solution of differential equations by pragyansri pathi graduate student department of chemical...
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Solution of Differential Equations
by
Pragyansri Pathi
Graduate student
Department of Chemical Engineering,
FAMU-FSU College of Engineering,FL-32310
Topics to be discussed
Solution of Differential equations
Power series method
Bessel’s equation by Frobenius method
POWER SERIES METHOD
Power series method
Method for solving linear differential equations with variable co-efficient.
)()()( ''' xfyxqyxpy
p(x) and q(x) are variable co-effecients.
Power series method (Cont’d)
Solution is expressed in the form of a power series.
Let ..........33
2210
0
xaxaxaaxaym
mm
so, ..........32 2321
1
1'
xaxaaxmaym
mm
..........3.42.32)1( 2432
2
2'' xaxaaxammym
mm
Power series method (Cont’d)
Substitute ''' ,, yyy
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.
Examples
Example 1: Solve 0' yy
Example 2: Solve xyy 2'
Example 3: Solve 0'' yy
Power series method (Cont’d)
• The general representation of the power series solution is,
......)()()( 202010
00
xxaxxaaxxaym
mm
Theory of power series method
Basic concepts
A power series is an infinite series of the form
(1) ......)()()( 202010
00
xxaxxaaxxam
mm
x is the variable, the center x0, and the coefficients a0,a1,a2 are real.
Theory of power series method (Cont’d)
(2) The nth partial sum of (1) is given as,
nnn xxaxxaxxaaxs )(......)()()( 0
202010
where n=0,1,2..
(3) The remainder of (1) is given as,
......)()()( 202
101
n
nn
nn xxaxxaxR
Convergence Interval. Radius of convergence
• Theorem: Let ......)()()( 202010
00
xxaxxaaxxaym
mm
be a power series ,then there exists some number ∞≤R≤0 ,called its radius of convergence such that the series is convergent for
Rxx )( 0 Rxx )( 0and divergent for
• The values of x for which the series converges, form an interval, called the convergence interval
Convergence Interval. Radius of convergence
What is R?
The number R is called the radius of convergence. It can be obtained as,
m
m
m a
aR
1lim
1
or m
mm
aR
lim
1
Examples (Calculation of R)
Example 1: 12n
nnx
n
xn
xn
xn
termn
termnn
n
n
n
.2
)1(
)(
2
2
)1(
)(
)1( 1
2
1
Answer
As n→∞,2.2
)1( x
n
xn
Examples (Calculation of R) Cont’d
Hence 2R
• The given series converges for
2)0( x
• The given series diverges for
2)0( x
Examples
• Example 2:
Find the radius of convergence of the following series,
0 2.nn
n
n
x
Examples
• Example 3:
Find the radius of convergence of the following series,
0 1)...3).(2).(1.(n
n
nnnn
x
Operations of Power series: Theorems
,)()(00
n
nan
n
nan xxbxxa
(1) Equality of power series
If with 0R
then nn ba ,for all n
Corollary
If
0
,0)(n
an xxa all an=0, for all n ,R>0
Theorems (Cont’d)
0n
nnxay
(2) Termwise Differentiation
If is convergent, then
derivatives involving y(x) such as
are also convergent.
If
0n
nnxa and
0n
nnxb are convergent
in the same domain x, then the sum also converges in that domain.
etcxyxy ),(),( '''
(3) Termwise Addition
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 x=x0 if it can be represented by a power series in powers of x-x0 with radius of convergence R>0.
Example
• Let’s try this
0)()()( ''' xyxxyxy
BESSEL’S EQUATION.
BESSEL FUNCTIONS Jν(x)
Application
• Heat conduction • Fluid flow• Vibrations• Electric fields
Bessel’s equation
• Bessel’s differential equation is written as
0)( 22'''2 ynxxyyx
or in standard form,
0)1(1
2
2''' y
x
nyx
y
Bessel’s equation (Cont’d)
• n is a non-negative real number.• x=0 is a regular singular point.• The Bessel’s equation is of the type,
)(
)(
)(
)(
)(
)( '''
xr
xfy
xr
xqy
xr
xpy
and is solved by the Frobenius method
Non-Analytic co-efficients –Methods of Frobenius
• If x is not analytic, it is a singular point.
)()()()( ''' xfyxqyxpyxr
→ )(
)(
)(
)(
)(
)( '''
xr
xfy
xr
xqy
xr
xpy
The points where r(x)=0 are called as singular points.
Non-Analytic co-efficients –Methods of Frobenius (Cont’d)
• The solution for such an ODE is given as,
0m
mm
r xaxy
Substituting in the ODE for values of y(x),
)(),( ''' xyxy , equating the co-efficient of xm and obtaining the roots gives the indical solution
Non-Analytic co-efficients –Methods of Frobenius (Cont’d)
We solve the Bessel’s equation by Frobenius method.
Substituting a series of the form,
0m
rmmxay
Indical solution
)0(1 nnr nr 2
General solution of Bessel’s equation
• Bessel’s function of the first kind of order n
is given as,
0
2
2
)!(!2
)1()(
mnm
mmn
n mnm
xxxJ
Substituting –n in place of n, we get
02
2
)!1(!2
)1()(
mnm
mmn
n mnm
xxxJ
Example
• Compute
)(0 xJ
• Compute
)(1 xJ
General solution of Bessel’s equation
• If n is not an integer, a general solution of Bessel’s equation for all x≠0 is given as
)(.)(.)( 21 xJcxJcxy nn
Properties of Bessel’s Function
J0(0) = 1
Jn(x) = 0 (n>0)
J-n(x) = (-1)n Jn(x)
)()( 1 xJxxJxdx
dn
nn
n
)()( 1 xJxxJxdx
dn
nn
n
Properties of Bessel’s Function (Cont’d)
)()(2
1)( 11 xJxJxJ
dx
dnnn
)(.)(..2)(. 11 xJxxJnxJx nnn
CxJxdxxJx nn
n
n )())((1
CxJxdxxJx nn
n
n
)())((1
Properties of Bessel’s Function (Cont’d)
)(.2)()( '11 xJxJxJ nnn
)(.2
)()( 11 xJx
nxJxJ nnn
xx
xJ sin2
)(2
1
xx
xJ cos2
)(2
1
Examples
• Example : Using the properties of Bessel’s functions compute,
)(3 xJ1.
2. dxxJx ))((2
2
3. )(2
3 xJ