Name of the University : Vardhman Mahaveer Open University
Degree : M.Sc
Department : Mathematics
Subject Code/Name : MAMT-09 – Integral Transforms and Integral Equations
Year : II
Document Type : Question Bank
Website : vmou.ac.in
Download Model/Sample Question Paper : https://www.pdfquestion.in/uploads/vmou.ac.in/3594.-MAMT-09_94.pdf
Integral Transforms & Integral Equations :
Section – A :
1. Define Error function and complementary Error function.
2. Find the Laplace transform
Related : VMOU MAMT06 Analysis and Advanced Calculus M.Sc Question Bank : www.pdfquestion.in/3592.html
3. Find x+2Y-
4. State Existence conditions of Laplace Transform.
5. Define Laplace transform and write the formula.
6. Define Integral transform and give the formula.
7. Define Null function & give an example.
8. Define the convolution of two functions.
9. Write Dirchlet’s conditions.
10. What is a boundary value problem-
11. Write two dimensional heat conduction equation-
12. Write 3-dimensional wave equation-
13. Define complex fourier transform and Inverse fourier transform.
14. Define fourier sine transfom and inverse fourier sine transform.
15. Define fourier cosine transform and Inverse fouries cosine transform.
16. Derive the relationship between fourier transform and Laplace transform.
17. What is convolution theorem for fourier transform-
18. State the Parseval’s Identity for fourier transform.
19. State the mellin inversion theorem.
20. State the convolution theorem for Mellintransfom.
21. Define Inverse Mellin transform.
22. Define Mellin Transform of –
23. Define Hankel Transform.
24. Define Bessel function of first kind & show that –
25. Write change of scale property for Hankel transform.
26. State the relation between Hankel and Laplace transform.
27. State inversion formula for the Hankel transform.
28. State Parseval’s theorem for Hankel transform.
28. If we want to remove the term –
29. If the differential equation ranges from -8 –
30. Derive the formula for –
31. Find fourier cosine transform of –
32. Write the Hankel transform of the derivative –
33. Define Integral Equation.
34. What is the difference between Linear and Non-linear Integral equation.
35. Define the term singular Integral equation.
36. Define volterra Integral equation of first and second kinds.
37. Define fredhilm integral equation of first nad second kinds.
38. What is the integral equaton of convolution type-
39. Define the terms :(i) Separable or Degenerate kernel.
40. Define eigen values of a kernel in the integral equation.
41. Define eigen values of a kernel in the integral equation.
42. Define the Abel integral equation.
43. Define Intergo-differential equation.
44. Define the term “separable kernel”.
45. Define the term “Orthogonal function”.
46. State Regularity conditions.
47. Define the Iterated kernel
48. Define the Resvent kernel
49. Define the Neumann series
50. Define the inner or Scalar product of two functions.
51. Define complex Hilbert space.
52. Define orthogonal system of functions.
53. Define the term “Orthonormal set”.
54. Define Schwarz inequality.
55. State Hilbert-Schmidt Theorem
Define the following terms :
56. Fredholm determinant
57. Fredholm minor
58. State Fredholm’s first fundamental theorem.
59. State first and second series for non-homogeneous fredholm integral equation of second kind
Advance Algebra :
1. Define external direct product.
2. Define internal direct product.
3. If O(H) = 2 and O(K) = 5 then find O(H×K)
4. If H and K are two sub groups of G such that G is an internal
5. Define kernel of homomorphism.
6. Define conjugate class.
7. Define normalize of an element in a group.
8. Define centre of a group.
9. Write class equation for the finite group.
10. Define commutator of two elements of a group.
11. Define derived subgroup of a group.
12. Define maximal normal subgroup of agroup.
13. Define composition series of a group.
14. Is – 6 a solvable group.
15. Define prime element in an integral domain.
16. Define Euclidean ring.
17. Define Unique factorization domain.
18. Is every Euclidean ring a principal ideal domain-
19. Is 5 an associate of 5 and -5 in (Z, +, .)
20. Define left module over a ring.
21. Define submodule.
22. Define module homomorphism.
23. Define cyclic module.
24. Is sum of two sub modules a sub module of an R-module-
25. Define linear transformation on vector spaces.
26. Define dual space of a vector space V over field F.
27. Define rank of a linear transformation on vector spaces.
28. Let V be a finite dimensional vector space and –
29. Is sum of two linear transformation a linear transformation-
30. Define field extension.