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Green's Function Method-Mathematical Physics-Assignment, Exercises of Mathematical Physics

Chennai Mathematical InstituteMathematical Physics

This assignment is for Mathematical Physics course. It was assigned by Saikiran Janak at Chennai Mathematical Institute. It includes: Green, Function, Method, Initial, Value, Problem, Construct, Operator, Boundary, Exist, Particular, Solution

Typology: Exercises

2011/2012

Uploaded on 07/17/2012

sarmistha
sarmistha 🇮🇳

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Mathematical Physics – 1 Assignment Number 2.
PAM -507 (Due on March 20, 2012)
Question 1.
Use the Green’s function method to solve the following initial value problems
(a) y// -y = 1 y(0) = 0 y/(0)=-1
(b) y// -3y/ -4y = e-x y(2) = 3 y/(2)=-1
(c) t2 y// + ty/ + 4y = sin (log t) y(1) = 1 y/(1)=0
(d) y// +y/ = δ(t-π)- δ(t-2π) y(0) = 1 y/(0)=1
(e) y// + y = 5δ(t-π/2)sin (t) y(0) = 0 y/(0)=2
Question 2.
Construct the Green’s function, if possible, for the following operator and boundary
conditions (where D= dt
d )
(a) L =D2 y(a)=0 and y(b)=0
(b) L=D2 y(0) = 0 y/(1)=0
(c) L= x2 D2 +2xD +1 y(1) = 0 y(e π/√3)=0
Question 3
Show that the Green’s function of the following ODE does not exist
y// + π2 y = π2x y(0) = 1 y(1)=0
Then find the solution.
Question 4
Use Green’s function method to find the particular solution of the following
ODE
y// -y = 3x2 y(0) = 0 y(1)=0
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Mathematical Physics – 1 Assignment Number 2.

PAM -507 (Due on March 20, 2012)

Question 1.

Use the Green’s function method to solve the following initial value problems (a) y //^ -y = 1 y(0) = 0 y /^ (0)=- (b) y //^ -3y /^ -4y = e-x^ y(2) = 3 y /(2)=- (c) t^2 y//^ + ty/^ + 4y = sin (log t) y(1) = 1 y/^ (1)= (d) y//^ +y /^ = δ(t-π)- δ(t-2π) y(0) = 1 y /^ (0)= (e) y//^ + y = 5 δ(t-π/2)sin (t) y(0) = 0 y /^ (0)=

Question 2.

Construct the Green’s function, if possible, for the following operator and boundary conditions (where D= (^) dtd^ )

(a) L =D^2 y(a)=0 and y(b)= (b) L=D^2 y(0) = 0 y /^ (1)= (c) L= x^2 D^2 +2xD +1 y(1) = 0 y(e π/√^3 )=

Question 3

Show that the Green’s function of the following ODE does not exist y//^ + π^2 y = π^2 x y(0) = 1 y(1)= Then find the solution.

Question 4

Use Green’s function method to find the particular solution of the following

ODE

y //^ -y = 3x 2 y(0) = 0 y(1)=

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Question 5

Construct the Green’s function for x^2 y//^ + x y /^ + (k^2 x^2 -1) y = 0 subject to the B.C y(0) = 0 y(1)=

Question 6

Soleve the ODE Y//(t) = - δ(t 2 -5t+6) +t 3 y(0) = 0 y/^ (0)=

Question 7

Determine the deflection curve of a string of unit length governed by y//^ = - q(x)/T y(0) = 0 y(1)= when the load q(x) is equally concentrated loads at P located at x=1/4 and x=3/4.

Question 8

Find the eigenfunctions and eigenvalues of the following equation y//(x)+λy(x)= for the BC y(0)=y(L) y /^ (0)=y/^ (L)

This is an example of periodic Sturm-Liouville equation.

Question 9

Construct the Green’s function for the boundary value problem and discuss the results y//^ +k 2 y/^ = f(x) y(0) = C 1 y(1)= C 2 and k ≠ 0

Question 10

Use Green’s function method to solve the following ODE y//^ + 2y /^ +y = 7 y(0) = 1 y(1)= -

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