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Roll No. ........................
2. δ-ξ is called :
Total No. of Questions : 18
[Total No. of Printed Pages : 05
(a) Dirac δ-function
(b) Jacobi δ-function
(c) Hamilton δ-function
(d) None of the above
OP-500
M. Sc. (Reg./Pvt./ATKT) Examination, 2024
3. Finite sine Fourier Transforms of the function
f(x) = 2x, 0 < x < 4 is :
(Fourth Semester)
(a)
32 - cos Pπx / Pπ
(b)
cos Pπ
(c)
-32 - cos Pπx / Pπ
(d) None of the above
MATHEMATICS
First Paper
4. One-dimensional heat equation is :
Partial Differential Equations-II
Time : 3 Hours]
[Maximum Marks : 85
Note : Attempt all Sections.
(α)
∂u / ∂t = C ∂²u / ∂x²
(b)
∂²u / ∂t² = C ∂²u / ∂x²
(c)
∂²u / ∂x² + ∂²u / ∂y² = 0
(d) None of the above
Section A
(Objective Type Questions)
Note : Attempt all questions.
5×3=15
1.
A Laplace transform exists when :
(a) the function is piece-wise continuous
(b) The function of exponential order
(c) The function piece discrete
(d) The functions is of differential order
5. What is the disadvantage of exponential fourier series :
...apply the divergence theorem to the sphere and show that :
(a) It is tough to calculate
(b) It is not easily visualized
(c) It cannot be easily visualized as sinusoids
(d) It is hard for manipulation
Δ²(1/r) = -4πδ(r)
where δ(r) is a Dirac delta function.
Or
State and prove final value theorem.
State and prove final value theorem.
Section B
(Short Answer Type Questions)
(Short Answer Type Questions)
Note : Attempt any Five questions. 5×5=25
1.
Show that the solution of diffusion equation is unique.
2.
What is Green's function for the wave equation ?
3.
Write statement of Helmholtz theorem.
4.
Find the Fourier Cosine transform of eā»Ė£.
5.
Discuss any two Fourier transform methods.
6.
Write the statement of Fourier integral theorem.
7.
Write the property of inverse Laplace transform.
8.
Find the fundamental solution of Bessel's potential.
2.
State and prove convolution theorem.
Or
Write a short note on Green function for Laplace transform.
Write a short note on Green function for Laplace transform.
3.
Explain Parseval's relation of Fourier transform ?
Explain by a suitable example.
Explain by a suitable example.
Or
Explain Mellin Fourier integral with suitable example.
Explain Mellin Fourier integral with suitable example.
4.
If f(t) is periodic function with period T, then prove that :
L{f(t),s} = ∫0T (e-st / (1-e-sT)) dt
Or
State and prove the complex inversion formula.
State and prove the complex inversion formula.
5.
Solve the following Boundary Value Problem (B.V.P.) using the Laplace transform technique :
ut = uxx, 0 < x < 1, t > 0
u(0, t) = 1, u(1, t) = 1, t > 0
u(x, 0) = 1 + sinπx, 0 < x < 1
Or
Explain Multiple Fourier transform and its properties with suitable example.
Explain Multiple Fourier transform and its properties with suitable example.