolute values will be the norm i
am just using definition of infinite norm
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nothing else
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just this is definition of infinite norm
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so you can show that this is <=
max of i am skipping in between steps you
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should fill them up just go back and look
at why this step comes from this
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you can prove this in equalities that is this
particular difference infinite norm of this
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difference is 00:21:22,110
to show that in this particular case you can
show that
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i am using here the fact that the elements
are drawn from the unit ball so that is why
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these types have been written and essentially
using this inequalities what you can show
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is that gi-gj uing these inequalities you
can also do analysis using the derivative
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of this and taking this infinite norm you
can also do analysis using derivative of this
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right hand side jacobian matrix and infinite
norm of the jacobian matrix that analysis
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is also possible
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in this particular case we have found that
if we apply g on any xi and xj then this inequality
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holds if this inequality holds what it means
is that this constant on the right hand side
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is <1 so this is strictly <1 so this g map
is a contraction if g map is a contraction
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i am guaranteed that there exist a solution
in this unit ball the solution is unique and
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starting anywhere in this unit ball this is
in reference to the infinite norm
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it will be a square it will look like a square
we have seen this how does the unit ball look
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like in different norms starting from any
initial guess within this the iterations will
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converge to the solution so this we are guaranteed
because we are able to prove this in equality
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here for this particular x=g(x) what is important
here is that just looking at or just developing
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this inequality this is infinite i am guaranteed
that a solution exist in the ball
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i am guaranteed that i start from anywhere
and i will reach the solution and this iteration
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scheme is going to work that is what i know
from this analysis just do not bother about
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these in between steps assume that this sequence
is true because our aim is not to do this
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algebra you can work on this algebra later
more important is that by doing this algebra
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i can show that infinite norm of gi-gj/xi-xj
for any i j i can prove this
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i take any 2 points in this ball apply g on
both the points the new points will have a
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distance which is closer than the original
2 points that is the main thing if that happens
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we are assured that the solution exist we
are assured that starting from x not we will
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reach the solution moreover from any initial
guess in this region if we start we will still
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reach the solution that is the important point
it is difficult to do this analysis for a
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very large scale nonlinear system
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nevertheless it is important to get this insight
that how does 1 look at analysis of convergence
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of iterative schemes for solving nonlinear
algebraic equations because most of the times
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you will be actually dealing with nonlinear
algebraic equations large scale in your computation
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work because most of the chemical engineering
problems 999% of them are nonlinear problems
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reactions of heavy transfer occurs and turbulence
and always things will make the life very
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very complex
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we have to work with a set of nonlinear algebraic
equations what is it that governs the convergence
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we can get some clues if you can show that
the iteration scheme that you have formed
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actually is a contraction map difficult to
show in general for large scale system but
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this does give you insight which is very very
important that is what you should carry i
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want to stop here i do not want to get into
too much details
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in the notes i have given some more detailed
discussion on newton’s method so there are
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special theorems for convergence of newton’s
method and more than the proof and the theorem
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statement i have tried to give some qualitative
insights as to how to interpret those theorems
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i have not included the proof the proof can
be found in any of the text books on nonlinear
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systems like one of the very well known textbooks
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so you can find proofs there but the interpretation
is quite important as to how do you make convergence
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occur so typically if you have formed iteration
scheme in this case i worked with i did not
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take a derivative but you could also try to
see for this particular system you can work
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this out
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you can try to see whether dou g/dou x infinite
norm if this is strictly <1 in the region
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where you are trying to operate or trying
to solve the problem or dou g/dou x 1 norm
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is strictly <1 if these conditions are met
then we are guaranteed that the solution exist
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and we will reach the solution these are some
why infinite norm and why 1 norm because they
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are easy to compute infinite norm and 1 norm
are easy to compute
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other norms like 2 norms will require eigen
value computation other than that 1 norm and
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infinite norm are easy to compute so you can
quickly make a judgment what is going wrong
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when you are solving the problem this brings
us to an end of methods for solving nonlinear
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algebraic equations we have looked at different
