LEGENDRE Differential Equation and Polynomial
LEGENDRE Differential Equation:
LEGENDRE Differential Equation and Polynomial has a great role in the area of Differential Equations , which is like a planet in the universe of Mathematics. This equation and polynomial is not only useful for solving mathematical differential equations but also used in physics.
The Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.
The equation
(1-x²)y" - 2xy' +p(p+1)y =0 .......(1)
Where p is a real constant is known as Legendre differential equation and it occurs in many areas of mathematics and physics . To obtain a solution of equation (1) we shall use the power series methode. For this purpose we observe that equation (1) can be put in the standard form as
y" -( 2x/1 - x²) y' + [p(p+1)/1 - x²] y =0 ...(2)
since -2x/1-x² and p(p+1)/1- x² are analytic for |x|<1, x=0 is an ordinary point of (1) .
Hence the equation (1) has a power series solution valid for |x|<1. Let us assume that
y = Σ cₙxⁿ for n = 0 to ∞ .........(3)
is a solution of equation(1).
Differentiating equation (3) and substituting for y,y' and y" in equation (1) we have
∞ ∞
(1 - x²) Σ n(n-1) cₙxⁿ⁻² - 2x Σ ncₙxⁿ⁻¹+p(p+1)
n=2 n=1
∞
Σ cₙxⁿ = 0
n=0
which on simplification
∞ ∞
Σ (n+2)(n+1)cₙ₊₂xⁿ- Σ n(n-1)cₙxⁿ
n=0 n=0
∞ ∞
-2 Σ ncₙxⁿ+ p(p+1) Σ cₙxⁿ=0
n=0 n=0
where in second sum we have taken the limit for n=0 instead of n=2 since the second sum corresponding to n= 0 and 1 vanish.
∞
i.e Σ [(n+2)(n+1)cₙ₊₂ + (p+n+1)(p-n)cₙ]xⁿ =0
n=0
Since this equation is valid for all x in the interval of convergence |x|<1 the coefficient of all powers of x in equation (3) must vanish and this yeilds the recursion formula:
cₙ₊₂ = -(p+n+1)(p-n)cₙ/( n+2)(n+1),n≥0 .......(4)
setting n=0,1,2,3 etc in (4) we have
c₂ = -(p+1)p c₀/ 2*1;
c₃=-(p+2)(p-1) c₁/3*2;
c₄ = -(p+3)(p-2)c₂/4*3;
= (p+3)(p+1)p(p-2)c₀/4! ;
c₅= -(p+4)(p-3)c₃/5*4
= (p+4)(p+2)(p-1)(p-3)c₁/5! ;
In general
c₂ₘ=[(-1)ᵐ(p+2m-1)(p+2m-3)...(p+1)p(p-2)....
(p-2m+2)]c₀/(2m)! ;
c₂ₘ₊₁=[(-1)ᵐ(p+2m)(p+2m-2)....(p+2)(p-1)(p-3)
........(p-2m+1)]c₁/(2m+1)! ;
where m = 1,2.......
Substituting these values in equation (3) we obtained the desired power series solution of equation (1) as :
y = c₀[1-(p+1)x²/2! + (p+3)(p+1)p(p-2)x⁴/4!....]
+c₁[x-(p+2)(p-1)x³/3!
+ (p+4)(p+2)(p-1)(p-3)x⁵/5!....]
or y= c₀y₁+c₁y₂ (say). .........(5)
where y₁and y₂ repesent the power series in the first and second brackets respectively. c₀ and c₁ are two arbitary constants ,equation (5) represents general solution of Legendre differential equation.
There are different values of y for different p . So when p=n where n is a positive integer then it gives Legender polynomial solution of degree n.
The polynomial solution Pₙ(x ) of Legendre equation is called Legender polynomial if it satisfies the condition Pₙ(1)=1 ,n=1,2,.......
Expression for Legender polynomial is given by
Pₙ(x)=1/2ⁿn! dⁿ/dxⁿ(x²-1)ⁿ .......(6)
This is all about Legendre differential equation and polynomial.
See also:
Legendre Polynomial and its creation.About Scientist:
Adrien-Marie Legendre was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.
Recommended Books :
2 . Differential Equation Theory By Mc Grew Hill.
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