Solution of Differential Equations

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Solution of Differential Equations
by Pragyansri Pathi Graduate student Department
of Chemical Engineering, FAMU-FSU College of
Engineering,FL-32310
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Topics to be discussed

• Solution of Differential equations
• Power series method
• Bessels equation by Frobenius method

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• POWER SERIES METHOD

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Power series method

• Method for solving linear differential equations
  with variable co-efficient.

p(x) and q(x) are variable co-effecients.
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Power series method (Contd)

• Solution is expressed in the form of a power
  series.
• Let

so,
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Power series method (Contd)

• Substitute

• Collect like powers of x
• Equate the sum of the co-efficients of each
  occurring power of x to zero.
• Hence the unknown co-efficients can be
  determined.

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Examples

• Example 1 Solve

Example 2 Solve
Example 3 Solve
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Power series method (Contd)

• The general representation of the power series
  solution is,

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Theory of power series method

• Basic concepts
• A power series is an infinite series of the form
• (1)

x is the variable, the center x0, and the
coefficients a0,a1,a2 are real.
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Theory of power series method (Contd)

• (2) The nth partial sum of (1) is given as,

where n0,1,2..
(3) The remainder of (1) is given as,
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Convergence Interval. Radius of convergence

• Theorem Let

be a power series ,then there exists some number
8R0 ,called its radius of convergence such that
the series is convergent for
and divergent for

• The values of x for which the series converges,
  form an interval, called the convergence interval

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Convergence Interval. Radius of convergence

• What is R?

The number R is called the radius of convergence.
It can be obtained as,
or
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Examples (Calculation of R)

• Example 1

Answer
As n?8,
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Examples (Calculation of R) Contd

• Hence

• The given series converges for

• The given series diverges for

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Examples

• Example 2

Find the radius of convergence of the following
series,
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Examples

• Example 3

Find the radius of convergence of the following
series,
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Operations of Power series Theorems
(1) Equality of power series

If
with
then
,for all n
Corollary
If
all an0, for all n ,Rgt0
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Theorems (Contd)
(2) Termwise Differentiation

If
is convergent, then
derivatives involving y(x) such as
are also convergent.
(3) Termwise Addition
If
and
are convergent
in the same domain x, then the sum also converges
in that domain.
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Existence of Power Series Solutions. Real
Analytic Functions

• The power series solution will exist and be
  unique provided that the variable co-efficients
  p(x), q(x), and f(x) are analytic in the domain
  of interest.

What is a real analytic function ?
A real function f(x) is called analytic at a
point xx0 if it can be represented by a power
series in powers of x-x0 with radius of
convergence Rgt0.
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Example

• Lets try this

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• BESSELS EQUATION.
• BESSEL FUNCTIONS J?(x)

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Application

• Heat conduction
• Fluid flow
• Vibrations
• Electric fields

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Bessels equation

• Bessels differential equation is written as

or in standard form,
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Bessels equation (Contd)

• n is a non-negative real number.
• x0 is a regular singular point.
• The Bessels equation is of the type,

and is solved by the Frobenius method
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Non-Analytic co-efficients Methods of Frobenius

• If x is not analytic, it is a singular point.

?
The points where r(x)0 are called as singular
points.
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Non-Analytic co-efficients Methods of Frobenius
(Contd)

• The solution for such an ODE is given as,

Substituting in the ODE for values of y(x),
, equating the co-efficient of xm and obtaining
the roots gives the indical solution
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Non-Analytic co-efficients Methods of Frobenius
(Contd)

• We solve the Bessels equation by Frobenius
  method.
• Substituting a series of the form,

Indical solution
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General solution of Bessels equation

• Bessels function of the first kind of order n
• is given as,

Substituting n in place of n, we get
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Example

• Compute

• Compute

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General solution of Bessels equation

• If n is not an integer, a general solution of
  Bessels equation for all x?0 is given as

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Properties of Bessels Function

• J0(0) 1
• Jn(x) 0 (ngt0)
• J-n(x) (-1)n Jn(x)

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Properties of Bessels Function (Contd)
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Properties of Bessels Function (Contd)
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Examples

• Example Using the properties of Bessels
  functions compute,

1.
2.
3.