concepts we have looked at how to solve them
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using different algorithms we just briefly
touched upon idea of condition number
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also we very very briefly touched upon the
idea of convergence of iterative schemes we
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have not gone deep into it but at least you
know about what is the tool or what is the
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machinery that is used for actually looking
at this problem let us move on to solving
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ordinary differential equations initial value
problems now what i want to do next is before
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i proceed again we go back to our global diagram
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so our global diagram was so we have this
original problem then we use approximation
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theory to come up with transformed problem
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so we have been calling it transformed computable
forms and then we said there are 4 tools 1
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is ax=0 this tool set which we will be using
and the other tool set was f(x)=0 so solving
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nonlinear algebraic equations solving linear
algebraic equations this is the second tool
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set that we have
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the third tool set that i am going to look
at is od-ivp because in many cases the transformed
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problem is an od initial value problem i talk
about a method later on how do you transform
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a boundary value problem into initial value
problem actually not just one initial value
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problem a series of initial value problems
which are then solved iteratively the fourth
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tool is stochastic methods but we are not
going to get into this
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so right now we have done this how to solve
ax=b we looked at many many methods we looked
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at many issues that are associated with this
we have looked at f(x)=0 and now we are moving
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to od-ivp all these after all is going to
give us approximate solution
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this is going to give an approximate solution
to the original problem so moving on to solving
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ordinary differential equations initial value
problem
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general form that the types of equation that
i am going to look at is of this type dx/dt=f(x
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t) where f is the function vector
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and x belongs to rn what i am given apart
from this differential equation model i am
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also given
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initial condition at time=0 now before i move
on let me explain one notational difference
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that we will have in this case if you are
dealing with vectors we will have to deal
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with 3 different attached indices with the
vector
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suppose x is my vector here i-th element of
the vector will be given by xi this notation
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we have been using even earlier bracket k
will indicate k-th iteration now additional
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complexity comes in we have time so time will
come here so there are 3 things attached to
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the vector in some cases you will have i-th
component of the vector you will have time
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t appearing here and you may have k-th iteration
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in some cases we do not need i and k we just
might work with x t x t means
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vector x at time t so now a third dimension
comes into picture here when you write in
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the notation sometimes there are schemes which
are iterative and you will need index sometimes
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you need to prefer to i-th component so you
need xi and t is time now what kind of equations
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i am worried about what kind of equations
i am going to look at
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you might say that well what is written here
is only a first order vector matrix equation
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dx/dt=f(x) i am writing only a first order
equation only first order derivatives and
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in your engineering problems you often come
across models which are second order third
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order fourth order and when you did your first
course in the differential equations you had
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n-th order differential equations and then
you had methods of solving n-th order differential
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equation
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so why am i doing things only for the first
order differential equation though the difference
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here is the vector differential equation earlier
we were looking at scalar differential equation
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what i am going to show that any n-th order
differential equation can be converted into
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n first order differential equations so this
form which i have written here is very very
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generic so let us begin by looking at this
conversion let us say you have this
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let us say i have this differential equation
in the scalar variable y so y is a scalar
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y is some mass fraction or some temperature
or whatever is the case you have some differential
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equation let us say this is n-th order differential
equation in general nonlinear differential
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equation we do not know i am just writing
a generic form could be anything this is in
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one variable an independent variable is time
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what i am going to do now i am going to define
new state variables so my state variable and
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what i am given together here to solve this
problem say initial value problem so what
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do i need to solve this problem i need a differential
equation and i need the initial conditions
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initial conditions are given for y(0) dy/dt
at 0 so we are given initial condition we
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are given initial y0 initial derivatives up
to order n-1 these are required to solve this
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differential equation
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with this differential equation together with
this initial condition will be initial value
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problem solving ordinary differential equation
