The presentation shows types of equations inside every topic along with its general form, generating formula, and other equations like recursion, frobenius, rodrigues etc for calculus. Its an overall explanation in a brief. You are at correct link to get your work done out of this in your engineering maths.
2. ๏ฑ Abstract
๏ฑ Keywords
๏ฑ Termology
๏ฑ Introduction
๏ฑ Power Series
๏ฑ Legendre Polynomial
๏ฑ Besselโs Equation
๏ฑ Conclusion
3. In this presentation , our aim
is to solve an infinite
convergent series using Power
series, Legendre Polynomial
and Besselโs Equation.
Throughout this presentation
we will gather the basic
knowledge about the topic .
This presentation is a team
work and every member is
associated with this
presentation has collected the
data from various trusted
sources.
5. ๏ฐ Power Series Solution:
The power series method is used to seek a power series solution to certain differential equations. In
general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into
the differential equation to find a recurrence relation for the coefficients.
๏ฐ Legendre Polynomial:
Legendre polynomials are a system of complete and orthogonal polynomials, with a vast number of
mathematical properties, and numerous applications.
๏ฐ Besselโs Equation:
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich
Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex
number ฮฑ, the order of the Bessel function.
6. Sometimes we face some differential equations which cannot be solved by the ordinary methods.
For such equations an infinite convergent series arranged in ascending power of independent
variables satisfying the equation approximately.
Those equations could be easier to solve if we use one the followings:
๏ฐ Power Series
๏ฐ Legendre Polynomial
๏ฐ Besselโs Equation
7. Validity of Series Solution:
Consider a second order linear differential equation of the form
๐0 ๐ฅ
โ 2๐ฆ
โ ๐ฅ2 + ๐1 ๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐2 ๐ฅ ๐ฆ = 0 (1)
where P0(x), P1(x), P2(x) are polynomials in x. As every differential equation of the form (1) may not
have series solution, we now find the conditions under which the above equation has series solution.
If P0(a) not equals to 0, then the point x=a is called an ordinary point; otherwise x=a is called a
singular point (1).
Equation (1) can be written as
โ 2๐ฆ
โ ๐ฅ2 + ๐ ๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐ ๐ฅ ๐ฆ = 0 (2)
where ๐ ๐ฅ =
๐1 ๐ฅ
๐0 ๐ฅ
and ๐ ๐ฅ =
๐2 ๐ฅ
๐0 ๐ฅ
If (x-a) P (x) and (x-a)2 Q(x) possess derivatives of all order in neighbourhood of x=a is called a
regular singular point.
8. Theorem 1:
When x=a is an ordinary point of (1), its solution can be expressed as a series of the form
y= a0 + a1 (x-a) + a2(x-a)2 + โฆโฆโฆ..
= ๐=0
โ
๐๐ ๐ฅ โ ๐ ๐
i.e differential equation has a series solution.
Theorem 2:
When x=a is a regular singular point of (1), its solution can be expressed as a series of the
form
y= xm (a0 + a1 (x-a) + a2 (x-a)2 + โฆโฆ.)
= ๐=0
โ
๐๐ ๐ฅ โ ๐ ๐+๐
9. General Method to Solve an ordinary differential equation:
Here we discuss two methods, depending on the nature of the point x=x0 , about which the series solution is
convergent.
Method 1:
Series solution or power series solution when x=x0 is an ordinary point.
Let y= ๐=0
โ
๐๐ ๐ฅ โ ๐ฅ0
๐
(1)
be the series solution of p.d.e
๐0 ๐ฅ
โ 2๐ฆ
โ ๐ฅ2 + ๐1 ๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐2 ๐ฅ ๐ฆ = 0 (2)
Then
โ ๐ฆ
โ ๐ฅ
= ๐=1
โ
๐๐๐ ๐ฅ โ ๐ฅ0
๐โ1 ;
โ 2๐ฆ
โ ๐ฅ2 = ๐=2
โ
๐ ๐ โ 1 ๐๐ ๐ฅ โ ๐ฅ0
๐โ2
Substituting these values in (2), we get an identity. Then equating the coefficient of xn to zero we get a relation
between an and an+2 which is called the recurrence relation. From this relation we obtain the coefficients of (1)
in terms of a0 and a1. These are substituted in (1) to get the required solution.
