Rtmnu Integral Equation Unit -IV (New).pptx

1. UNIT-IV
Hilbert transform. Finite Hilbert transforms.
Miscellaneous integral transforms. Approximate methods
of solutions for linear integral equations. Approximate
evaluation of Eigen values and Eigen functions.
It contains : Collocation method of approximation, Galerkin Method,
Kellogs Methos

2. Hilbert Transform
• Definition: Infinite Hilbert Transform

6. Finite Hilbert Transform
• Definition:

7. Derivation of various form of the Hilbert Transform pairs from Eqs
(1) and (2)
• I. The first from of the Hilbert transform pairs:

20. II. Second Alternative Form of Hilbert Trasnfrom pairs

22. III. Third Alternative Form of Hilbert Transform pairs

29. Method of Approximation
• Collocation method of approximation,
• Galerkin Method of approximation
• Kellogs Methos of approximation

32. Collocation Method of Approximation

42. Some problems on Collocation Methods
Ans: C1 = 1 , C2= -1.441, C3= 0.0130
(ii) ∅1(x)= Sinx , ∅2(x)= Cosx

44. The Galerkin Approximation method

58. Some Problems on Galerkin Method
Ex. (1) Use the Galerkin method to find an approximate solution of the
Fredholm Integral equation of second kind
𝑢 𝑥 = 𝑥 + −1
1
𝑥 𝑡 𝑢 𝑡 𝑑𝑡 in terms of
(i) Three Linearly Independent functions ∅1 (x) = 1, ∅2 (x) = x,
∅3 (x) =
3𝑥2 −1
2
Ans: 𝑆3 𝑥 = 3𝑥
(ii) Three Linearly Independent functions ∅1 (x) = 1, ∅2 (x) = sinx,
∅3 (x) = cosx

59. 𝑥2

60. Kellogg’s Method

65. • Examples : Using the Kellogg method, find the smallest characterstic
numbers of the following kernels:

66. UNIT-IV END