initial value problem this is what i get now
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what i am going to do now is to start defining
a new set of variables
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my new variable x1t-yt x2t=dy/dt x3t=d2y/dt
square up to xnt=dn-1/dt n-1-th derivative
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i am defining new variable x1 to xn now you
can see that these variables are related to
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first order differential equations i can very
easily say that dx1/dt=x2 dx2/dt=x3 so i have
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such n-1 equations
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this is my equation number 1 equation number
2 and this is my equation number n-1 i have
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n-1 such relationships between the variables
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all of them are first order differential equations
the last 1 is now just the equation that we
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have so the last equation n-th equation
this is dxn/dt this is nothing but d/dt of
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d n-1y this is my definition this is = fx1
x2… xnt i have an n-th order differential
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equation which got converted into n first
order differential equations this is my first
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equation second equation n-1-th equation and
the last equation came from the original n-th
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order differential equation
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x1 x2 x3 …xn are the new state variables
that we have defined so what i have actually
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done is a scalar n-th order differential equation
i have converted into n first order differential
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equations in new variables so if i have n-th
order equation i can convert it into n first
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order equations if i have 2 simultaneous equations
1 n-th order in 1 variable other m-th order
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in other variable first 1 will give me n first
order equations
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second 1 will give me m first order equations
you can stake them together into a bigger
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vector you will still get this form so this
is the very very generic form i am not doing
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any compromise any n-th order equation or
any set of n-th order equations n-th m-th
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order equations can be combined into finally
this form this is the very very generic form
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so do not worry about why are we looking at
only first order vector differential equation
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so all the advanced books on nonlinear differential
equations will worry about this generic form
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because anything can be converted to the generic
form that is the first thing to understand
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so all the methods that we will develop are
for this if you have n-th order equations
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you know how to convert them into n-th first
order equations and write it like this so
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what will be f(x) in this particular case
what will be the f vector
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let us go back and write that
in this particular case my f vector after
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a transformation actually
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my equations are d/dt of x1 x2 x3 …xn=x2
x3 …xn and f(x1 x2 …xnt) this is my f(x)
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this is the transform problem this is my f(x)
and i am given the initial condition so i
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am given initial condition x not which is
whatever this is y0 dy0/dt all these are given
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to me this is my x not this is given to me
this is my f(x) the original equation will
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appear as 1 scalar nonlinear function in a
function vector this my function vector
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this is a transform problem i do not have
to worry about n-th order equations i am not
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going to do separate methods in the first
course of differential equation you have second
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order differential equations 1 chapter on
second order differential equations then you
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will look at n-th order equations we are not
going to separate we are just going to look
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at n differential equations which are coupled
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if you are trained to solve dynamic simulation
of a chemical plant there will be 1000s of
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differential equations which are solved simultaneously
together in fact they might be differential
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and algebraic equations not differential equations
so we are worried about right now to begin
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with solving large number of differential
equations simultaneously together in 1 shot
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that is my aim this form is very generic applicable
to any set
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other way of getting these kind of equations
we have already seen where do you get these
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kind of equations in problem discretization
where did you find them finite difference
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method orthogonal collocations of partial
differential equations that involve time and
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space we discretize in space we got differential
equation in time we got n differential equations
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they were first order if those are all second
order you can convert them into 2 first order
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equations
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all that is possible that is not difficult
so converting n-th order equation into first
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order equations is not a problem we are going
to look at the generic form this could be
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arising from any of the sources this could
be arising from the 1 which we have done right
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now it could be arising from discretization
of a pde it might be arising from some other
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context we already have studied about in what
context this kind of problems will come
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we will look at only how to solve this abstract
form of vector differential equation the other
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thing which you might worry about is that
where does this time t come into picture most
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of the times the differential equations that
you get an exercise that i have given you
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to solve differential equations for 1 particular
system and i had given you a program which
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solves differential equations for a cstr i
suppose you remember to submit assignment
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soon
