solution of differential equations
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/1.jpg)
Solution of Differential Equations
by
Pragyansri Pathi
Graduate student
Department of Chemical Engineering,
FAMU-FSU College of Engineering,FL-32310
![Page 2: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/2.jpg)
Topics to be discussed
Solution of Differential equations
Power series method
Bessel’s equation by Frobenius method
![Page 3: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/3.jpg)
POWER SERIES METHOD
![Page 4: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/4.jpg)
Power series method
Method for solving linear differential equations with variable co-efficient.
)()()( ''' xfyxqyxpy
p(x) and q(x) are variable co-effecients.
![Page 5: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/5.jpg)
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
![Page 6: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/6.jpg)
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.
![Page 7: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/7.jpg)
Examples
Example 1: Solve 0' yy
Example 2: Solve xyy 2'
Example 3: Solve 0'' yy
![Page 8: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/8.jpg)
Power series method (Cont’d)
• The general representation of the power series solution is,
......)()()( 202010
00
xxaxxaaxxaym
mm
![Page 9: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/9.jpg)
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.
![Page 10: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/10.jpg)
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
![Page 11: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/11.jpg)
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
![Page 12: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/12.jpg)
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
![Page 13: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/13.jpg)
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
![Page 14: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/14.jpg)
Examples (Calculation of R) Cont’d
Hence 2R
• The given series converges for
2)0( x
• The given series diverges for
2)0( x
![Page 15: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/15.jpg)
Examples
• Example 2:
Find the radius of convergence of the following series,
0 2.nn
n
n
x
![Page 16: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/16.jpg)
Examples
• Example 3:
Find the radius of convergence of the following series,
0 1)...3).(2).(1.(n
n
nnnn
x
![Page 17: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/17.jpg)
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
![Page 18: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/18.jpg)
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
![Page 19: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/19.jpg)
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.
![Page 20: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/20.jpg)
Example
• Let’s try this
0)()()( ''' xyxxyxy
![Page 21: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/21.jpg)
BESSEL’S EQUATION.
BESSEL FUNCTIONS Jν(x)
![Page 22: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/22.jpg)
Application
• Heat conduction • Fluid flow• Vibrations• Electric fields
![Page 23: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/23.jpg)
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
![Page 24: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/24.jpg)
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
![Page 25: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/25.jpg)
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.
![Page 26: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/26.jpg)
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
![Page 27: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/27.jpg)
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
![Page 28: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/28.jpg)
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
![Page 29: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/29.jpg)
Example
• Compute
)(0 xJ
• Compute
)(1 xJ
![Page 30: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/30.jpg)
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
![Page 31: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/31.jpg)
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
![Page 32: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/32.jpg)
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
![Page 33: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/33.jpg)
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
![Page 34: Solution of Differential Equations](https://reader031.vdocument.in/reader031/viewer/2022012314/56813e18550346895da7fbf3/html5/thumbnails/34.jpg)
Examples
• Example : Using the properties of Bessel’s functions compute,
)(3 xJ1.
2. dxxJx ))((2
2
3. )(2
3 xJ