#31|| Problem#3 || Construction of interpolation polynomial|| Newton’s general interpolation ||

National Science Day-2024

Solve for p 𝒑^𝟐+𝟐𝒑𝒚𝒄𝒐𝒕𝒙=𝒚^𝟐

Solve 𝟑𝟐 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 (𝟎, 𝟏) given that h=0.25, upto t=5 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖=(𝒙,𝟎).

Solve 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 (𝟎, 𝟏) at t=0.002 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖(𝒙,𝟎)={(𝟐𝒙 (𝟎, 𝟎.𝟓) @𝟐(𝟏−𝒙), (𝟎.𝟓, 𝟏)

Solve 𝝏𝒖/𝝏𝒕=(𝝏^𝟐 𝒖)/𝝏𝒙^𝟐, 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟏,𝒕), 𝒖(𝒙,𝟎)=𝒔𝒊𝒏𝝅𝒙 by taking h=0.2, (𝟎, 𝟎.𝟏) #matheconnect

Find the numerical solution (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =𝟐 𝝏𝒖/𝝏𝒕 , 𝒖=(𝟎,𝒕)=𝟎=𝒖(𝟒,𝒕), 𝒖(𝒙,𝟎)=𝒙(𝟒−𝒙) ,h=1,t=4 steps.

Numerical Solution of the one dimensional Heat equation & working procedure.

Show that: 𝒊)𝑷_𝟐(𝒄𝒐𝒔𝜽)=1/4 (𝟏+𝟑𝒄𝒐𝒔𝜽), 𝒊𝒊) 𝑷_𝟑 (𝒄𝒐𝒔𝜽)=𝟏/𝟖 (𝟑𝒄𝒐𝒔𝜽+𝟓𝒄𝒐𝒔𝟑𝜽).

Express 𝒇(𝒙)=𝒙^𝟒−𝟑𝒙^𝟑−𝒙^𝟐+𝟓𝒙 in terms of Legendre polynomial.

Express 𝟐𝒙^𝟑−𝒙^𝟐−𝟑𝒙+𝟐 in terms of Legendre polynomial

If 𝒙^𝟑+𝟐𝒙^𝟐−𝒙+𝟏=𝒂𝑷_𝟎 (𝒙)+𝐛_𝑷 𝟏( 𝒙)+c𝑷 _𝟐 (𝒙) +d𝑷_𝟑 (𝒙) .Find the value of a, b, c, d.

Express 𝒙^𝟑+𝟐𝒙^𝟐−𝟒𝒙+𝟓 in terms of Legendre polynomial

Rodrigues Formula & Results.

Series Solution of Legendre s Differential Equation

Orthogonality of Bessel s Functions

Prove that : 𝑱((−𝟏)⁄𝟐) (𝒙)=√(𝟐/𝝅𝒙) 𝒄𝒐𝒔𝒙.

Prove that : 𝑱(𝟏⁄𝟐) (𝒙)=√(𝟐/𝝅𝒙) 𝒔𝒊𝒏𝒙.

Solve the wave equation (𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 =(𝝏^𝟐 𝒖)/〖𝝏𝒕〗^𝟐 𝒖(𝒙,𝟎)=𝒔𝒊𝒏𝝅𝒙 (0,1), h=0.2, k=0.2 at 𝒕=𝟏.

Solve the wave equation 25 (𝝏^𝟐 𝒖)/𝝏𝒙^𝟐 =(𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 𝒖(𝒙,𝟎)={(𝟐𝟎𝒙 𝟎≤𝒙≤𝟏, 𝟓(𝟓−𝒙) 𝟏≤𝒙≤𝟓) h=1, 𝟎≤𝒕≤𝟏.

Solve the Wave equation (𝝏^𝟐 𝒖)/(𝝏𝒙^𝟐) =𝟎.𝟎𝟔𝟓 (𝝏^𝟐 𝒖)/𝝏𝒕^𝟐 , 𝒖(𝒙,𝟎)=𝒙^𝟐 (𝒙−𝟓), h=1, k=0.5 ,4 steps.

Solve the wave equation (𝝏^𝟐 𝒖)/〖𝝏𝒕〗^𝟐 =𝟒 (𝝏^𝟐 𝒖)/〖𝝏𝒙〗^𝟐 𝒖(𝒙,𝟎)=𝒙(𝟒−𝒙) , h=1, k=0.5 upto 4 steps.

Working procedure for one dimensional wave equation

21MAT31: Numerical solution of the one dimensional wave equation.

21MAT31:Numerical Method PDE, Classification of PDE of second order & Finite difference approximate.