10. Method 2:
(Method of Frobenius) Series Solution when x=x0 is a Regular Singular Point:
Let ๐ฆ = ๐=0
โ
๐๐ ๐ฅ โ ๐ฅ0
๐+๐
, ๐0 โ 0 (3)
be the series solution of p.d.e
๐0 ๐ฅ
โ 2๐ฆ
โ ๐ฅ2 + ๐1 ๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐2 ๐ฅ ๐ฆ = 0 (1)
Then
โ ๐ฆ
โ ๐ฅ
= ๐=0
โ
๐๐ ๐ + ๐ ๐ฅ โ ๐ฅ0
๐+๐โ1
โ 2๐ฆ
โ ๐ฅ2 = ๐=0
โ
๐๐ ๐ + ๐ ๐ + ๐ โ 1 ๐ฅ โ ๐ฅ0
๐+๐โ2
Substituting these values in (1) we get an identity. Then the coefficient of the lowest power
of x from both side we get a quadratic equation in m, known as indicial equation.
11. The differential equation of the form
1 โ ๐ฅ2 โ 2๐ฆ
โ ๐ฅ2 โ 2๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐ ๐ + 1 ๐ฆ = 0
is called Legendreโs equation where n is a positive integer.
It can also be written as,
โ
โ ๐ฅ
1 โ ๐ฅ2 โ ๐ฆ
โ ๐ฅ
+ ๐ ๐ + 1 ๐ฆ = 0
12. Definition of Pn(๐) and Qn(๐)
The solution of Legendreโs equation is called Legendreโs function or Legendreโs polynomials.
When n is a positive integer and ๐0 =
1.3.5โฆ 2๐โ1
๐
, then solution denoted by Pn(๐ฅ) and is called Legendreโs
function of the first kind or Legendreโs polynomials of degree n.
โด Pn ๐ฅ =
1.3.5โฆ 2๐โ1
๐!
๐ฅ๐ โ
๐ ๐โ1
2 2๐โ1
๐ฅ๐โ2 +
๐ ๐โ1 ๐โ2 ๐โ3
2.4 2๐โ1 2๐โ3
ร ๐ฅ๐โ4 โ โฆ .
= ๐=0
(
๐
2
)
โ1 ๐ 2๐โ๐!
2๐ ๐ ๐โ2๐! ๐โ๐!
๐ฅ๐โ2๐
Where (
๐
2
) =
๐
2
๐๐ ๐ ๐๐ ๐๐ฃ๐๐
๐โ1
2
๐๐ ๐ ๐๐ ๐๐๐
Here Pn(๐ฅ) is a terminating series, as n is a positive integer.
Again When n is a positive integer and ๐0 =
๐!
1.3.5โฆ(2๐+1)
, the solution is denoted by Qn(๐ฅ) and is called
Legendreโs function of the second kind.
โด Qn ๐ฅ =
๐!
1.3.5..(2๐+1)
๐ฅโ๐โ1
+
(๐+1) ๐+2
2. 2๐+3
๐ฅโ๐โ3
+
๐+1 ๐+2 ๐+3 (๐+4)
2.4 2๐+3 2๐+5
๐ฅโ๐โ5
+ โฆ .
Here Qn(๐ฅ) is an infinite or non-terminating series, as n is a positive integer.
13. Generating function for Legendre polynomials
Theorem. Pn(๐ฅ) is the coefficient of โ๐ in the expansion of (1-2๐ฅh+โ2)
โ
1
2 in ascending powers of h, where
๐ฅ โค 1, |โ| < 1.