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that equation is of this form dx/dt=f(x u)
there are some free variables x are dependent
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variables and there are some free variables
like feed flow coolant flow coolant temperature
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inlet concentration all these are this u variables
so in that particular problem cstr problem
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x corresponds to concentration of a and temperature
and u corresponds to inlet flow rate cooling
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water flow rate inlet concentration cooling
water temperature at inlet and so on
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so these are the free variables but if you
go back and look at the problem statement
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these manipulated variables or input variables
have been defined as a function of time this
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is sinusoidal this is whatever we have defined
these as some functions of time once these
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are given as functions of time we can substitute
them here as some function of time and then
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once
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these are specified functions of time then
only we can solve the initial value problem
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for those specified functions of time this
problem has been transformed to dx/dt=f(x
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t) because u will be function of only time
some specified function of time a ramp function
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step function sinusoidal function or whatever
whatever you want to study the dynamics of
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the particular system you are specified this
free inputs and then this becomes a problem
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which again is the generic form
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so this parameter or these input variables
we assume that we already know them and then
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we want to solve the problem for the known
inputs how does the dynamics evolves in time
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that is what we want to solve that is why
we are looking at in general dx/dt=f(x t)
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how this is specified as a function of time
let us not worry about that right now it could
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be an operator who is giving these values
it could be a controller which is finding
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out these values
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it could be some environmental conditions
which define the cooling water inlet temperature
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we do not bother about that right now we want
to solve the problem when this is specified
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how do you actually find out x as a function
of time i want to find out given these input
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trajectories in time i want to find out x
trajectory that is concentration trajectory
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starting from time 0 to whatever final time
you want and temperature trajectory as solution
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of this problem is going to be not 1 vector
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when you are solving nonlinear algebraic equations
you got 1 vector as a solution the fixed point
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now the solution is going to be a trajectory
in time trajectory in time over the finite
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if we are solving over a finite time or whatever
t goes to infinity if you want to look at
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now linear differential equations of this
type you probably have already looked at in
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some other course wherever we need them we
will visit them
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those of you who have not done the other course
on analytical methods in chemical engineering
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i will briefly mention those results which
we need here we are going to look at the problem
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when this f(x) on the right hand side is nonlinear
not when it is linear that is very very crucial
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we will use the results for linear later on
to get some insights into the convergence
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properties under what conditions the methods
that you have proposed will converge
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that is why we will use some linear system
results but in general what we are going to
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look at is methods for solving nonlinear ordinary
differential equations given initial conditions
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how do you get trajectories in time or it
could be trajectories in space we have seen
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that for example method of lines for converting
laplace equation you discretize only in 1
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spatial direction the other 1 is stated as
a differential equation so you get instead
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of differential equations in time or space
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you want to integrate the differential equations
so t here in general need not be time alone
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t here is treated as independent variable
in some context it could be space so maybe
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i should write a generic form that neta so
neta is some independent variable it could
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be time on space depending upon the context
and initial condition at neta=0 is given and
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you want to integrate this set of differential
equations
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the way that we are going to proceed will
briefly peak into the issue of existence of
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solution very very briefly and then move on
to the different methods of doing numerical
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00:55:15,359 --> 00:55:25,359
integration again what is going to help us
taylor series approximation and polynomial
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00:55:25,359 --> 00:55:30,690
approximations we are going to meet our old
friends taylor and weierstrass again and use
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them repeatedly to solve these problems
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what i want to stress here is that the same
ideas are used again and again to form the
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solution methods there are few fundamental
ideas which if you understand those ideas
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and if you know how to apply them you can
almost do everything from scratch same idea
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is repeatedly used if you get this viewpoint
then i think you have learnt a lot next class
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onwards we will begin with how to solve ordinary
differential equations and algorithms
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and then finally we will move on to the convergence
properties under what conditions these converge
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try to get some insights into relative behavior
of different methods and so on