Proof. Since |๐ฅ|โค 1, |โ| < 1, we have
(1-2๐ฅh+โ2
)
โ
1
2 = [1-h(2๐ฅ โ โ)] โ
1
2
= 1+
1
2
โ 2๐ฅ โ โ +
1.3
2.4
โ2
2๐ฅ โ โ 2 +โฏ +
1.3โฆ 2๐โ3
2.4โฆ 2๐โ2
ร โ๐โ1
2๐ฅ โ โ ๐โ1
+
1.3โฆ 2๐โ1
2.4โฆ2๐
โ๐
2๐ฅ โ โ ๐
+ โฏ
Therefore coefficient of โ๐
=
1.3โฆ(2๐โ1)
2.4โฆ2๐
(2๐ฅ)๐
+
1.3โฆ 2๐โ3
2.4โฆ 2๐โ2
๐ โ 1๐ถ1
โ1 . (2๐ฅ)๐โ2
+
1.3โฆ(2๐โ5)
2.4โฆ(2๐โ4)
๐ โ 2๐ถ2
(โ1)2
. (2๐ฅ)๐โ4
+ โฏ โฆ (1)
=
1.3โฆ(2๐โ1)
๐!
๐ฅ๐ โ
2๐
2๐โ1
๐ โ 1 .
๐ฅ๐โ2
22 +
2๐ 2๐โ1
2๐โ1 2๐โ3
ร
๐โ2 ๐โ3
2!
.
๐ฅ๐โ4
24 โ โฏ
=
1.3โฆ(2๐โ1)
๐!
๐ฅ๐ โ
๐ ๐โ1
2 2๐ฅโ1
๐ฅ๐โ2 +
๐ ๐โ1 ๐โ2 ๐โ3
2.4 2๐โ1 2๐โ2
๐ฅ๐โ4 โ โฏ
= Pn(๐ฅ)
Thus we can say that
๐=0
โ
โ๐Pn(๐ฅ) = (1โ2๐ฅh+โ2)
โ
1
2
15. General solution of Besselโs equation:
The general solution of Besselโs equation is
๐ฆ = ๐ด๐ฝโ๐ ๐ฅ + ๐ต๐ฝ๐ ๐ฅ
where A, B are two arbitrary constants.
The differential equation of the form:
๐ฅ2 โ 2๐ฆ
โ ๐ฅ2 + ๐ฅ
โ ๐ฆ
โ ๐ฅ
+ ๐ฅ2 + ๐2 ๐ฆ = 0
or,
โ 2๐ฆ
โ ๐ฅ2 +
1
๐ฅ
โ ๐ฆ
โ ๐ฅ
+ 1 โ
๐2
๐ฅ2 ๐ฆ = 0
is called Besselโs equation of order n, n being a non-negative constant.
16. Recurrence formulae:
I.
โ ๐ฆ
โ ๐ฅ
๐ฅ๐๐ฝ๐ = ๐ฅ๐๐ฝ๐โ1 ๐๐ ๐ฅ๐ฝ๐
โฒ ๐ฅ = โ๐๐ฝ๐ ๐ฅ + ๐ฅ๐ฝ๐โ1 ๐ฅ .
II.
โ
โ ๐ฅ
๐ฅโ๐
๐ฝ๐ ๐ฅ = โ๐ฅโ๐
๐ฝ๐+1 ๐ฅ ๐๐ ๐ฅ๐ฝ๐
โฒ
๐ฅ = ๐๐ฝ๐ ๐ฅ โ ๐ฅ๐ฝ๐+1 ๐ฅ
III. 2๐ฝ๐
โฒ ๐ฅ = ๐ฝ๐โ1 ๐ฅ โ ๐ฝ๐+1 ๐ฅ
IV. 2๐๐ฝ๐ ๐ฅ = ๐ฅ ๐ฝ๐โ1 ๐ฅ + ๐ฝ๐+1 ๐ฅ
Definition: Besselโs function of 2nd kind of order n or Neumann function is (given by)
๐๐ ๐ฅ = ๐ฝ๐ ๐ฅ
โ ๐ฅ
๐ฅ ๐ฝ๐ ๐ฅ 2
17. This project has been beneficial for us , as it enabled us to gain a lot of knowledge about the use of
Power Series , Legendre Polynomial and Besselโs Equation as well as its applications. It also helped
us to develop a better coordination among us as we shared different perspectives and ideas
regarding the sub-topics and equations we organized in this project .